1 Introduction
The aim of this paper is to study iterations of rational maps in one complex variable near degeneration. The emphasis is on a further exploration of the interplay between iterate maps and measures of maximal entropy, initiated by DeMarco [Reference DeMarco2, Reference DeMarco3] and continued by DeMarco and Faber [Reference DeMarco and Faber4], Favre [Reference Favre8], Favre and Gong [Reference Favre and Gong9] and Okuyama [Reference Okuyama15, Reference Okuyama16].
The space of rational functions
$f \in \mathbb {C}(z)$
of degree
$d \ge 2$
, denoted by
${\operatorname {Rat}_d}$
, is naturally identified with a Zariski open subset of
$\mathbb {P}^{2d+1}$
. This identification furnishes
${\operatorname {Rat}_d}$
with the compactification
$\overline {\operatorname {Rat}}_d \equiv \mathbb {P}^{2d+1}$
. Elements of the codimension one algebraic set
$\partial {\operatorname {Rat}_d}:= \overline {\operatorname {Rat}}_d \setminus {\operatorname {Rat}_d}$
are called degenerate rational maps of degree d. See §2.2.
For
$n \ge 2$
, the n-th iterate map
$\Phi _n: {\operatorname {Rat}_d} \to {\operatorname {Rat}}_{d^n}$
, sending f to its n-th iterate
$f^{\circ n}$
, extends to a rational map between the corresponding projective spaces
$\overline {\operatorname {Rat}}_d$
and
$\overline {\operatorname {Rat}}_{d^n}$
. According to DeMarco [Reference DeMarco2], the map
$\Phi _n$
has a nonempty indeterminacy locus
$I(d) \subset \partial {\operatorname {Rat}_d}$
which is independent of n. In particular,
$\Phi _n$
fails to have a continuous extension at any point of
$I(d)$
. To resolve this situation, it is convenient to consider the embedding
$$ \begin{align*}\begin{array}{cccc} \Phi:&{\operatorname{Rat}_d}&\hookrightarrow&\Pi_{n\ge1} \overline{\operatorname{Rat}}_{d^n}\\ & f & \mapsto& (f, f^{\circ 2},\cdots). \end{array} \end{align*} $$
This yields a larger compactification of
${\operatorname {Rat}_d}$
:
$$ \begin{align*}\widehat{{\operatorname{Rat}}}_d := \operatorname{closure} (\Phi({\operatorname{Rat}_d})) \subset \Pi_{n\ge1} \overline{\operatorname{Rat}}_{d^n}.\end{align*} $$
Trivially, every iterate map
$\Phi _n$
extends continuously to
$\widehat {{\operatorname {Rat}}}_d$
as the projection to the n-th coordinate.
On the other hand, every rational map
$f \in {\operatorname {Rat}_d}$
has a unique measure of maximal entropy
$\mu _f$
, see [Reference Freire, Lopes and Mañé10, Reference Lyubich12]. This measure is also characterized as the unique probability measure such that
$$ \begin{align*}\dfrac{1}{d^n} (f^{\circ n})^* \mu \to \mu_f,\end{align*} $$
for all probability measures
$\mu $
(see §2.7) having no mass on the exceptional set of f. The support of
$\mu _f$
is the Julia set of f, denoted by 
. Moreover,
$\mu _f \in M^1(\overline {\mathbb {C}})$
depends continuously on
$f\in {\operatorname {Rat}_d}$
, where
$M^1(\overline {\mathbb {C}})$
denotes the space of probability measures endowed with the weak*-topology, see [Reference Mañé13].
Given
$g \in \partial {\operatorname {Rat}_d}$
, the pull-back operator on (nonexceptional) measures extends continuously to g. More precisely, the pull-back
$g^*$
can be defined by the property that, for all nonexceptional probability measures
$\mu $
for g, as
$f\in {\operatorname {Rat}_d}$
converges to g, we have that
$f^* \mu $
converges to
$g^* \mu $
, (see §2.6).
Our first result, extending DeMarco’s and DeMarco-Faber’s results ([Reference DeMarco2, Theorem 0.1] and [Reference DeMarco and Faber4, Theorem A]), is about the limiting behavior at
$\partial {\operatorname {Rat}}_d$
of the measure of maximal entropy:
Theorem A. For any
$\mathbf {g}=(g_n)_{n\ge 1} \in \widehat {{\operatorname {Rat}}}_d$
, there exists a probability measure
$\mu _{\mathbf {g}}\in M^1(\overline {\mathbb {C}})$
such that if
$\mu \in M^1(\overline {\mathbb {C}})$
is nonexceptional for
$\mathbf {g}$
, then as
$n \to \infty $
,
$$ \begin{align*}\dfrac{1}{d^n}(g_n)^* \mu \to \mu_{\mathbf{g}}.\end{align*} $$
Moreover, the map
$$ \begin{align*}\begin{array}{cccc} \widehat{{\operatorname{Rat}}}_d&\to& M^1(\overline{\mathbb{C}})\\ \mathbf{g} & \mapsto & \mu_{\mathbf{g}} \end{array} \end{align*} $$
is continuous.
Note that
$\overline {\operatorname {Rat}}_d \setminus I(d)$
can be identified with a subset of
$\widehat {{\operatorname {Rat}}}_d$
via the continuous extension of
$\Phi : {\operatorname {Rat}_d} \to \widehat {{\operatorname {Rat}}}_d$
. After this identification, the previous theorem for
$\mathbf {g} \in \overline {\operatorname {Rat}}_d \setminus I(d)$
was established by DeMarco in [Reference DeMarco2, Theorem 0.1].
The limiting behavior of measures at the indeterminacy locus
$I(d)$
was further studied by DeMarco and Faber in [Reference DeMarco and Faber4]. They showed that any limit of the measures
$\mu _f$
as
${\operatorname {Rat}_d} \ni f \to g \in \partial {\operatorname {Rat}_d}$
is purely atomic, see [Reference DeMarco and Faber4, Theorem A]. We build on their techniques to prove Theorem A.
Degeneration along holomorphic families can be studied with the aid of non-Archimedean dynamics. A holomorphic family of maps
$f_t \in \overline {\operatorname {Rat}}_d$
, parameterized by t in a neighborhood of
$0 \in \mathbb {C}$
, is degenerate if
$f_t \in \partial {\operatorname {Rat}_d}$
only for
$t=0$
. DeMarco and Faber showed that along any degenerate holomorphic family the measures of maximal entropy converge, see [Reference DeMarco and Faber4, Theorem B]. Moreover, they proved that the limit measure is closely related to an equidistribution measure associated to a non-Archimedean dynamical system. Related approaches studying the limit of measures and the corresponding Lyapunov exponents were implemented by Favre [Reference Favre8], Favre-Cong [Reference Favre and Gong9], and Okuyama [Reference Okuyama15, Reference Okuyama16]. It is worth mentioning that, for every degenerate family
$f_t$
, there exists
$\mathbf {g}=(g_n) \in \widehat {{\operatorname {Rat}}}_d$
such that
$f_t^{\circ n} \to g_n$
, as
$t \to 0$
, for all n. Thus, convergence of the measure of maximal entropy along degenerate holomorphic families can be deduced from Theorem A. Although the proof of Theorem A profits from non-Archimedean dynamics intuition, it does not rely on non-Archimedean tools.
There is a corresponding discussion in the moduli space, which was also initiated by DeMarco, in [Reference DeMarco3]. Consider the quotient
${\operatorname {rat}_d}:={\operatorname {Rat}_d}/\operatorname{PSL}(2,\mathbb{C})$
of
${\operatorname {Rat}_d}$
that identifies Möbius conjugate rational maps. Following Silverman [Reference Silverman17], its GIT-compactification
$\overline {\operatorname {rat}}_d$
is a projective variety and the boundary
$\partial {\operatorname {rat}_d}:=\overline {\operatorname {rat}}_d\setminus {\operatorname {rat}_d}$
consists of the GIT-conjugacy classes of semistable maps in
$\overline {\operatorname {Rat}}_d$
. For
$n\ge 1$
, the n-th iterate map
$\Psi _n: {\operatorname {rat}}_d\to {\operatorname {rat}}_{d^n}$
, sending
$[f]$
to
$[f^{\circ n}]$
, extends to a rational map
$\overline {\operatorname {rat}}_d \dashrightarrow \overline {\operatorname {rat}}_{d^n}$
with non trivial indeterminacy locus that depends on n, see [Reference DeMarco3, Theorem 5.1] and [Reference Kiwi and Nie11, Theorem A]. To resolve this situation, as above, we consider
$$ \begin{align*} \begin{array}{cccc} \Psi:&{\operatorname{rat}_d}&\hookrightarrow&\Pi_{n\ge1} \overline{\operatorname{rat}}_{d^n}\\ & [f] & \mapsto& ([f],[f^{\circ 2}],\cdots),\end{array} \end{align*} $$
and let
$$ \begin{align*}\widehat{\operatorname{rat}}_d := \operatorname{closure} (\Psi({\operatorname{rat}_d})) \subset \Pi_{n\ge1} \overline{\operatorname{rat}}_{d^n}.\end{align*} $$
Iterate maps
$\Psi _n$
extend continuously to
$\widehat {{\operatorname {rat}}}_d$
as the projection to the corresponding coordinate.
In order to consider measures for elements in the moduli space, following DeMarco [Reference DeMarco3], it is convenient to work with conformally barycentered probability measures, see §5. The space of conformally barycentered probability measures modulo the push-forward action by
$\operatorname {SO}(3)$
is locally compact. We denote by
$\overline {M}^1_{dm}(\overline {\mathbb {C}})$
its one point (
$=[\infty ]$
) compactification.
For any measure
$\mu \in M^1(\overline {\mathbb {C}})$
with no atoms of weight
$\ge 1/2$
, there exists a Möbius transformation such that
$\nu = A_\ast \mu $
is conformally barycentered. Moreover,
$\nu $
is unique up to push-forward by an element in
$\operatorname {SO}(3)$
. Thus, we have a well defined class
$[\mu ] :=[\nu ] \in \overline {M}^1_{dm}(\overline {\mathbb {C}})$
. If
$\mu $
has an atom of weight
$\ge 1/2$
, we let
$[\mu ] := [\infty ]\in \overline {M}^1_{dm}(\overline {\mathbb {C}})$
. Since the measure of maximal entropy
$\mu _f$
of any rational map f is atom free, there exists a continuous map
$$ \begin{align*}\begin{array}{ccc} {\operatorname{rat}_d}&\rightarrow&\overline{M}^1_{dm}(\overline{\mathbb{C}})\\ f & \mapsto& [\mu_{f}]. \end{array} \end{align*} $$
We show this map extends continuously to
$\widehat {{\operatorname {rat}}}_d$
, answering positively the question posed by DeMarco in [Reference DeMarco3, Section 10]:
Theorem B. For any
$[\mathbf {g}]=([g_n])_{n\ge 1} \in \widehat {{\operatorname {rat}}}_d$
, there exists
$x_{[\mathbf {g}]} \in \overline {M}^1_{dm}(\overline {\mathbb {C}})$
such that if
$\mu \in M^1(\overline {\mathbb {C}})$
is nonexceptional for
$(g_n)_{n\ge 1}$
, then as
$n \to \infty $
,
$$ \begin{align*}\left[\dfrac{1}{d^n}(g_n)^* \mu\right] \to x_{[\mathbf{g}]}.\end{align*} $$
Moreover, the map
$$ \begin{align*}\begin{array}{cccc} \widehat{{\operatorname{rat}}}_d&\to& \overline{M}^1_{dm}(\overline{\mathbb{C}})\\ {[\mathbf{g}]} & \mapsto & x_{[\mathbf{g}]} \end{array} \end{align*} $$
is continuous.
The space
$\widehat {{\operatorname {rat}}}_d$
coincides with the compactification of
${\operatorname {rat}_d}$
constructed by taking the inverse limit of the closure of
${\operatorname {rat}_d}$
in
$\Pi _{i=1}^n\overline {\mathrm {rat}}_{d^i}$
under the iterate map
${\operatorname {rat}_d}\to \Pi _{i=1}^n\overline {\mathrm {rat}}_{d^i}$
. On the other hand, by considering the closure of the graph of the map
${\operatorname {rat}_d}\rightarrow \overline {M}^1_{dm}(\overline {\mathbb {C}})$
in
$\overline {\operatorname {rat}}_d\times \overline {M}^1_{dm}(\overline {\mathbb {C}})$
, DeMarco constructed a compactification
$\overline {\overline {\operatorname {rat}}}_d$
of
${\operatorname {rat}_d}$
and established that, in the quadratic case, the compactifications
$\widehat {{\operatorname {rat}}}_2$
and
$\overline {\overline {\operatorname {rat}}}_2$
are canonically homeomorphic [Reference DeMarco3, Theorem 1.1]. However, for higher degrees, the compactifications
$\widehat {{\operatorname {rat}}}_d$
and
$\overline {\overline {\operatorname {rat}}}_d$
are not homeomorphic and, in particular, there cannot exist a continuous map
$\overline {\overline {\operatorname {rat}}}_d\to \widehat {{\operatorname {rat}}}_d$
which restricts to the identity on
${\operatorname {rat}_d}$
, see [Reference DeMarco3, Section 10]. Our Theorem B immediately implies that there exists a continuous map
$\widehat {{\operatorname {rat}}}_d\to \overline {\overline {\operatorname {rat}}}_d$
which restricts to the identity on
${\operatorname {rat}_d}$
. In other words, the GIT limits of the iterates of an unbounded sequence in
${\operatorname {rat}_d}$
determine the limit of the barycentered measures of maximal entropy.
Along a degenerating sequence of rational maps
$\{f_k\}$
, the choice of coordinates in the domain and in the range become of central importance. The key to Theorems A and B is to analyze the case in which an iterate of
$f_k$
uniformly converges to a nondegenerate rational map, modulo changes of coordinates in the domain and the range. We obtain a “Structure Theorem” that describes the limiting dynamics, in this case, see §3. The Structure Theorem yields, as a byproduct of independent interest, Theorem C, stated below.
Recall that a polynomial-like map of degree d is a triple
$(U,V,\psi )$
, where U and V are open topological discs contained in
$\overline {\mathbb {C}}$
with U relatively compact in V and
$\psi :U\to V$
is a proper holomorphic map of degree d. Any polynomial-like map of degree d is quasi-conformally equivalent to a degree d polynomial, see [Reference Douady and Hubbard6, Theorem 1].
Theorem C. Let
$\{f_k\}$
be a sequence in
${\operatorname {Rat}_d}$
such that
$\{[f_k]\}$
converges to an element in
$\partial {\operatorname {rat}}_d$
. Assume there exist
$\{A_k\}$
and
$\{B_k\}$
such that
$B_{k}\circ f_k^{\circ 3}\circ A_{k}^{-1}$
converges in
${\operatorname {Rat}}_{d^3}$
, as
$k\to \infty $
. Then for any sufficiently large k, there exist
$U_k$
and
$V_k$
such that
$(U_k,V_k, f_k)$
is a polynomial-like map of degree d. Moreover, 
and
$\overline {\mathbb {C}}\setminus U_k$
is contained in a completely invariant attracting Fatou component of
$f_k$
.
In our context, the assumption that
$f_k^{\circ 3}$
converges to a nondegenerate map, modulo changes of coordinates in the domain and range, is sharp for the Structue Theorem to hold (see Remark 3.4). However, we do not know if
$3$
can be replaced by
$2$
in the statement of Theorem C.
Overview of the Proofs of Theorems A–C
We now outline the main ideas behind the proofs of our principal results.
To prove Theorem A, given
$(g_n)$
in
$\widehat {{\operatorname {Rat}}}_d \setminus {\operatorname {Rat}}_d$
and a nonexceptional probability measure
$\mu $
, define
$$ \begin{align*}\lambda_n:=\dfrac{1}{d^n} (g_n)^* \mu. \end{align*} $$
Consider a sequence
$\{f_k\}$
of degree d rational maps such that
$\{f_k^{\circ n}\}_k$
converges to the degenerate map
$g_n \in \partial {\operatorname {Rat}}_{d^n}$
for all n. Let
$\mu _k$
be the equilibrium measure of
$f_k$
and assume, after passing to a subsequence, that
$\mu _k \to \mu _\infty $
.
Theorem A claims that
$\{\lambda _n\}$
also converges to
$\mu _\infty $
. We may further assume that
$[f_k]\subset \mathrm {rat}_d$
diverge in moduli space, since the nondivergent case is not difficult to understand (see §4.3).
Away from a finite subset of
$\overline {\mathbb {C}}$
, called the holes of
$g_n$
, the sequence
$\{f_k^{\circ n}\}_k$
converges locally uniformly to a rational map
$\tilde {g}_n$
of degree at most
$d^n-1$
, called the reduction of
$g_n$
(see §2.1). The reduction
$\tilde {g}_n$
may be constant, and in general one has
$\deg \tilde {g}_n = o(d^n)$
. Degenerate maps
$g_n$
with constant reduction encode little information about the limiting dynamics of
$\{f_k\}$
and constitute one of the main difficulties in the study of measure convergence (cf. [Reference DeMarco2]).
A key idea is to regard the spheres in the domain and range of
$f_k^{\circ n}$
as distinct spheres
$\overline {\mathbb {C}}_0$
and
$\overline {\mathbb {C}}_n$
, and to introduce a dynamically meaningful normalization in the range. Indeed, a basic principle of degenerate rational dynamics applies: after passing to a subsequence, there exists a choice of coordinates
$A_{n,k}$
in the image sphere such that
$A_{n,k} \circ f_k^{\circ n}$
converges to a map
$\varphi _n \in \overline {\operatorname {Rat}}_{d^n}$
having nonconstant reduction
$\tilde {\varphi }_n:\overline {\mathbb {C}}_0 \to \overline {\mathbb {C}}_n$
(Proposition 2.1). The limit
$\varphi _n$
is unique modulo post-composition by a Möbius transformation. If the equilibrium measures
$\mu _k$
converge to
$\mu _\infty $
and
$(A_{n,k})_*\mu _k$
converges to
$\nu _n$
, then by DeMarco and Faber [Reference DeMarco and Faber4] (see Proposition 2.10) we have
$$ \begin{align} \mu_\infty = \frac{1}{d^n}(\varphi_n)^* \nu_n. \end{align} $$
At this point, the behavior of
$\deg \tilde {\varphi }_n$
becomes decisive. We distinguish two cases:
$\deg \tilde \varphi _n = o(d^n)$
or
$\deg \tilde \varphi _n \neq o(d^n)$
. In the former case, the techniques developed by DeMarco and Faber apply. More precisely, the right-hand side of (1.1) differs from
$\lambda _n$
by a measure of mass at most
$\deg \tilde {\varphi }_n/d^n$
(see Proposition 4.3
Footnote 1). Therefore
$\lambda _n \to \mu _\infty $
.
Thus, the proof of Theorem A reduces to the remaining case
$\deg \tilde {\varphi }_n \neq o(d^n)$
, which requires a new ingredient: the Structure Theorem 3.1. To discuss it, we need to introduce fully ramified times.
The maps
$\tilde {\varphi }_n:\overline {\mathbb {C}}_0 \to \overline {\mathbb {C}}_{n+1}$
inherit a natural decomposition from iteration. For each n, there exists a rational map
$\tilde {\varphi }_{n,n+1}:\overline {\mathbb {C}}_n \to \overline {\mathbb {C}}_{n+1}$
of degree at most d such that
$\tilde {\varphi }_{n+1} = \tilde {\varphi }_{n,n+1} \circ \tilde {\varphi }_n$
. When
$d= \deg \tilde {\varphi }_{n,n+1}$
, we say that n is a fully ramified time. In the case
$\deg \tilde {\varphi }_n \neq o(d^n)$
, all sufficiently large times are fully ramified.
The Structure Theorem asserts that the presence of just
$3$
consecutive fully ramified times, say
$m, m+1$
and
$m+2$
, imposes severe restrictions on the limiting dynamics. More precisely, it forces the spheres
$\overline {\mathbb {C}}_m, \overline {\mathbb {C}}_{m+1}, {\overline {\mathbb {C}}_{m+2}}, {\overline {\mathbb {C}}_{m+3}}, \dots $
, which a priori form a tree of bouquets of spheres (see Appendix A), to organize into a linear concatenation obtained by identifying
$0 \in \overline {\mathbb {C}}_n$
with
$\infty \in \overline {\mathbb {C}}_{n+1}$
for all
$n \ge m$
, in suitable coordinates. Moreover,
$\tilde {\varphi }_{n,n+1}$
is a polynomial for all
$n> m$
, and a monomial whenever both n and
$n+1$
are fully ramified times.
In the proof of Theorem A, the Structure Theorem implies that the Julia sets of
$f_k$
converge to
$\infty \in \overline {\mathbb {C}}_n$
for all n sufficiently large. In particular,
$(A_{n,k})_*\mu _k \to \delta _\infty $
(see Corollary 3.15), which provides the key input in §4.4 to complete the argument in the case
$\deg \tilde {\varphi }_n \neq o(d^n)$
.
The location of the Julia sets of
$f_k$
is clarified by extracting polynomial-like maps. The mechanism enabling this extraction is provided by Proposition 3.7, which is also used in the proof of the Structure Theorem to rule out certain configurations of the initial spheres
$\overline {\mathbb {C}}_m, \overline {\mathbb {C}}_{m+1}, \overline {\mathbb {C}}_{m+2}$
.
In this language, and modulo conjugacy, the hypothesis of Theorem C is that the sequence
$\{f_k\}$
has fully ramified times
$m = 0,1,2$
. Its proof relies on Lemmas 3.5 and 3.6, which determine the configuration of spheres and transition maps associated to an arbitrary number, at least
$3$
, of consecutive fully ramified times. Although to prove Theorem C we only need to study the initial spheres
$\overline {\mathbb {C}}_1,\overline {\mathbb {C}}_2, \overline {\mathbb {C}}_3$
, the lemmas are proved in greater generality without raising the level of difficulty.
The proof of Theorem B is also based on the Structure Theorem but is more subtle, since the arguments take place in quotient spaces formed by equivalence classes of maps and measures. Let
$\{[f_k]\} \subset {\operatorname {rat}_d}$
be such that
$[f_k^{\circ n}]$
converges to
$[g_n] \in \partial {\operatorname {rat}}_{d^n}$
for all
$n \ge 1$
. When the sequence
$\{[\mu _k]\}$
, where
$\mu _k$
is the equilibrium measure of
$f_k$
, is bounded away from infinity, Theorem A applies almost directly. Otherwise, the setting of Theorem B leads to a family of sequences
$\{f_{n,k}\} \subset {\operatorname {Rat}}_d$
, one for each iterate n. That is, for each n there exists a sequence of coordinate changes
$\{B_{n,k}\}$
such that
$f_{n,k}^{\circ n}$
converges to
$g_n$
, where
$$ \begin{align*}f_{n,k} := B_{n,k} \circ f_k \circ B_{n,k}^{-1}. \end{align*} $$
Passing to subsequences, for each n there exists
$\{A_{n,k} \}$
such that
$A_{n,k} \circ f_{n,k}^{\circ n} \to \psi _n$
, where
$\psi _n \in \overline {{\operatorname {Rat}}}_{d^n}$
has nonconstant reduction
$\tilde {\psi }_n$
. At this stage, the basic distinction now arises between the cases
$\deg \tilde \psi _n \neq o(d^n)$
and
$\deg \tilde \psi _n = o(d^n)$
. In the former case, there exist arbitrarily large n for which
$\{f_{n,k}\}$
exhibits
$3$
consecutive fully ramified times, and the Structure Theorem applies. In the latter case, a more delicate analysis is required, using the associated bouquet of trees of spheres described in Appendix A.
Outline
The paper is organized as follows. In §2, we discuss the basic properties of degenerate rational maps including their action on measures via pull-back. In §3, based on the trees of bouquets of spheres from Appendix A, we establish our Structure Theorem for the limiting dynamics of “fully ramified sequences.” As a byproduct, we obtain Theorem C. We prove Theorem A in §4 and Theorem B in §5. In Appendix A, we discuss maps acting on trees of bouquet of spheres. This is closely related to the work by Arfeux [Reference Arfeux1] regarding maps acting on trees of spheres.
2 Degenerate rational maps
A degenerate rational map
$f \in \partial {\operatorname {Rat}_d}$
has a well defined action outside a finite set of points, called the holes of f. The action, known as the reduction
$\tilde {f}$
of f, is a lower degree rational map, possibly constant. Intuitively, the holes “map,” under f, onto
$\overline {\mathbb {C}}$
with a degree, called its depth. The reduction
$\tilde {f}$
, the set of holes
$\operatorname {Hole}(f)$
and their depth
$d_h(f)$
, completely prescribe the degenerate map f. We discuss these definitions from [Reference DeMarco2] in §2.1. The parameter space of rational maps, degenerate or not, is discussed in §2.2. A well known formula for the depth of the holes of compositions is written in §2.3. As rational maps converge to a degenerate map, their action is better understood by changing coordinates, or scales, at the domain and the range. In §2.4, we consider the space of rational maps, possibly degenerate but with nonconstant reduction, modulo post-composition by Möbius maps. It is a compact space. In §2.5, we present a formula for the depth of holes in the setting of §2.4 as an analogue of a formula obtained by the authors in [Reference Kiwi and Nie11], and also furnish a formula to compute the depth counting preimages. In §2.6, we recall the definitions of the pull-back of measures by a rational map and, following DeMarco [Reference DeMarco2], by a degenerate rational map. We show that the degenerate pull-back is the continuous extension of the pull-back (Proposition 2.9). In §2.7, we summarize some general properties of the measures of maximal entropy. Continuity of the pull-back of measures yields a key result established by DeMarco and Faber (see [Reference DeMarco and Faber4, Theorem 2.4]), which provides us with an equation satisfied by suitable limits of the measures of maximal entropy.
2.1 Holes and reduction
Although we mostly work in nonhomogeneous coordinates, to define reductions, holes and depths, homogeneous coordinates are convenient. A rational map
$f: \mathbb {P}^1 \to \mathbb {P}^1 $
of degree
$d\ge 2$
can be written as
$$ \begin{align*}f: [z:w] \mapsto [P(z,w):Q(z,w)],\end{align*} $$
where
$P,Q$
are relatively prime degree d homogeneous polynomials. For concreteness, one may write
$$ \begin{align} \begin{aligned} P(z,w) & = a_0 w^d + a_1 z w^{d-1} + \cdots + a_d z^d,\\ Q(z,w) & = b_0 w^d + b_1 z w^{d-1} + \cdots + b_d z^d. \end{aligned} \end{align} $$
Then
$f = [P:Q]$
is a degree d rational map if and only if
$P,Q$
are relatively prime.
A degree d degenerate rational map f is an expression
$[P:Q]$
such that P and Q are not relatively prime. Consistent with the notation, for all
$\lambda \in \mathbb {C} \setminus \{0\}$
, we regard
$[P:Q]$
and
$[\lambda P: \lambda Q]$
as the same map. Let
$H_f$
be a common factor of P and Q of maximal degree. Then, there exist polynomials
$\tilde {P}$
,
$\tilde {Q}$
, such that
$$ \begin{align*}P = H_f \tilde{P}\ \ \ \ \text{and}\ \ \ \ Q= H_f \tilde{Q}.\end{align*} $$
We say that
$$ \begin{align*}\operatorname{Hole} (f) := \{ h \in \mathbb{P}^1 : H_f(h) =0\}\end{align*} $$
is the set of holes of f. The depth
$d_h(f)$
of a hole h is the multiplicity of h as a root of
$H_f$
; that is, modulo multiplication by a constant,
$$ \begin{align*}H_f = \prod_{h \in \operatorname{Hole}(f)} L_h^{d_h(f)},\end{align*} $$
where
$L_h$
is a homogeneous linear form vanishing at h; moreover, for notational convenience, we let
$d_h(f):=0$
whenever
$h \in \mathbb {P}^1\setminus \operatorname {Hole}(f)$
. We say that
$$ \begin{align*}\tilde{f} := [\tilde{P}: \tilde{Q}]\end{align*} $$
is the reduction of f. Note that
$\tilde {f}$
is a well defined self-map of
$\mathbb {P}^1 $
. We write
$f=H_f \cdot \tilde {f}$
.
Given a nondegenerate rational map
$\tilde {f}$
of degree
$\deg \tilde {f} < d$
and a set of holes
$\operatorname {Hole}(f)$
with prescribed depths such that the total number of holes, counted with depths, is
$d-\deg \tilde {f}$
, there exists a unique degree d degenerate map f compatible with this information.
For notational simplicity, we mostly work with nonhomogeneous coordinates, after identification of
$\mathbb {P}^1 $
with
$\overline {\mathbb {C}} := \mathbb {C}\cup \{\infty \}$
. Thus, we write
$f(z) = P(z,1)/Q(z,1)$
or simply
$f(z) = P(z)/Q(z)$
.
2.2 Spaces of rational maps
The space of possibly degenerate rational maps
$\overline {\operatorname {Rat}}_d$
is the
$(2d+1)$
-dimensional complex projective space
$\mathbb {P}^{2d+1} $
. Its elements are identified with expressions of the form
$f = [P:Q]$
by
$$ \begin{align*}\begin{array}{ccc} \overline{\operatorname{Rat}}_d & \leftrightarrow & \mathbb{P}^{2d+1} \\ f = [P:Q] & \leftrightarrow & [a_0: \cdots : a_d : b_0 : \dots :b_d], \end{array} \end{align*} $$
where
$P,Q$
are polynomials of the form (2.1).
The degree d homogeneous resultant of
$P,Q$
is a homogeneous polynomial in the coefficients. Its vanishing locus
$\partial {\operatorname {Rat}_d} := \{ \operatorname {Res} = 0\}$
is the set of degenerate maps in
$\overline {\operatorname {Rat}}_d$
. Hence, the Zariski open subset
${\operatorname {Rat}}_d := \overline {\operatorname {Rat}}_d \setminus \partial {\operatorname {Rat}_d}$
is the space of rational maps of degree d. Convergence in
${\operatorname {Rat}_d} \subset \mathbb {P}^{2d+1}$
coincides with uniform convergence of rational maps.
Möbius transformations form the space
${\operatorname {Rat}}_1$
of degree
$1$
rational maps that is naturally identified with
$\operatorname {PSL(2,\mathbb {C})}$
. Each element of
$\partial {\operatorname {Rat}}_1$
is a degenerate Möbius transformation: a map with constant reduction and one hole, counted with depth.
2.3 Composition of degenerate maps
Given
$d, d' \ge 1$
, consider the map
$$ \begin{align*}\begin{array}{cccc} \circ :& {\operatorname{Rat}_d} \times {\operatorname{Rat}}_{d'} & \to & {\operatorname{Rat}}_{dd'}\\ & ( f, g) & \mapsto & f \circ g. \end{array} \end{align*} $$
It extends to a rational map from the projective variety
$\overline {\operatorname {Rat}}_d \times \overline {\operatorname {Rat}}_{d'}$
into
$\overline {\operatorname {Rat}}_{dd'}$
with indeterminacy locus
$$ \begin{align*}I(d,d') := \{ (f,g) : \tilde{g} \equiv h \in \operatorname{Hole}(f) \}.\end{align*} $$
Following [Reference DeMarco3, Lemma 2.6], for all
$(f,g) \in \overline {\operatorname {Rat}}_d \times \overline {\operatorname {Rat}}_{d'} \setminus I(d,d')$
, we have that
$f \circ g$
is the unique element of
$\overline {\operatorname {Rat}}_{dd'}$
with reduction
$\tilde {f} \circ \tilde {g}$
and holes
$h\in \tilde {g}^{-1} (\operatorname {Hole}(f)) \cup \operatorname {Hole}(g)$
with depths
$$ \begin{align} d_h (f\circ g) = d_h (g) \cdot d + \deg_h (\tilde{g}) \cdot d_{\tilde{g}(h)} (f). \end{align} $$
2.4 Left classes of rational maps
To deal with degenerations towards maps with constant reduction, the key is the following “scaling” result:
Proposition 2.1 [Reference DeMarco and Faber4, Lemma 2.1].
Let
$\{f_k\}$
be a sequence in
${\operatorname {Rat}_d}$
. Then there exists a subsequence
$\{f_{k_i} \}$
and a sequence
$\{A_i\}$
in
$\operatorname {PSL(2,\mathbb {C})}$
such that
$$ \begin{align*}A_i \circ f_{k_i} \to \varphi \in \overline{\operatorname{Rat}}_d,\end{align*} $$
and
$\tilde {\varphi }$
is not constant. Moreover, if
$\{B_i\}$
is another such sequence, then
$\{B_i^{-1} \circ A_i\}$
converges in
$\operatorname {PSL(2,\mathbb {C})}$
.
This result may be understood as the compactness of certain orbit space. Let us denote the maps with nonconstant reduction by
$$ \begin{align*}{\operatorname{Rat}}_d^*:=\{ f \in \overline{\operatorname{Rat}}_d : \deg \tilde{f} \neq 0 \}.\end{align*} $$
Then post-composition by
$\operatorname {PSL(2,\mathbb {C})}$
leaves
${\operatorname {Rat}}_d^*$
invariant. An equivalent formulation of the above proposition is the following:
Corollary 2.2. Each
$\operatorname {PSL(2,\mathbb {C})}$
-orbit, under post-composition, is closed in
${\operatorname {Rat}}_d^*$
. The quotient topological space 
is compact.
Remark 2.3. In fact,
${\operatorname {Rat}}_d^*$
is the set of GIT-stable elements of
$\overline {\operatorname {Rat}}_d$
, under post-composition by
$\operatorname {PSL(2,\mathbb {C})}$
. There are no strictly semi-stable elements for this action. Hence, 
is a geometric quotient which is, naturally, a projective variety.
We denote the
$\operatorname {PSL(2,\mathbb {C})}$
-orbit of
$f \in {\operatorname {Rat}}_d^*$
under the post-composition action by
Let
$\{f_k\}\subset {\operatorname {Rat}}_d$
be a sequence such that 
converges to 
in 
for each
$n\ge 1$
. Omitting the dependence on the choice of
$\varphi _n$
, we say that
$\{ A_{n,k}: n,k \ge 1 \} \subset {\operatorname {Rat}}_1$
is a collection of scalings for
$\{f_k\}$
if for all
$n \ge 1$
, as
$k \to \infty $
,
$$ \begin{align*}A_{n,k} \circ f^{\circ n}_k \to \varphi_n.\end{align*} $$
The following result is similar to [Reference DeMarco and Faber4, Lemma 2.5].
Lemma 2.4. Assume that 
converges to 
in 
for each
$n\ge 1$
. Then for all
$n\ge 1$
, there exist
$\varphi _{n,n+1} \in {\operatorname {Rat}}_d^*$
such that
$$ \begin{align*}\varphi_{n+1} = \varphi_{n,n+1} \circ \varphi_n.\end{align*} $$
Moreover, if
$\{A_{n,k}\}$
is a collection of scalings for
$\{f_k\}$
, then
$$ \begin{align*}A_{n+1,k} \circ f_k \circ A^{-1}_{n,k} \to \varphi_{n,n+1}.\end{align*} $$
Proof. For any
$n \ge 1$
, let
$\{ A_{n,k}\} \subset {\operatorname {Rat}}_1$
be a collection of scalings for
$\{f_k\}$
. Modulo passing to a subsequence, by Proposition 2.1 applied to
$\{ f_k \circ A^{-1}_{n,k}\}$
, there exists
$\{B_{n+1,k}\}$
such that, as
$k \to \infty $
,
$$ \begin{align*}B_{n+1,k} \circ f_k \circ A^{-1}_{n,k} \to \psi_{n,n+1},\end{align*} $$
for some
$\psi _{n,n+1} \in {\operatorname {Rat}}^*_{d}$
. Continuity of composition (§2.3) implies that
$B_{n+1,k} \circ f_k^{\circ n+1} \to \psi _{n,n+1} \circ \varphi _n$
. Again by Proposition 2.1, we conclude that
$A_{n+1,k} \circ B^{-1}_{n+1,k} \to M_{n+1}$
, for some Möbius transformation
$M_{n+1} \in {\operatorname {Rat}}_1$
. Let
$\varphi _{n,n+1} := M_{n+1} \circ \psi _{n,n+1}$
. Thus, a map
$\varphi _{n,n+1} \in \overline {\operatorname {Rat}}_d$
as in the statement of the Lemma exists.
Note that any accumulation point
$\varphi \in \overline {\operatorname {Rat}}_d$
of
$A_{n+1,k} \circ f_k \circ A^{-1}_{n,k}$
satisfies that
$\varphi \circ \varphi _n = \varphi _{n+1}$
. To finish the proof, it suffices to show that
$\varphi $
is uniquely determined by this property, that is,
$\varphi = \varphi _{n,n+1}$
. Indeed, surjectivity of
$\tilde {\varphi }_n$
yields that
$\tilde {\varphi } = \tilde {\varphi }_{n,n+1}$
. Moreover, the depth
$d_h(\varphi )$
, for any
$h \in \overline {\mathbb {C}}$
, is completely determined by the equation
$$ \begin{align*}d_h(\varphi_{n+1}) = d_h(\varphi_n) \cdot d + \deg_h \tilde{\varphi}_n \cdot d_{\tilde{\varphi}_n(h)}(\varphi).\end{align*} $$
Therefore,
$\varphi = \varphi _{n,n+1}$
.
2.5 Depth formulas
The following result is the complex analytic analogue of [Reference Kiwi and Nie11, Lemma 2.9].
Lemma 2.5. Let
$\{f_k\} \subset {\operatorname {Rat}_d}$
be a sequence converging to
$g\in \partial {\mathrm {Rat}}_d$
with constant reduction. Assume that 
. Then there exists
$a \in \overline {\mathbb {C}}$
such that
$$ \begin{align*}\operatorname{Hole}(g) = \operatorname{Hole}(\varphi) \cup \tilde{\varphi}^{-1} (a).\end{align*} $$
Moreover,
$$ \begin{align*}d_h(g) = \begin{cases} d_h(\varphi), & \text{ if } h \notin \tilde{\varphi}^{-1}(a), \\ d_h(\varphi) + \deg_h \tilde{\varphi}, & \text{ otherwise.} \end{cases} \end{align*} $$
Furthermore, if
$A_k \circ f_k \to \varphi $
in
${\operatorname {Rat}}_d^*$
, then
$A_k \to A\in \partial {\operatorname {Rat}}_1$
with
$\tilde {A}=a$
and
$\operatorname {Hole}(A)=\{c\}$
where
$\tilde {g} \equiv c$
.
Proof. Consider a sequence
$\{A_k\} \subset {\operatorname {Rat}}_1$
such that
$A_k \circ f_k \to \varphi $
. Passing to a subsequence, suppose
$A_{k_i} \to A \in \partial {\operatorname {Rat}}_1$
and write
$a:=\tilde {A}$
. Then
$\operatorname {Hole}(A) = \{c\}$
where
$\tilde {g} \equiv c$
; for otherwise
$A_{k_i} \circ f_{k_i}(z)$
would converge to a outside
$\operatorname {Hole}(\varphi )$
, and hence
$\tilde {\varphi }$
would be constant, contrary to our hypothesis.
It follows that
$A^{-1}_{k_i}$
converges to some
$B\in \partial {\operatorname {Rat}}_1$
with reduction c and hole a, that is,
$H_B(z) = z-a$
. Hence,
$$ \begin{align*}A^{-1}_{k_i} \circ (A_{k_i} \circ f_{k_i}) \to (H_B \circ \tilde{\varphi}) H_\varphi \cdot c = g.\end{align*} $$
Thus, the first two assertions hold. Since a is independent of the subsequence, the last assertion also holds.
Given
$g \in \partial {\operatorname {Rat}}_d$
, we define the algebraically exceptional set
$Ex_g\subset \overline {\mathbb {C}}$
by
$$ \begin{align*}Ex_g:= \begin{cases} \{\tilde g\}\ \ &\text{if}\ \deg\tilde g=0,\\ \emptyset\ \ &\text{if}\ \deg\tilde g\ge 1. \end{cases}\end{align*} $$
The next lemma provides a way to compute depths by counting preimages of nonexceptional points. It states precisely the sense in which holes “map onto
$\overline {\mathbb {C}}$
” by a degenerate map (c.f. [Reference DeMarco2, Lemmas 4.5 and 4.6]).
Lemma 2.6. Let
$\{f_k\} \subset {\operatorname {Rat}_d}$
be a sequence converging to
$g\in \partial {\mathrm {Rat}}_d$
and assume that 
. Pick
$z_0\in \overline {\mathbb {C}}$
. Then for any
$w \in \overline {\mathbb {C}}$
, there exists an arbitrarily small neighborhood U of w, such that the following hold for all sufficiently large k:
-
1. If
$z_0\not \in Ex_g$
, then, counting with multiplicities,
$$ \begin{align*}\#(f_k^{-1} (z_0) \cap U) = \begin{cases} d_w(g) & \text{ if } \tilde{g}(w) \neq z_0, \\ d_w(g) + \deg_w \tilde{g} & \text{ if } \tilde{g}(w) = z_0. \end{cases} \end{align*} $$
-
2. Counting with multiplicities,
$$ \begin{align*}\#(f_k^{-1} (z_0) \cap U)\le d_w(\varphi)+\deg_w\tilde\varphi\le d_w(g)+\deg_w\tilde\varphi.\end{align*} $$
Proof. Let
$P,Q \in \mathbb {C}[z]$
be two degree d polynomials such that
$g=P/Q$
. Let H be a common divisor of P and Q with maximal degree. Then
$P=H\tilde {P}$
,
$Q= H \tilde {Q}$
and
$\tilde {g} =\tilde {P}/\tilde {Q}$
for some relatively prime polynomials
$\tilde {P},\tilde {Q}$
. If
$z_0 \notin Ex_g$
, we have that
$\tilde {g} \not \equiv z_0$
. Hence, the multiplicity of w as a solution of
$$ \begin{align*}0=P-z_0Q=H(\tilde{P}-z_0 \tilde{Q})\end{align*} $$
is
$d_w(g)$
if
$\tilde {g}(w) \neq z_0$
, and
$d_w(g) + \deg _w \tilde {g}$
if
$\tilde {g}(w)=z_0$
. Thus writing
$f_k=P_k/Q_k$
and observing that the number of solutions of
$P_k - z_0Q_k$
in a small neighborhood of w converge to the multiplicity of w as a solution of
$P-z_0Q=0$
, we obtain statement (1).
In view of the above, for statement (2), it suffices to consider the case where
$\deg \tilde g=0$
. Let
$\{A_k\}\subset {\operatorname {Rat}}_1$
such that
$A_k\circ f_k\to \varphi $
. Passing to a subsequence in k, we can assume that
$A_k(z_0)\to a$
. Then given a small neighborhood U of w, we have that, for all sufficiently large k,
$$ \begin{align*}\#(f_k^{-1} (z_0) \cap U)=\#((A_k\circ f_k)^{-1}(A_k(z_0))\cap U)= \begin{cases} d_w(\varphi)+\deg_w\tilde\varphi\ &\text{if}\ a=\tilde\varphi(w),\\ d_w(\varphi) \ &\text{if}\ a\not=\tilde\varphi(w),\\ \end{cases}\end{align*} $$
counted with multiplicities. Then statement (2) follows from Lemma 2.5.
2.6 Pull-back of measures
Given a continuous function
$\varphi : \overline {\mathbb {C}} \to \mathbb {R}$
and a rational map
$f:\overline {\mathbb {C}}\to \overline {\mathbb {C}}$
, the push-forward of
$\varphi $
by f is the continuous function:
$$ \begin{align*}f_* \varphi (z) := \begin{cases} \sum\limits_{w \in f^{-1}(z)} \deg_w f \cdot \varphi(w)\ \ &\text{if}\ \deg f \ge 1,\\ \ \ \ 0 \ &\text{if}\ \deg f=0. \end{cases}\end{align*} $$
Clearly,
$f_*$
is a nonnegative bounded linear operator on the space of continuous function with the
$\sup $
-norm. Its dual is the pull-back
$f^*$
, whose action on a (positive Radon) measure
$\mu $
in
$\overline {\mathbb {C}}$
is determined by the property that
$$ \begin{align*}\int \varphi(w) d f^* \mu (w) = \int f_* \varphi (z) d \mu(z),\end{align*} $$
for all continuous functions
$\varphi $
. Taking
$\varphi \equiv 1$
, yields
$f^* \mu (\overline {\mathbb {C}}) = \deg f \cdot \mu (\overline {\mathbb {C}})$
. All measures considered here are nonnegative Radon measures.
For any
$g \in \partial {\operatorname {Rat}}_d$
, we can obtain a natural measure of total mass
$d-\deg {\tilde {g}}$
induced by the depths of its holes:
Definition 2.7. Given
$g \in \partial {\operatorname {Rat}}_d$
, we say that the measure
$$ \begin{align*}\eta_g := \sum_{h \in \operatorname{Hole}(g)} d_h(g) \delta_h\end{align*} $$
is the depth measure of g.
Consider
$g \in \overline {\operatorname {Rat}}_d$
. If
$g\in \partial {\operatorname {Rat}_d}$
, recall from §2.5 the algebraically exceptional set
$Ex_g\subset \overline {\mathbb {C}}$
for g. If
$g \in {\operatorname {Rat}_d}$
, then the dynamically exceptional set
$E_g$
of g, is the one formed by all finite grand orbits, under g. We say that a measure
$\mu $
in
$\overline {\mathbb {C}}$
is nonexceptional for
$g \in \overline {\operatorname {Rat}}_d$
if
$\mu (Ex_g) =0$
when
$g \in \partial {\operatorname {Rat}_d}$
and
$\mu (E_g)=0$
when
$g \in {\operatorname {Rat}_d}$
. Moreover, given an element in
$\Pi _{i=1}^\infty \overline {\operatorname {Rat}}_{s_i}$
, we say a measure
$\mu $
in
$\overline {\mathbb {C}}$
is nonexceptional for this element if
$\mu $
is nonexceptional for each coordinate. Following DeMarco [Reference DeMarco2, Section 3], we introduce the pull-back of measures by g.
Definition 2.8. Given
$g \in \partial {\operatorname {Rat}}_d$
and a nonexceptional measure
$\mu $
for g, we say that
$$ \begin{align*}g^* \mu = \tilde{g}^* \mu + \mu(\overline{\mathbb{C}}) \cdot \eta_g\end{align*} $$
is the pull-back of
$\mu $
under g.
Given
$g \in \partial {\operatorname {Rat}}_d$
and a continuous function
$\varphi : \overline {\mathbb {C}} \to \mathbb {R}$
, we define the push-forward of
$\varphi $
by g
$$ \begin{align*}g_* \varphi := \tilde{g}_* \varphi + \sum_{h \in \operatorname{Hole}(g)} d_h(g) \varphi(h);\end{align*} $$
then for any a nonexceptional measure
$\mu $
for g, one obtains that
$$ \begin{align*}\int \varphi d g^* \mu = \int g_* \varphi d \mu.\end{align*} $$
Although not explicitly discussed in DeMarco [Reference DeMarco2] and DeMarco-Faber [Reference DeMarco and Faber4], a key fact is that the pull-back defined above is the continuous extension of the pull-back of (nonexceptional) measures:
Proposition 2.9. Let
$g \in \partial {\operatorname {Rat}_d}$
and
$\mu $
be a nonexceptional measure for g. Then
$f^*\nu $
converges to
$g^* \mu $
, as
$f \in {\operatorname {Rat}}_d$
converges to g and
$\nu $
converges to
$\mu $
.
Proof. Let
$W\subset \overline {\mathbb {C}}$
be a small neighborhood of the point in
$Ex_g$
if
$Ex_g\not =\emptyset $
; and set
$W=\emptyset $
if
$Ex_g=\emptyset $
. Consider
$z \in \overline {\mathbb {C}} \setminus W$
. As
${\operatorname {Rat}_d} \ni f \to g$
, the set
$f^{-1}(z)$
converges to
$\operatorname {Hole}(g) \cup \tilde {g}^{-1}(z)$
; each point is counted with multiplicity. More precisely, given
$w \in \operatorname {Hole}(g) \cup \tilde {g}^{-1}(z)$
, let
$m_w := d_w(f)$
if
$w \notin \tilde {g}^{-1}(z)$
, and
$m_w := d_w(f) + \deg _w \tilde {g}$
otherwise. By Lemma 2.6 (1), given any small neighborhood of w, for all f sufficiently close to g, this neighborhood contains
$m_w$
preimages of z under f, counted with multiplicities.
Now, consider a nonnegative continuous test function
$\varphi :\overline {\mathbb {C}}\to \mathbb {R}$
. From the previous paragraph, as
$f \to g$
, we have that
$f_* \varphi : \overline {\mathbb {C}} \setminus W \to \mathbb {R} $
pointwise converges to the continuous function
$g_*\varphi :\overline {\mathbb {C}} \setminus W \to \mathbb {R}$
. Since
$\overline {\mathbb {C}} \setminus W$
is compact, the convergence is, in fact, uniform.
Given
$\varepsilon>0$
, assume that W is sufficiently small with
$\mu (\partial W)=0$
and
$\nu $
is sufficiently close to
$\mu $
so that both
$\mu (W)$
and
$\nu (W)$
are less than
$\varepsilon $
. Then
$$ \begin{align*} \left|\int_{\overline{\mathbb{C}}} \varphi (f^* d\nu - g^* d\mu) \right| & = \left|\int_{\overline{\mathbb{C}}} f_* \varphi d\nu - \int_{\overline{\mathbb{C}}} g_* \varphi d\mu \right|\\ & = \left|\int_{\overline{\mathbb{C}} \setminus W} f_* \varphi d\nu - \int_{\overline{\mathbb{C}} \setminus W} g_* \varphi d\mu + \int_{W} f_* \varphi d\nu - \int _{W}g_* \varphi d\mu \right|\\ &\le \left|\int_{\overline{\mathbb{C}} \setminus W} f_* \varphi d\nu - \int_{\overline{\mathbb{C}} \setminus W} g_* \varphi d\mu\right| + \left|\int_{W} f_* \varphi d\nu\right|+\left| \int _{W}g_* \varphi d\mu \right|\\ &\le\left|\int_{\overline{\mathbb{C}} \setminus W} f_* \varphi d\nu - \int_{\overline{\mathbb{C}} \setminus W} f_* \varphi d\mu\right|+\left| \int_{\overline{\mathbb{C}} \setminus W} (f_* \varphi-g_* \varphi) d\mu\right| + d \|\varphi\|_\infty 2 \varepsilon, \end{align*} $$
where the last inequality follows from the fact that
$\|F_* \varphi \|_\infty \le d \|\varphi \|_\infty $
for any
$F \in \overline {\operatorname {Rat}}_d$
. The convergence of
$\nu $
to
$\mu $
and the uniform convergence of
$f_* \varphi $
to
$g_* \varphi $
in
$\overline {\mathbb {C}} \setminus W$
, as
$f \to g$
, yield that
$f^* \nu \to g^* \mu $
as
$(f,\nu ) \to (g,\mu )$
.
2.7 Measures of maximal entropy
Recall that
$M^1(\overline {\mathbb {C}})$
denotes the space of probability measures in
$\overline {\mathbb {C}}$
endowed with the weak* topology. Given a rational map
$f \in {\operatorname {Rat}_d}$
, there exists a unique probability measure
$\mu _f$
such that
$f^* \mu _f = d \mu _f$
and 
where 
is the exceptional set for f (i.e., the elements of
$\overline {\mathbb {C}}$
with finite grand orbit). Moreover, for any
$\mu \in M^1(\overline {\mathbb {C}})$
satisfying 
,
$$ \begin{align*}\dfrac{1}{d^n} (f^{\circ n})^\ast \mu \to \mu_f.\end{align*} $$
Furthermore,
$\mu _f$
is supported on the Julia set 
of f and is the unique measure of maximal entropy for f. This measure
$\mu _f$
varies continuously with f. See [Reference Freire, Lopes and Mañé10, Reference Lyubich12, Reference Mañé13].
The fundamental property of an accumulation point
$\mu $
of measures
$\mu _f$
as f approaches a degenerate map was formulated by DeMarco and Faber [Reference DeMarco and Faber4, Section 2.4] in terms of pairs of probability measures
$(\mu ,\nu )$
that take into account the changes of scales involved. More precisely, let
$(\mu ,\nu )$
be a pair of probability measures on
$\overline {\mathbb {C}}$
. For a sequence
$\{A_k\}\subset {\operatorname {Rat}}_1$
, we say that a sequence
$\{\mu _k\}\subset M^1(\overline {\mathbb {C}})$
converges
$\{A_k\}$
-weakly to
$(\mu , \nu )$
if
$\mu _k$
converges to
$\mu $
and
$(A_k)_\ast \mu _k$
converges to
$\nu $
.
Proposition 2.10 [Reference DeMarco and Faber4, Theorem 2.4].
Let
$\{f_k\}$
be a sequence in
$\mathrm {Rat}_d$
and let
$\{A_k\}$
be a sequence in
${\operatorname {Rat}}_1$
such that
$A_k\circ f_k$
converges to a map
$\varphi \in {\operatorname {Rat}}_d^*$
. Assume that the
$\{A_k\}$
-weak limit of
$\{\mu _{f_k}\}$
is
$(\mu ,\nu )$
. Then
$$ \begin{align*}\mu =\frac{1}{d}\varphi^\ast\nu.\end{align*} $$
Proof. For all
$k \ge 1$
,
$$ \begin{align*}(A_k \circ f_k)^* (A_k) _* \mu_k = d \cdot \mu_{f_k}.\end{align*} $$
Continuity of the pull-back at
$(\varphi ,\nu )$
(Proposition 2.9) yields
$\varphi ^* \nu = d \cdot \mu .$
3 Limiting dynamics of fully ramified sequences
Recall from §2.4 that
${\operatorname {Rat}}^*_d$
denotes the space of degree d rational maps with nonconstant reduction. Möbius transformations act by post-composition on
${\operatorname {Rat}}^*_d$
. Its orbit space, denoted by 
, is a compact Hausdorff space, see Corollary 2.2.
Here we consider sequences
$\{f_k\} \subset {\operatorname {Rat}_d}$
such that 
converges to 
, for all n. From Lemma 2.4, there exists
$\varphi _{n,n+1} \in {\operatorname {Rat}}_d^*$
such that
$\varphi _{n+1} = \varphi _{n,n+1} \circ \varphi _n$
, for all n. We say that
$n_0$
is a fully ramified time if
$\varphi _{n_0,n_0+1}$
is a nondegenerate map of degree d.
In this section, we analyze the limiting dynamics for sequences having several successive fully ramified times. Our analysis is based on the trees of bouquet of spheres discussed in Appendix A and closely related to the work by Arfeux [Reference Arfeux1]. We establish the Structure Theorem 3.1 for sequences
$\{f_k\}$
diverging in moduli space
${\operatorname {rat}_d}$
and possessing at least
$3$
successive fully ramified times. We also deduce Corollaries 3.15 and 3.16 which are crucial in our proofs of Theorems A and B. Moreover, we obtain Theorem C as a byproduct of the Structure Theorem 3.1.
Recall that we should regard
$\tilde {\varphi }_{n,n+1}$
as a map between two spheres:
$\overline {\mathbb {C}}_n$
and
$\overline {\mathbb {C}}_{n+1}$
. Assuming that for some
$m \ge 0$
, the times
$m,m+1$
and
$m+2$
are fully ramified, the Structure Theorem asserts that the spheres
$\overline {\mathbb {C}}_m, \overline {\mathbb {C}}_{m+1}, \dots $
are organized in a linear concatenation and describes the maps
$\tilde {\varphi }_{n,n+1}$
for
$n \ge m$
. The main tools are the results from Appendix A and Proposition 3.7. The general idea is to combine the simple connectivity, given by Proposition A.1, with the formula for the multiplicities of the critical points of
$\tilde {\varphi }_{n,n+1}$
, given by Proposition A.3, to establish the linear concatenation for the initial segment of fully ramified times. Then, to show that the linear concatenation respects the order on the integers, we rely on the extraction of polynomial-like maps, given by Proposition 3.7. Finally, to extend our description beyond fully ramified times, we count preimages with the aid of Proposition A.2.
3.1 Statement of the Structure Theorem
Let
$\{f_k\} \subset {\operatorname {Rat}_d}$
be such that 
converges to 
in 
for each
$n\ge 1$
. Recall from §2.4 that
$\{ A_{n,k}: n, k \ge 1 \} \subset {\operatorname {Rat}}_1$
is a collection of scalings for
$\{f_k\}$
if for any
$n \ge 1$
, as
$k \to \infty $
,
$$ \begin{align*}A_{n,k} \circ f^{\circ n}_k \to \varphi_n.\end{align*} $$
Such a collection
$\{A_{n,k}\}$
has convergent changes of coordinates if for all
$n \neq n'$
,
$$ \begin{align*}A_{n',k} \circ A^{-1}_{n,k} \to A_{(n,n')} \in \overline{\operatorname{Rat}}_1.\end{align*} $$
When clear from context,
$A_{(n,n')} \in \overline {\operatorname {Rat}}_1$
will be the limit of the corresponding coordinate changes
$A_{n',k} \circ A^{-1}_{n,k}$
and, if
$A_{(n,n')} \in \partial {\operatorname {Rat}}_1$
, write
$$ \begin{align*}a_{n,n'}:= \tilde{A}_{(n,n')}\in\overline{\mathbb{C}}.\end{align*} $$
In the case that
$A_{(n,n')} \in \partial {\operatorname {Rat}}_1$
for all
$n \neq n'$
, we say that
$\{A_{n,k}: n,k \ge 1\}$
is a collection of independent scalings. Note that
$a_{n',n}$
is the unique hole of
$A_{(n,n')}$
. By definition, independent collections of scalings have convergent changes of coordinates.
Let
$m \ge 1$
and consider a collection
$\{A_{n,k}: n \ge m \}$
of independent scalings, where we abuse of notation and omit writing
$k \ge 1$
and, sometimes, simply write
$\{A_{n,k}\}$
. In view of Appendix A, given
$\ell \ge m$
, the collection
$\{A_{n,k}: m \le n \le \ell \}$
is, in a certain sense, a tree of bouquets of spheres. More precisely, for each integer
$n \ge 1$
, consider a copy
$\overline {\mathbb {C}}_n$
of the Riemann sphere. Let
$\sim $
be the equivalence relation in
$\overline {\mathbb {C}}_m \sqcup \cdots \sqcup \overline {\mathbb {C}}_\ell $
that identifies
$a_{n,n'} \in \overline {\mathbb {C}}_{n'}$
with
$a_{n',n} \in \overline {\mathbb {C}}_n$
if and only if
$$ \begin{align*}a_{n,n"} = a_{n',n"}\end{align*} $$
for all
$n" \neq n, n'$
such that
$m \le n"\le \ell $
. All other
$\sim $
-classes are trivial. Let
By Proposition A.1, the space 
is simply connected. For each n, we have the retraction
which maps
$\overline {\mathbb {C}}_{n'}$
onto
$a_{n',n}$
, for all
$n'\neq n$
. It is convenient to identify
$\overline {\mathbb {C}}_n$
with the corresponding subset of 
.
Collapsing 
to a point produces a continuous map 
. The image of
$\overline {\mathbb {C}}_{\ell +1}$
is the common point in 
of all the spheres
$\overline {\mathbb {C}}_j$
that intersect
$\overline {\mathbb {C}}_{\ell +1}$
in 
. We say that the inverse limit of 
is the space 
associated to the collection of scalings
$\{A_{n,k} : n \ge m\}$
.
In view of Lemma 2.4,
$A_{n+1,k} \circ f_k \circ A_{n,k}^{-1}$
converges to a map
$\varphi _{n,n+1} \in {\operatorname {Rat}}^*_d$
. Recall that we regard
$\tilde \varphi _{n,n+1}$
as a map from
$\overline {\mathbb {C}}_n$
onto
$\overline {\mathbb {C}}_{n+1}$
, and also regard
$A_{n,k}$
as a map from
$\overline {\mathbb {C}}$
onto
$\overline {\mathbb {C}}_n$
. We will show that, under certain conditions, the maps
$\tilde {\varphi }_{n,n+1}:\overline {\mathbb {C}}_n \to \overline {\mathbb {C}}_{n+1}$
produce a well defined map from 
to 
.
Theorem 3.1 (Structure Theorem).
Let
$\{f_k\}\subset {\operatorname {Rat}_d}$
be a sequence such that 
for each
$n\ge 1$
, and
$[f_k]\to [g]\in \partial {\operatorname {rat}}_d$
, as
$k\to \infty $
. Let
$\{A_{n,k}\}$
be a collection of scalings for
$\{f_k\}$
having convergent changes of coordinates.
Assume that there exists
$m \ge 1$
such that
$m, m+1$
and
${m+2}$
are fully ramified times. Then
$\{A_{n,k} :n \ge m\}$
is a collection of independent scalings and the following statements hold:
-
1. The space
associated to
$\{A_{n,k}\}$
is the linear concatenation of
$\overline {\mathbb {C}}_{m}, \overline {\mathbb {C}}_{m+1}, \cdots $
. That is, in suitable coordinates, employed for the rest of the statement, is obtained by identifying
$0 \in \overline {\mathbb {C}}_n$
to
$\infty \in \overline {\mathbb {C}}_{n+1}$
, for all
$m \le n < \ell $
(see Figure 1).Figure 1Structure Theorem.
-
2. The map
is well defined and continuous. -
3. For all
$n> m$
,
$$ \begin{align*}\deg_{\infty} \tilde{\varphi}_{n,n+1}=\deg \tilde\varphi_{n,n+1}=\deg_0 \tilde{\varphi}_{n-1,n}.\end{align*} $$
-
4. If
$n>m$
and
$n+1$
is a fully ramified time, then
$\tilde {\varphi }_{n,n+1}$
is a monomial of degree d. -
5. If
$n>m$
is not a fully ramified time, then
$z=\infty $
is the unique hole of
$\varphi _{n,n+1}$
.
associated to 
is obtained by identifying 
Continuity of 
is equivalent, once (1) is established, to
$\tilde {\varphi }_{n,n+1}(\infty )=\infty $
and
$\tilde {\varphi }_{n-1,n}(0)=0$
for
$n>m$
. Statement (3) yields that
$\deg \tilde {\varphi }_{n,n+1}$
is a nonincreasing function of
$n>m$
. Moreover, from statements (1)-(3),
$\tilde {\varphi }_{n,n+1}$
is a polynomial for all
$n>m$
.
The map 
should be regarded as a limiting action of the sequence
$\{f_k\}$
. In general, let us assume that
$\{A_{n,k}: n \ge m\}$
is a collection of independent scalings for a sequence
$\{f_k\}$
. For any
$\ell \ge m$
, let
$$ \begin{align*}J_\ell:=\{n : m \le n \le \ell \}\end{align*} $$
and consider the embedding
From Appendix A, the space 
is naturally identified with the Hausdorff limit of the embedded copies 
of
$\overline {\mathbb {C}}$
. Similarly, for
$\ell +1$
, the space 
is the limit of 
. Then,
$f_k:\overline {\mathbb {C}} \to \overline {\mathbb {C}}$
acts on the embedded sphere as
Roughly speaking, Proposition A.2 allows us to count preimages under
$F_k$
. That is, consider 
and 
. Then, for sufficiently large k and for any 
close to
$\rho ^{-1}_{\ell +1,n+1}(w_0)$
, there exists
$\delta $
preimages of w, under
$F_k$
, close to
$\rho ^{-1}_{\ell ,n}(z_0)$
where
$$ \begin{align*}\delta = \begin{cases} d_{z_0} (\varphi_{n,n+1}), & \text{ if } \tilde{\varphi}_{n,n+1}(z_0)\neq w_0,\\ d_{z_0} (\varphi_{n,n+1})+ \deg_{z_0} \tilde{\varphi}_{n,n+1}(z_0), & \text{ if } \tilde{\varphi}_{n,n+1}(z_0)= w_0. \end{cases} \end{align*} $$
See Proposition A.2 for a precise statement.
Remark 3.2. In Berkovich space language, consider a type II point
$x:=x_n$
that maps onto
$y:=x_{n+1}$
by a rational function F, and let
$D_x(z_0), D_y(w_0)$
be directions at x and y, respectively. Then the number of preimages in
$D_x(z_0)$
of a given point
$w \in D_y(w_0)$
is prescribed by a well known formula that depends on the surplus multiplicity and multiplicity of the direction
$D_x(z_0)$
(see [Reference Faber7, Section 3]). The formula above is its Archimedean analogue. In this analogy,
$x_n, x_{n+1}$
correspond to
$\overline {\mathbb {C}}_n, \overline {\mathbb {C}}_{n+1}$
.
Remark 3.3. A basic example to consider is
$$ \begin{align*}f_k(z) = c_k z^2 + c_k \in {\operatorname{Rat}}_2,\end{align*} $$
where
$c_k \to \infty $
. For
$n \ge 1$
, let
$$ \begin{align*}A_{n,k} (z) := c_k^{-2^{n-1}} z.\end{align*} $$
Note that
$A_{1,k} \circ f_k (z)$
converges to
$ z^2 + 1$
, and for
$n \ge 2$
,
$$ \begin{align*}A_{n,k} \circ f_k^{\circ n} (z) \to z^{2^{n-1}},\end{align*} $$
uniformly in compact subsets of
$\mathbb {C}$
as
$k \to \infty $
. Consequently, all times are fully ramified for
$\{f_k\}$
. It is not difficult to construct similar examples of polynomials in any degree, not necessarily unicritical. In a certain sense, the Structure Theorem says that a degenerating sequence of rational maps
$\{f_k\}$
with at least three succesive fully ramified resembles a degeneration along polynomial maps (c.f. Theorem C).
Remark 3.4. If we have just two consecutive ramified times
$m=1,2$
, the spheres
$\overline {\mathbb {C}}_1, \overline {\mathbb {C}}_2$
and
$\overline {\mathbb {C}}_3$
are not necessarily organized in a linear concatenation. Indeed, consider the family of quadratic rational maps, parametrized by
$0<t \ll 1$
:
$$ \begin{align*}f_t(z) :=\dfrac{1 + t^3 z^2}{t z^2} + t^{-1/2}.\end{align*} $$
Let
$A_{1,t} (z) := t z$
,
$A_{2,t}(z):= -t^{-3/2} + z t^{-1}$
, and
$A_{3,t} (z) = - 2^{-1} t^{-3/2} (z-t^{-1/2} -1)$
. As
$t \to 0$
, for a generic z, we have
$A_{1,t} \circ f_t(z)\to z^{-2}$
,
$A_{2,t} \circ f_t \circ A_{1,t}^{-1} (z) \to z^{-2}$
, and
$A_{3,t} \circ f_t \circ A_{2,t}^{-1} (z) \to z $
. Thus,
$m=0,1$
are fully ramified times for any sequence
$\{f_{t_k}\}$
such that
$t_k \searrow 0$
. However, the space associated to the scalings
$\{ A_{1,t_k}, A_{2,t_k},A_{3,t_k}\}$
is a bouquet of three spheres with a common point. Note that, for k large,
$f_k$
has an attracting fixed point
$z_k = t_k^{-1/2} + 1 + o(1)$
, and therefore, a quadratic-like restriction.
3.2 Proof of the Structure Theorem
We work under the hypothesis of the Structure Theorem and consider
$n_0 \ge m+2$
such that n is a fully ramified time if
$m \le n \le n_0$
. In §3.2.1 we establish that
$\overline {\mathbb {C}}_m, \dots , \overline {\mathbb {C}}_{n_0}$
are organized in a linear concatenation respecting the order of the integers, and deduce Theorem C. The proof of the Structure Theorem will be by induction on
$\ell \ge n_0$
. In §3.2.2, we reformulate the assertions of the theorem in a manner suitable for induction and establish the base case. In §3.2.3, the inductive step for
$\ell \ge n_0$
is proven. We rely on a series of lemmas, stated and established, under the assumptions and notation of Theorem 3.1.
3.2.1 Fully ramified times
Choose
$n_0\ge m+2$
and assume that n is a fully ramified time for all n such that
$m+2 \le n \le n_0$
.
The spheres
$\overline {\mathbb {C}}_m, \dots , \overline {\mathbb {C}}_{n_0+1}$
are pairwise distinct:
Lemma 3.5. If
$m\le n < n' \le n_0+1$
, then
$A_{(n',n)} \in \partial {\operatorname {Rat}}_1$
.
Proof. By contradiction, suppose that
$A_{(n',n)}$
is nondegenerate. Then
$$ \begin{align*}[f^{\circ n'-n}_k]=[ (A_{n,k} \circ A_{n',k}^{-1}) \circ (A_{n',k} \circ f^{\circ n'-n}_k \circ A_{n,k})] \to [A_{(n',n)} \circ \varphi_{n'-1,n'} \circ \cdots \circ \varphi_{n,n+1}] \in {\operatorname{rat}}_{d^{n'-n}},\end{align*} $$
which contradicts that
$[f_k] \to [g] \in \partial {\operatorname {rat}}_d$
, since the
$(n'-n)$
-th iterate map is proper (see [Reference DeMarco3, Proposition 4.1]).
Let 
be the space associated to
$\{A_{n,k}: m+1\le n\le n_0+1\}$
. According to Proposition A.3,
given by
for
$z \in \overline {\mathbb {C}}_n$
and
$n \in J_{n_0}$
, is continuous. Moreover, there exists a finite set 
of critical points, each with an assigned multiplicity, such that the total number of critical points is
$2d-2$
, counted with multiplicities. Furthermore, 
has the following property: if
$n \in J_{n_0}$
and
$z \in \overline {\mathbb {C}}_n$
, then the multiplicity of z as a critical point of
$\tilde \varphi _{n,n+1}$
is the number of elements of 
in
$\rho _{{n_0},n}^{-1} (z)$
, counted with multiplicities. A priori, 
might contain intersection points of distinct spheres.
Lemma 3.6. There exists a total order
$\prec $
in
$J_{n_0}$
such that
$$ \begin{align*}\begin{array}{ccc} J_{n_0-1} &\to & J_{n_0} \\ n &\mapsto & n+1 \end{array} \end{align*} $$
is a monotone map. Moreover, given
$n \neq n' \in J_{n_0}$
, for 
, the following hold:
-
1.
$\overline {\mathbb {C}}_n \cap \overline {\mathbb {C}}_{n'} \neq \emptyset $
if and only if n and
$n'$
are
$\prec $
-consecutive. -
2. If
$x \in \overline {\mathbb {C}}_{n}\cap \overline {\mathbb {C}}_{n'}$
, then
$\deg _x \varphi _{n,n+1} =d$
. -
3. If
$n,n'$
are
$\prec $
-consecutive in
$J_{n_0-1}$
, then
$n+1,n'+1$
are
$\prec $
-consecutive in
$J_{n_0}$
.
Proof. To lighten notation, when possible, we omit sub-indices and write 
,
$J=J_{n_0}$
and 
. Let us first show that given
$n \in J$
, there are at most two spheres, distinct from
$\overline {\mathbb {C}}_n$
, that intersect
$\overline {\mathbb {C}}_n$
. Suppose on the contrary that there are three distinct numbers
$n_1, n_2, n_3 \neq n$
in J such that the corresponding spheres have nonempty intersection with
$\overline {\mathbb {C}}_n$
at
$p_1, p_2, p_3$
, respectively. Simple connectivity of 
yields that
$X_j:=\overline {\mathbb {C}}_{n_j} \setminus \{p_j\}$
are pairwise disjoint for
$j=1,2,3$
, and hence 
are also pairwise disjoint for
$j=1,2,3$
. Since
$\varphi _{n_j,n_j+1}$
has degree d, its local degree at
$p_j$
is at most d. Thus
$X_j$
contains at least
$d-1$
critical points of
$\varphi _{n_j,n_j+1}$
, counted with multiplicities. Therefore, 
contains at least
$d-1$
points of 
, counted with multiplicity, by Proposition A.3. Then 
has at least
$3d-3> 2d -2$
points, counted with multiplicities, which contradicts Proposition A.3. A similar argument shows that three distinct spheres do not share a common point.
To define the order in J. Declare
$m \prec m +1$
. Consider the graph 
with vertex set J and edges given by all pairs
$\{n,n'\}$
such that
$\overline {\mathbb {C}}_n \cap \overline {\mathbb {C}}_{n'} \neq \emptyset $
in 
. Since there are no triple intersection points and 
is simply connected, by Proposition A.1, we conclude that 
is a connected tree. Moreover, the previous paragraph also implies that each vertex of 
has valence at most
$2$
. Thus 
is isomorphic to an interval graph: it has exactly two valence
$1$
vertices, say a and b. Without loss of generality,
$m+1$
belongs to the subtree connecting m with b. Declare
$n \prec n'$
if and only if n belongs to the connected component of 
containing a; equivalently, generic elements of
$\overline {\mathbb {C}}_{n}$
and
$\overline {\mathbb {C}}_a$
lie in the same connected component of 
.
Suppose that
$n \prec n'$
are two consecutive elements of
$J_{n_0}$
. Denote by 
the intersection point of
$\overline {\mathbb {C}}_n$
and
$\overline {\mathbb {C}}_{n'}$
. We claim that the local degree of
$\varphi _{n,n+1}$
at x is d. Indeed, the critical multiplicity of
$\varphi _{n,n+1}$
at x is the number of elements of 
, counted with multiplicity, in
$\rho _{n_0,n}^{-1}(x) \supset \rho _{n_0,n}^{-1}(\overline {\mathbb {C}}_{n'}\setminus \{x\})$
, which is at least
$d-1$
since
$\deg \varphi _{n',n'+1} = d$
. Therefore,
$\deg _x \varphi _{n,n+1}=d$
. A similar argument shows that
$\deg _x \varphi _{n',n'+1} =d$
.
Finally, if
$n,n' \in J_{n_0-1}$
are
$\prec $
-consecutive, then
$n +1$
and
$n'+1$
are consecutive in
$J\setminus \{m\}$
, since 
lies in the intersection of the corresponding spheres. Moreover, the translation map
$n \mapsto n+1$
is injective and, being consecutive preserving, must also be monotone.
We conclude that if the translation by
$1$
is
$\prec $
-monotone increasing, then
$$ \begin{align*}m \prec m+1 \prec \cdots \prec n_0-1 \prec n_0.\end{align*} $$
That is, the order
$\prec $
coincides with the standard order of the integers. Otherwise, if the translation by
$1$
is
$\prec $
- monotone decreasing, then either
$$ \begin{align*} m \prec m+2 \prec \cdots \prec m+3 \prec m+1, \end{align*} $$
or
$$ \begin{align*} \cdots\prec m+2 \prec m \prec m+1 \prec m+3 \prec \cdots. \end{align*} $$
However, studying the location of the Julia set 
of
$f_k$
for sufficiently large k, we will obtain an obstruction for translation to be decreasing. The location of the Julia set is controlled by the following general result whose statement is illustrated in Figure 2.
Illustration of Proposition 3.7.
Proposition 3.7. Consider
$\{h_k\} \subset {\operatorname {Rat}_d}$
. Assume that
$\{B_{n,k}:n=0,1 \}$
are a pair of independent scalings such that the associated space is obtained from the identification of
$ a \in \overline {\mathbb {C}}_0$
with
$b \in \overline {\mathbb {C}}_{1}$
, and fix a point
$w\in \overline {\mathbb {C}}_0$
. Suppose that the following hold:
-
1.
$B_{1,k} \circ h_k \circ B^{-1}_{0,k}$
converges uniformly to
$ \varphi \in {\operatorname {Rat}_d}$
. -
2.
$\deg _w \varphi = d$
. -
3.
$\varphi (w)\begin {cases} \neq b\ \text {if}\ w=a,\\ = b\ \text {if}\ w\not =a. \end {cases}$
Given a neighborhood V of b, for any k sufficiently large, the following statements hold:
-
(i)
and
$\overline {\mathbb {C}} \setminus B_{1,k}^{-1}(V)$
is contained in a completely invariant attracting Fatou component of
$h_k$
. -
(ii) There exist Jordan domains
$U_k'$
and
$U_k:=h_k(U^{\prime }_k)$
such that
$(U_k',U_k,h_k)$
is a degree d polynomial-like map with the modulus of
$U_k\setminus \overline {U^{\prime }_k}$
tending to
$\infty $
, as
$k\to \infty $
. Moreover,-
(a)
$B_{0,k}(U^{\prime }_k)$
converges to a Jordan domain
$U'$
in
$\overline {\mathbb {C}}_0$
, -
(b) if
$w=a$
, then
$\overline {\mathbb {C}}_0 \setminus \overline {U'}$
is a small neighborhood of a; -
(c) if
$w\not =a$
, then
$U'$
is a small neighborhood of w.
-
and 
As illustrated in Figure 2, we will choose a domain
$U'$
in
$\overline {\mathbb {C}}_0$
that is mapped onto a domain
$\varphi (U')$
in
$\overline {\mathbb {C}}_1$
with degree d by
$\varphi $
. One should think that
$U'$
maps under
$\varphi $
onto a domain that “properly contains”
$U'$
; although
$\varphi (U')$
is in a different sphere. The idea is that for k large, in the sphere
$\overline {\mathbb {C}}$
where
$h_k$
acts, the domain
$U_k':=B_{0,k}^{-1}(U')$
is, in fact, compactly contained in
$U_k:=h_k(U^{\prime }_k)$
since
$B_{1,k}(U_k)$
converges to
$\varphi (U')$
.
Proof. Changing
$B_{1,k}$
by
$B \circ B_{1,k}$
for a suitable Möbius transformation B, we may assume that
$b=\infty $
and
$\varphi (a) =0$
. Similarly, changing
$B_{0,k}$
, we may assume that
$a=0$
and, in the case
$w\not =a$
, we may also assume
$w=\infty $
Let
$C_k:=B_{0,k}\circ B^{-1}_{1,k}$
. Since
$b=\infty \in \overline {\mathbb {C}}_1$
is identified with
$a=0\in \overline {\mathbb {C}}_0$
, it follows that
$C_k$
converges to
$C \in \partial {\operatorname {Rat}}_1$
such that
$\tilde {C}=0$
and the hole of C is at
$\infty $
. Write
$T_k:=B_{1,k} \circ h_k \circ B_{0,k}^{-1}$
and
$H_k:= B_{0,k}\circ h_k \circ B^{-1}_{0,k}$
.
In view of (2) and (3), let D be a Jordan neighborhood of
$z=0$
(
$=\varphi (a)$
) so that
$D':=\varphi ^{-1}(D)$
is a Jordan neighborhood of
$z=0$
(
$=a$
) which maps d-to-
$1$
onto D, under
$\varphi $
. Note that
$\overline {\mathbb {C}}_1\setminus \overline D$
is a small neighborhood of
$z=\infty (=b)$
if
$w\not =a$
. The uniform convergence of
$T_k$
to
$\varphi $
implies that
$D^{\prime }_k:=T_k^{-1}(D)$
is a Jordan neighborhood of
$z=0$
converging to
$\varphi ^{-1}(D)$
in the Hausdorff topology. We will show that
$U' = \overline {\mathbb {C}}_0 \setminus \varphi ^{-1}(\overline {D})$
is the Jordan domain claimed in (ii-a). Note that
$H_k=C_k \circ T_k $
maps
$D^{\prime }_k$
onto
$D_k:= C_k(D)$
with degree d. The uniform convergence of
$C_k$
to
$0$
, in D, yields that
$D_k \Subset D^{\prime }_k$
for k large. Then
$H_k: \overline {\mathbb {C}}_0 \setminus \overline {D^{\prime }_k} \to \overline {\mathbb {C}}_0 \setminus \overline {D}_k$
is a degree d polynomial-like map. Moreover, the modulus
$\text {Mod}(D^{\prime }_k\setminus \overline {D_k})$
tends to infinity, as
$k\to \infty $
, since
$ C_k^{-1}(\overline {D_k})=\overline D$
and
$\overline {\mathbb {C}}_1\setminus C_k^{-1}(D^{\prime }_k)$
converges, in the Hausdorff topology, to
$\{\infty \}$
. After conjugacy by
$B^{-1}_{0,k}$
, the polynomial-like map
$(\overline {\mathbb {C}}_0 \setminus \overline {D^{\prime }_k}, \overline {\mathbb {C}}_0 \setminus \overline {D}_k, H_k)$
becomes the desired polynomial-like restriction of
$h_k$
. Thus, statement (ii) holds. Note that
$D_k'$
must be contained in a completely invariant attracting Fatou component of
$H_k$
. Since
$C_k^{-1}$
converges uniformly to
$\infty $
outside
$D^{\prime }_k$
, it follows that 
converges, in the Hausdorff topology, to
$\{\infty \}$
. In particular, for k large, 
, for any given neighborhood of
$b=\infty $
. Similarly,
$C^{-1}_{k}(D_k')$
contains
$\overline {\mathbb {C}}\setminus V$
for k large; therefore,
$\overline {\mathbb {C}} \setminus B_{1,k}^{-1}(V)$
is contained in a completely invariant attracting Fatou component of
$h_k$
.
We remark that the above polynomial-like map is extracted from a sequence
$\{h_k\}$
with just
$1$
fully ramified time. However, this extraction requires that assumptions (2) and (3) are satisfied. To prove Theorem C, we will show that assumptions (2) and (3) always hold for sequences with
$3$
consecutive fully ramified times.
Now we are ready to rule out monotone decreasing translations. Note that translation by
$2$
is always monotone increasing from
$J_{n_0-2}$
into
$J_{n_0}$
.
Lemma 3.8. The map
$n \mapsto n +1$
is
$\prec $
-monotone increasing in
$J_{n_0-1}$
.
Proof. We proceed by contradiction and suppose that
$n \mapsto n +1$
is
$\prec $
-orientation reversing. Then one of following concatenations of spheres occurs:
-
(i)
$\overline {\mathbb {C}}_{m} \prec \overline {\mathbb {C}}_{m+2}\prec \overline {\mathbb {C}}_{m+1}. $
-
(ii)
$\overline {\mathbb {C}}_{m+2}\prec \overline {\mathbb {C}}_{m} \prec \overline {\mathbb {C}}_{m+1}.$
Set
$a:=a_{m+1,m}, b:=a_{m,m+1}, c:=a_{m+1,m+2}$
, and consider small neighborhoods
$W \subset \overline {\mathbb {C}}_m$
of a,
$V \subset \overline {\mathbb {C}}_{m+1}$
of b and
$U \subset \overline {\mathbb {C}}_{m+2}$
of c. See Figure 3 for an illustration of these and other neighborhoods that we introduce below.
Illustration of proof of Lemma 3.8: case (i) on the left and case (ii) on the right.
Let
$W_k := A_{m,k}^{-1} (W)$
,
$V_k := A_{m+1,k}^{-1}(V)$
and
$U_k := A_{m+2,k}^{-1}(U)$
. We will freely assume that k is sufficiently large for the assertions below to hold.
We first show that, in both cases,
$$ \begin{align*}\tilde\varphi_{m,m+1}(a)=b \ \ \text{and}\ \ \tilde\varphi_{m+1,m+2}(b)=c,\end{align*} $$
by counting preimages of generic points in
$\overline {\mathbb {C}}_{m+2}$
and in
$\overline {\mathbb {C}}_{m+1}$
.
Indeed, if
$\tilde \varphi _{m,m+1}(a)\not =b$
, then for a generic point
$z\in \overline {\mathbb {C}}_{m+2}$
, under
$f_k$
, the point
$A_{m+2,k}^{-1}(z)$
has at least
$1$
preimage outside
$W_k$
. However,
$\deg \tilde \varphi _{m+1,m+2}=d$
, so all d preimages of
$A_{m+2,k}^{-1}(z)$
lie in
$W_k$
. It is a contradiction.
Similarly, if
$\tilde \varphi _{m+1,m+2}(b)\not =c$
, then for a generic point
$w\in \overline {\mathbb {C}}_{m+1}$
, under
$f_k$
, the point
$A_{m+1,k}^{-1}(w)$
has at least
$1$
preimage outside
$V_k$
. However,
$\deg \tilde \varphi _{m,m+1}=d$
, so all d preimages of
$A_{m+1,k}^{-1}(w)$
lie in
$V_k$
. This is again a contradiction.
With
$ \tilde \varphi _{m,m+1}(a)=b$
and
$\tilde \varphi _{m+1,m+2}(b)=c$
, we now consider cases (i) and (ii) separately.
Case (i): Let
$U'$
be a small neighborhood of
$c'=a_{m,{m+2}}$
and
$U^{\prime }_k:=A_{m+2,k}^{-1}(U')$
. By Proposition 3.7, applied to
$w=a$
and
$\varphi =\varphi _{m+1,m+2} \circ \varphi _{m,m+1}$
, we have that 
converges to
$c' \in \overline {\mathbb {C}}_{m+2}$
. Viewed in
$\overline {\mathbb {C}}_{m+1}$
, the Julia sets converge to b. More precisely, 
. But,
$\varphi _{m+1,m+2}(b)=c$
, so 
. This is impossible because
$U_k$
and
$U^{\prime }_k$
are disjoint. Hence (i) cannot occur.
Case (ii): To proceed with the argument, we first fix some notation. Let
$a':=a_{m+2,m}$
and note that
$b':=\tilde \varphi _{m,m+1}(a')\not = b$
. Choose a small neighborhood
$W'$
of
$a'\in \overline {\mathbb {C}}_m$
such that
$\tilde \varphi _{m,m+1}(W')$
, a neighborhood of
$b'$
, is properly contained in
$V^c:=\overline {\mathbb {C}}_{m+1}\setminus \overline V$
. Let
$W^{\prime }_k:= A_{m,k}^{-1}(W')$
, and
$V^c_k := A_{m+1,k}^{-1}(V^c)$
. Keep in mind that
$f_k(W^{\prime }_k) \Subset V_k^c$
.
By Proposition 3.7, for a choice of a small neighborhood W of
$a \in \overline {\mathbb {C}}_m$
, the map
$g_k:=(W_k,N_k,f_k^{\circ 2})$
is a degree
$d^2$
polynomial-like map where
$N_k:=f_k^{\circ 2}(W_k)$
. Moreover, the modulus
$\text {Mod}(N_k\setminus \overline {W_k})$
diverges to
$\infty $
, as
$k\to \infty $
. By shrinking V if necessary, we may assume that
$V_k$
is properly contained in
$f_k(W_k)$
.
We now show that the filled Julia set 
of
$g_k$
is connected. Since
$c \notin \varphi _{m+1,m+2}(V^c) \subset \overline {\mathbb {C}}_{m+2}$
, we have that
$f_k(V^c_k) \subset W_k'$
. Therefore,
$f_k^{\circ 2} (V_k^c) \Subset V^c_k$
, which yields that
$V_k^c$
is in the basin of an attracting fixed point
$z_k$
of
$g_k$
. We claim that all the critical values of
$g_k$
lie in this basin. The set of critical values of
$g_k$
is the union of
$X_k:=g_k(\operatorname {Crit}(f_k) \cap W_k)$
and
$Y_k:=f_k(\operatorname {Crit}(f_k)\cap f_k(W_k))$
. Note that, counting multiplicities,
$d-1$
critical points of
$f_k$
are contained in
$V_k^c \subset W_k$
and,
$d-1$
in
$W_k'$
. Thus,
$X_k \subset V_k^c$
and
$Y_k \subset f_k(W_k') \subset V_k^c$
. Therefore, all critical values of
$g_k$
are contained in the basin of
$z_k$
and
$g_k$
has connected filled Julia set.
Composing with appropriate coordinate changes and passing to a subsequence, if necessary, we claim that the sequence
$f_k^{\circ 2}$
converges to a degree
$d^2$
polynomial. Consider a scaling
$B_k$
such that 
has Euclidean diameter
$1$
and contains
$0$
. Then
$$ \begin{align*}(B_k(W_k), B_k(N_k), B_k\circ f_k^{\circ 2}\circ B_k^{-1})\end{align*} $$
is a polynomial-like map whose filled Julia set is connected, has Euclidean diameter
$1$
and contains
$0$
. Since the modulus of
$B_k(N_k)\setminus \overline {B_k(W_k)}$
tends to
$\infty $
, as
$k\to \infty $
, by [Reference McMullen14, Theorem 5.8] and a diagonal argument, the sequence
$B_k\circ f_k^{\circ 2}\circ B_k^{-1}:B_k(W_k)\to B_k(N_k)$
has a convergent subsequence whose limit is a polynomial-like map with infinite modulus. Thus, this limit is a polynomial of degree
$d^2$
.
Therefore, passing to a subsequence if necessary,
$[f_k^{\circ 2}]$
converges in
${\operatorname {rat}}_{d^2}$
. Since the iterate map
${\operatorname {rat}}_d\to {\operatorname {rat}}_{d^2}$
is proper, it follows that
$[f_k]$
converges in
${\operatorname {rat}}_d$
. This contradicts the assumption that
$[f_k]$
accumulates at a point of
$\partial {\operatorname {rat}}_d$
. Hence (ii) cannot occur, and the proof is complete.
We are now ready to deduce Theorem C.
Proof of Theorem C.
Assume that
$\{f_k\} \subset {\operatorname {Rat}}_d$
,
$\{A_k\}$
and
$\{B_k\}$
are as in the statement of the theorem. We proceed by contradiction. After passing to a subsequence, we suppose that
$f_k$
lacks of a degree d polynomial-like restriction for all k. Then
$h_k:=A_k \circ f_k \circ A_k^{-1}$
also has no degree d polynomial-like restriction. Passing to a further subsequence, 
converges to some 
for all n. Moreover,
$\varphi _3 \in {\operatorname {Rat}}_{d^3}$
. From Lemma 2.4, the sequence
$\{h_k\}$
has fully ramified times
$n=1,2,3$
. Let
$\{A_{n,k}\}$
be the collection of scalings such that
$A_{n,k} \circ h_k^{\circ n} \to \varphi _n$
. Then we are under the hypothesis of Theorem 3.1 for
$\{h_k\}$
and
$m=1$
. By Lemma 3.8, 
is the concatenation of
$3$
spheres
$\overline {\mathbb {C}}_1,\overline {\mathbb {C}}_2,\overline {\mathbb {C}}_3$
. Moreover,
$\varphi _{2,3}$
maps
$\overline {\mathbb {C}}_2$
onto
$\overline {\mathbb {C}}_3$
by a degree d map. By Lemma 3.6 (2), the intersection point of
$\overline {\mathbb {C}}_2$
with
$\overline {\mathbb {C}}_1$
maps, by
$\varphi _{2,3}$
, to the intersection point of
$\overline {\mathbb {C}}_3$
with
$\overline {\mathbb {C}}_2$
with local degree d. Thus, we are under the hypothesis of Proposition 3.7, for
$\{A_{n,k}: n=1,2\}$
, that yields the existence of a degree d polynomial-like restriction of
$h_k$
and, therefore, of
$f_k$
.
3.2.2 Inductive hypothesis
Given
$\ell \ge n_0$
, consider the following assertions:
-
(0.ℓ) For all
$m \le n , n' \le \ell $
, if
$n \neq n'$
, then
$A_{(n,n')} \in \partial {\operatorname {Rat}}_1$
. -
(1.ℓ)
is obtained by identifying
$a_{n+1,n} \in \overline {\mathbb {C}}_n$
with
$a_{n,n+1} \in \overline {\mathbb {C}}_{n+1}$
for
$m \le n < \ell $
. Moreover,
$a_{n-1,n} \neq a_{n+1,n}$
for
$m < n < \ell $
. -
(2.ℓ)
$ \tilde \varphi _{n,n+1}(a_{n-1,n}) = a_{n,n+1}$
if
$m < n < \ell $
, and
$\tilde \varphi _{n,n+1}(a_{n+1,n}) = a_{n+2,n+1}$
if
$m \le n < \ell -1$
. -
(3.ℓ)
$ \deg \tilde \varphi _{n,n+1}=\deg _{a_{n,n-1}} \tilde \varphi _{n-1,n}$
for
$m<n < \ell $
. -
(4.ℓ) If
$m<n<\ell $
and
$n+1$
is a fully ramified time, then the local degree of
$\varphi _{n,n+1}$
at
$a_{n\pm 1,n}$
is d. -
(5.ℓ) For
$m < n <\ell $
,
$$ \begin{align*}d= d_{a_{n-1,n}} (\varphi_{n,n+1})+ \deg_{a_{n-1,n}} \tilde{\varphi}_{n,n+1}.\end{align*} $$
is obtained by identifying 
In other words,
$(0.\ell )$
states that, for
$m \le n \le \ell $
, the scalings
$\{A_{k,n}\}$
are independent, so the corresponding spheres
$\overline {\mathbb {C}}_m, \dots , \overline {\mathbb {C}}_{\ell }$
are distinct. Then
$(1.\ell )$
says that these spheres are organized in a linear concatenation 
. Accordingly,
$(2.\ell )$
claims that the induced map on this linear concatenation 
is continuous. The inductive hypothesis
$(4.\ell )$
asserts that the maps
$\tilde {\varphi }_{n,n+1}$
are monomials in appropriate coordinates if n and
$n+1$
are fully ramified times. Finally,
$(3.\ell )$
and
$(5.\ell )$
are formulas describing the degree of
$\tilde {\varphi }_{n,n+1}$
and the depth of the holes of
$\varphi _{n,n+1}$
, which encode assertions (3) and (5) of the theorem, as explained below:
Proof of Theorem 3.1 assuming (0.
$\ell $
)–(5.
$\ell $
) for all
$\ell \ge n_0$
.
Clearly (0.
$\ell $
) for all
$\ell $
yields the independence of
$\{A_{n,k}:n\ge m\}$
. Moreover, (1.
$\ell $
) implies statement (1) of Theorem 3.1. For statement (2), we only need to check continuity at the intersection of consecutive spheres, which is the content of (2.
$\ell $
). Provided we normalize
$a_{n,n-1}=0 \in \overline {\mathbb {C}}_{n-1}$
and
$a_{n-1,n} = \infty \in \overline {\mathbb {C}}_n$
, for all n, statement (3.
$\ell $
) for all
$\ell $
is the second equality in (3). Statement (4) is exactly (4.
$\ell $
) for all
$\ell $
. For statement (5) and the remaining part of statement (3), note that
$d-\deg \tilde {\varphi }_{n,n+1}$
is the total number of holes of
$\varphi _{n,n+1}$
, counted with depths. Therefore,
$$ \begin{align*}d_{a_{n-1,n}} (\varphi_{n,n+1}) \le d-\deg\tilde{\varphi}_{n,n+1}\le d- \deg_{a_{n-1,n}} \tilde{\varphi}_{n,n+1}.\end{align*} $$
Thus, (5.
$\ell $
) for all
$\ell $
yields equality between all these quantities. Equivalently,
${a_{n-1,n}}$
is the unique hole of
$\varphi _{n,n+1}$
and the local degree of
$\tilde {\varphi }_{n,n+1}$
at this point is
$\deg \tilde {\varphi }_{n,n+1}$
. Statements (3) and (5) now follow.
Lemma 3.9. Statements (0.
$n_0$
)–(5.
$n_0$
) hold.
Proof. As observed above, Lemma 3.5 is (0.
$n_0$
). By Lemmas 3.6 and 3.8, the space 
is the concatenation of spheres indexed by
$J_{n_0}$
in the standard order of the integers. Therefore, statement (1.
$n_0$
) holds. Statement (2.
$n_0$
) follows from the continuity of 
, furnished by Proposition A.3. Statement (3.
$n_0$
) is trivially true and statement (4.
$n_0$
) is a direct consequence of Lemma 3.6 (2). Finally, statement (5.
$n_0$
) follows since
$\varphi _{n,n+1}$
has no holes and the local degree at
$a_{n-1,n}$
is d.
3.2.3 Proof of the Structure Theorem for not fully ramified times
We continue with the induction. Assume that (0.
$\ell $
)–(5.
$\ell $
) from §3.2.2 hold for some
$\ell \ge n_0$
. The proofs of the corresponding statements for
$\ell +1$
are organized in a series of lemmas below. Specifically, Lemma 3.10 shows that the scaling
$\{A_{\ell +1,k}\}$
is independent of the ones corresponding to the spheres
$\overline {\mathbb {C}}_m, \dots , \overline {\mathbb {C}}_\ell $
. Thus, 
consists of pairwise distinct spheres
$\overline {\mathbb {C}}_m,\dots , \overline {\mathbb {C}}_{\ell +1}$
, which are shown to be organized in a linear concatenation in Lemma 3.11. Then, Lemma 3.12 shows that the transition map
$\tilde {\varphi }_{\ell ,\ell +1}\colon \overline {\mathbb {C}}_\ell \to \overline {\mathbb {C}}_{\ell +1}$
is a polynomial with a unique hole at the intersection point with
$\overline {\mathbb {C}}_{\ell -1}$
. In Lemma 3.13, continuity of the induced map on 
is established. Finally, Lemma 3.14 studies the degree of
$\tilde {\varphi }_{\ell ,\ell +1}$
. We end this subsection by summarizing how all these lemmas yield the inductive step.
We employ the definitions and results from Appendix A. Let
$J := \{ n : m \le n \le \ell \}$
and consider the collection of independent scalings
$\mathbf {A}:= \{A_{n,k}: n \in J \}$
indexed by J. The space associated to
$\mathbf {A}$
is 
, the concatenation of the spheres
$\overline {\mathbb {C}}_m,\dots ,\overline {\mathbb {C}}_\ell $
. Denote by 
the corresponding embedding of
$\overline {\mathbb {C}}$
into
$\overline {\mathbb {C}}^{J}$
and by
$\rho _{\ell ,j}$
the retraction of 
onto
$\overline {\mathbb {C}}_j$
. For all
$n \neq j$
, note that
$\rho _{\ell ,j}(\overline {\mathbb {C}}_n) = \{a_{n,j}\}\subset \overline {\mathbb {C}}_j$
.
Lemma 3.10.
$A_{(\ell +1,j)}$
is degenerate for all
$j \in J$
.
Proof. By contradiction, suppose
$A_{(\ell +1,j)}$
is nondegenerate. Without loss of generality, we may assume that
$A_{\ell +1,k} = A_{j,k}$
, for all k. In the language of Appendix A, the sequence
$\{f_k\}$
maps
$\mathbf {A}$
to
$\mathbf {A}$
by
$\tau :J\to J$
where
$\tau (n) = n+1$
for
$n < \ell $
and
$\tau (\ell )=j$
. Consider 
defined by
In view of the inductive hypothesis (5.
$\ell $
) and Proposition A.2 applied to
$j-1$
, every point in 
, contained in a small neighborhood
$W_0$
of
$\rho _{\ell ,j}^{-1}(a_{j-1,j})= \overline {\mathbb {C}}_{m} \cup \cdots \cup \overline {\mathbb {C}}_{j-1}$
, has d preimages under
$G_k$
, close to
$\rho _{\ell ,j-1}^{-1} (a_{j-2,j-1})=\overline {\mathbb {C}}_m \cup \dots \cup \overline {\mathbb {C}}_{j-2}$
, which is bounded away from
$\overline {\mathbb {C}}_{\ell }$
. But
$W_0$
contains points
$w_0$
in
$\overline {\mathbb {C}}_{j}$
close to
$a_{j-1,j}$
with a preimage
$z_0$
in
$\overline {\mathbb {C}}_{\ell }$
, under
$\tilde \varphi _{\ell ,j}= \tilde \varphi _{\ell ,\ell +1}$
. It follows that points in 
, close to
$w_0$
, simultaneously have a
$G_k$
-preimage close to
$z_0 \in \overline {\mathbb {C}}_{\ell }$
and d preimages close to
$\rho _{\ell ,j-1}^{-1} (a_{j-2,j-1})$
, which is impossible.
Let
$J':=\{ j : m \le n \le \ell +1 \}$
. From the previous lemma,
$\mathbf {B} := \{ A_{n,k} : n \in J'\}$
is a collection of independent scalings indexed by
$J'$
. It follows that
$\{f_k\}$
maps
$\mathbf {A}$
to
$\mathbf {B}$
by
$\tau (n)=n+1$
. For all
$j \in J'$
, recall that
$\rho _{\ell +1,j}$
is the retraction from 
, the space associated to
$\mathbf {B}$
, onto
$\overline {\mathbb {C}}_j$
. Let 
be the corresponding embedding of
$\overline {\mathbb {C}}$
into
$\overline {\mathbb {C}}^{J'}$
and
Lemma 3.11.
$a_{\ell +1,\ell } \neq a_{\ell -1,\ell } \in \overline {\mathbb {C}}_{\ell }$
. Moreover, 
is obtained from the union of
$\overline {\mathbb {C}}_m,\dots ,\overline {\mathbb {C}}_{\ell +1}$
by identifying
$a_{n+1,n}$
with
$a_{n,n+1}$
for
$m\le n < \ell +1$
.
Proof. By contradiction, suppose that
$a_{\ell +1,\ell } = a_{\ell -1,\ell }\in \overline {\mathbb {C}}_{\ell }$
. Then
$\overline {\mathbb {C}}_{\ell -1}, \overline {\mathbb {C}}_{\ell }, \overline {\mathbb {C}}_{\ell +1}$
intersect at
$a_{\ell ,\ell -1} \in \overline {\mathbb {C}}_{\ell -1}$
, which is identified with
$a_{\ell -1,\ell } \in \overline {\mathbb {C}}_\ell $
. Thus
$\rho _{\ell +1,\ell }^{-1} (a_{\ell -1,\ell })$
contains
$\overline {\mathbb {C}}_{\ell +1}$
. Any point in 
close to a generic point
$w_0$
of
$\overline {\mathbb {C}}_{\ell +1}$
has a
$F_k$
-preimage in 
close to 
. Proposition A.2 and (5.
$\ell $
) yield a contradiction, since the d preimages of
$w_0$
are close to
$\rho _{\ell , \ell -1}^{-1} (a_{\ell -2,\ell -1})= \overline {\mathbb {C}}_{m} \cup \cdots \cup \overline {\mathbb {C}}_{\ell -2}$
, which is bounded away from
$\overline {\mathbb {C}}_{\ell }$
by (1.
$\ell $
). Thus,
$a_{\ell +1,\ell } \neq a_{\ell -1,\ell }\in \overline {\mathbb {C}}_{\ell }$
We conclude that in 
the sphere
$\overline {\mathbb {C}}_{\ell +1}$
has nonempty intersection only with
$\overline {\mathbb {C}}_\ell $
; for otherwise, if
$\overline {\mathbb {C}}_{\ell +1} \cap \overline {\mathbb {C}}_n \neq \emptyset $
for some
$n \le \ell -1$
, then
$a_{\ell +1,\ell } = a_{\ell -1,\ell }$
. Therefore, 
is the claimed concatenation of spheres.
According to the next lemma, in coordinates for
$\overline {\mathbb {C}}_\ell $
and
$\overline {\mathbb {C}}_{\ell +1}$
where
$a_{\ell -1,\ell }$
and
$a_{\ell ,\ell +1}$
are at
$\infty $
, the map
$\tilde {\varphi }_{\ell ,\ell +1}$
becomes a polynomial. Moreover,
$a_{\ell -1,\ell }$
is the unique hole of
$\varphi _{\ell , \ell +1}$
, if
$\varphi _{\ell ,\ell +1}\in \partial {\operatorname {Rat}}_d$
. In particular, (5.
$\ell +1$
) and the first assertion of (2.
$\ell +1$
) hold.
Lemma 3.12. Let
$a :=a_{\ell -1,\ell }$
,
$a':= a_{\ell ,\ell +1}$
and
$\varphi :={\varphi }_{\ell ,\ell +1}$
. Then
$\tilde {\varphi }^{-1}(a') = a$
and
$$ \begin{align*}d= \deg_a \tilde{\varphi}+ d_a (\varphi).\end{align*} $$
Proof. Let z be such that
$\tilde {\varphi }(z)=a'$
. Since
$\rho ^{-1}_{\ell +1,\ell +1}(a')$
contains
$\overline {\mathbb {C}}_n$
for all
$n \le \ell $
, in particular, it contains
$\overline {\mathbb {C}}_{\ell -1}$
. If 
is sufficiently close to a generic point of
$\overline {\mathbb {C}}_{\ell -1}$
, then it has a preimage close to
$\rho ^{-1}_{\ell ,\ell }(z)$
. However, all the
$F_k$
-preimages of
$w_k$
are close to of
$\overline {\mathbb {C}}_m \cup \dots \cup \overline {\mathbb {C}}_{\ell -2}$
, by (5.
$\ell $
) and Proposition A.2. It follows that
$\rho ^{-1}_{\ell ,\ell }(z)$
is contained in
$\overline {\mathbb {C}}_m \cup \dots \cup \overline {\mathbb {C}}_{\ell -2}$
, whose
$\rho _{\ell ,\ell }$
-image is
$\{a\}$
. Thus
$z=a$
is the unique
$\tilde {\varphi }_{\ell ,\ell +1}$
-preimage of
$a'$
. Moreover, since all the
$F_k$
-preimages of
$w_k$
are close to
$\rho ^{-1}_{\ell ,\ell } (a)$
, by Proposition A.2, the second assertion holds.
Now we prove the second assertion of (2.
$\ell +1$
):
Lemma 3.13. Let
$b:=a_{\ell , \ell -1}$
,
$b':=a_{\ell +1,\ell }$
and
$\varphi :={\varphi }_{\ell -1,\ell }$
. Then
$\tilde {\varphi } (b) = b'$
.
Proof. By contradiction, suppose that
$\tilde {\varphi }(b) \neq b'$
. Then
$\tilde {\varphi }^{-1}(b')$
consists of finitely many points in
$\overline {\mathbb {C}}_{\ell -1}$
bounded away from the rest of the spheres, by (1.
$\ell $
) and (2.
$\ell $
). Thus,
$\rho _{\ell ,\ell -1}^{-1}(z)=\{z\}$
for all
$z \in \tilde {\varphi }^{-1}(b')$
. Therefore, the
$F_k$
-preimages of points in 
close to a generic point of
$\overline {\mathbb {C}}_{\ell +1}=\rho _{\ell +1,\ell }^{-1}(b')$
are close to the finite set
$\tilde {\varphi }^{-1}(b') \subset \overline {\mathbb {C}}_{\ell -1}$
far from
$\overline {\mathbb {C}}_\ell $
. This is impossible since
$\tilde {\varphi }$
maps
$\overline {\mathbb {C}}_{\ell }$
onto
$\overline {\mathbb {C}}_{\ell +1}$
.
We also have the following result on degrees, which gives (3.
$\ell +1$
):
Lemma 3.14. Let
$b:=a_{\ell , \ell -1}$
. Then
$\deg \tilde {\varphi }_{\ell -1,\ell } \ge \deg _b \tilde {\varphi }_{\ell -1,\ell } = \deg \tilde {\varphi }_{\ell ,\ell +1}$
.
Proof. Let
$b':=a_{\ell +1,\ell }$
. Note that b is not a hole of
$\tilde {\varphi }_{\ell -1,\ell }$
by (5.
$\ell $
). Let
$w_0$
be a generic point of
$\overline {\mathbb {C}}_{\ell +1}=\rho ^{-1}_{\ell +1,\ell }(b')$
. There are exactly
$\deg _b \tilde {\varphi }_{\ell -1,\ell }$
preimages, under
$F_k$
, of every point in 
close to
$w_0$
, which are close to
$\overline {\mathbb {C}}_\ell = \rho ^{-1}_{\ell ,\ell -1}(b)$
. But there are
$\deg \tilde {\varphi }_{\ell ,\ell +1}$
preimages of
$w_0$
in
$\overline {\mathbb {C}}_\ell $
, so
$\deg \tilde {\varphi }_{\ell ,\ell +1}= \deg _b \tilde {\varphi }_{\ell -1,\ell } \le \deg \varphi _{\ell -1,\ell }.$
We now conclude the induction by certifying that (0.
$\ell +1$
)–(5.
$\ell +1$
) follow from the above lemmas. Indeed, (0.
$\ell +1$
) is a consequence of Lemma 3.10, (1.
$\ell +1$
) is Lemma 3.11. Lemma 3.12 implies the first assertion of (2.
$\ell +1$
) together with (5.
$\ell +1$
). The second assertion of (2.
$\ell +1$
) is obtained from Lemma 3.13. Statement (3.
$\ell +1$
) is contained in Lemma 3.14. Suppose that
$n_0+1=\ell $
is not a fully ramified time. Then (4.
$\ell +1$
) reduces to (4.
$n_0$
), which we have already proven (Lemma 3.6 (2)). This finishes the proof of Theorem 3.1.
3.3 Measures and depths
Consider a sequence
$\{f_k\}$
with a collection of scaling
$\{A_{n,k}\}$
for which the Structure Theorem 3.1 applies. Proposition 3.7 says that the Julia set 
, viewed in
$\overline {\mathbb {C}}_n$
for
$n>m$
, converges to
$a_{n-1,n}$
. Measure-theoretically we have the following consequence:
Corollary 3.15. Under the assumption and notation of Theorem 3.1, for any
$n>m$
, let
$A_n:=\lim A_{n,k}$
and
$a_n:= \tilde {A}_n$
. Then the following hold:
-
1.
$a_n=a_{n-1,n}$
. -
2.
converges in the Hausdorff topology on compact sets to
$\{a_n\}$
. In particular,
$$ \begin{align*}(A_{n,k})_* \mu_{f_k} \to \delta_{a_n}.\end{align*} $$
converges in the Hausdorff topology on compact sets to 
Proof. Consider
$n> m$
. By Theorem 3.1,
$$ \begin{align*}T_k:=A_{n+m,k} \circ f_k^{\circ m} \circ A_{n,k}^{-1}\end{align*} $$
converges to a map
$\varphi \in {\operatorname {Rat}}^*_{d^m}$
with a unique hole at
$a_{n-1,n}=\infty $
. If
$a_n \neq a_{n-1,n}$
, then
$A_{n+m,k} \circ f_k^{\circ m} = T_k \circ A_{n,k}$
converges to
$\tilde {\varphi }(a_n)$
for generic
$z \in \overline {\mathbb {C}}$
. By Theorem 3.1 (4), the unique preimage of
$a_{m,n+m}=\infty $
under the polynomial
$\tilde {\varphi }$
is
$a_{n-1,n}=\infty $
. Thus,
$\tilde {\varphi } (a_n) \neq \infty $
. However,
$A_{m,k}\circ f_k^{\circ m}$
converges to map with nonconstant reduction
$\tilde {\varphi }_m$
. We conclude that, generically,
$$ \begin{align*}(A_{m+n,k}\circ A_{m,k}^{-1})\circ (A_{m,k}\circ f_k^{\circ m})(z)\to a_{m,m+n}=\infty.\end{align*} $$
This is a contradiction. Hence statement (1) holds.
For statement (2), consider a small neighborhood W of
$a_n$
. By statement (1), we have that
$a_n=a_{n-1,n}$
and, by Proposition 3.7 and Theorem 3.1, we conclude that W contains 
for all sufficiently large k. The conclusion follows since 
is the support of
$(A_{n,k})_* \mu _{f_k}$
.
Sequences with
$3$
consecutive fully ramified times have well-behaved limits in
$\widehat {{\operatorname {Rat}}}_d$
:
Corollary 3.16. Under the assumption and notation in Theorem 3.1, if
$f_k^{\circ n}$
converges to
$g_n \in \overline {\operatorname {Rat}}_{d^n}$
for all n (i.e.,
$\{ f_k \}$
converges to
$(g_n)$
in
$\widehat {{\operatorname {Rat}}}_d$
), then for any given
$n>m$
the following hold:
-
1.
$\eta _{g_n} = \eta _{g_{m+1}}$
; -
2.
$\tilde g_n=\tilde g_{m+1}$
is a constant in
$\overline {\mathbb {C}}$
; and -
3. for any
$z\in \overline {\mathbb {C}}$
,
$$ \begin{align*}\frac{d_z(g_n)}{d^n}=\frac{d_z(g_{m+1})}{d^{m+1}}.\end{align*} $$
Proof. Consider
$n> m$
. Recall that, in convenient coordinates, we put
$a_{n-1,n}$
at
$\infty $
. Let us first prove statement (2). The hole h of
$A_n = \lim A_{n,k}$
is
$\tilde g_n$
. By Corollary 3.15 (1), generically,
$A_{n,k}$
converges to
$a_n = a_{n-1,n}=\infty \in \overline {\mathbb {C}}_n$
. Off
$a_n$
, we have that
$A_{n,k}^{-1}$
converges to h. Since
$A_n \circ A_{n+1}^{-1}$
converges generically to
$a_{n+1,n} \neq a_n$
, it follows that
$A_{n+1,k}^{-1} = A_{n,k}^{-1} \circ (A_n \circ A_{n+1}^{-1})$
also converges to h. Thus the hole of
$A_{n+1}$
is also h and
$\tilde {g}_{n+1} =h$
.
We now show statement (3), which immediately implies statement (1). It is sufficient to show that
$d_z(g_{n+1})=d_z(g_n)\cdot d$
for all
$z \in \overline {\mathbb {C}}$
. By Lemma 2.5, if
$\tilde {\varphi }_{m+1} (z) \neq a_{m, m+1}$
, then
$d_z (g_n) = d_z (\varphi _n)$
; and if otherwise,
$d_z (g_n) = d_z (\varphi _n)+ \deg _z \tilde {\varphi }_n$
. To compute
$d_z(g_{n+1})$
, we consider two cases according to whether
$\tilde {\varphi }_{m+1} (z) = a_{m,m+1}$
or not, equivalently,
$\tilde {\varphi }_n (z) = a_{n-1,n} =\infty \in \overline {\mathbb {C}}_n$
or not since all transition maps are polynomials (Theorem 3.1 (4)).
Suppose that
$\tilde {\varphi }_{m+1} (z) \neq a_{m,m+1}$
. That is,
$\tilde {\varphi }_n(z) \neq a_{n-1,n}=a_n$
. Then
$\tilde {\varphi }_n(z)$
is not a hole of
$\varphi _{n,n+1}$
and
$\tilde {\varphi }_{n+1} (z) \neq a_{n+1}$
. Therefore, using the formula (2.2), we obtain that
$$ \begin{align*}d_z(g_{n+1}) = d_z (\varphi_{n+1}) = d_z (\varphi_{n}) \cdot d = d_z(g_n) \cdot d.\end{align*} $$
Suppose that
$\tilde {\varphi }_{m+1} (z) = a_{m,m+1}$
. That is,
$\varphi _n(z) = a_{n-1,n}=a_n$
. Then
$\varphi _n(z)$
is the hole of
$\varphi _{n,n+1}$
, which has depth
$d - \deg \tilde {\varphi }_{n,n+1}$
and the local degree at
$a_n$
is
$\deg \tilde {\varphi }_{n,n+1}$
. It follows that
$$ \begin{align*} d_z(g_{n+1}) & = d_z (\varphi_{n+1}) + \deg_z \tilde{\varphi}_{n+1}\\ &= d_z(\varphi_n) \cdot d + \deg_z \tilde{\varphi}_{n} \cdot d_{a_n} (\varphi_{n,n+1}) + \deg_z \tilde{\varphi}_{n} \cdot \deg_{a_n} \tilde{\varphi}_{n,n+1}\\ &= d_z(\varphi_n) \cdot d + \deg_z \tilde{\varphi}_{n} \cdot d\\ &= d_z(g_{n}) \cdot d.\\[-37pt] \end{align*} $$
4 Iterations and measures for degenerate maps
In this section, we prove Theorem A. Given
$\mathbf {g} = (g_n) \in \widehat {{\operatorname {Rat}}}_d$
, we aim at proving the existence of a probability measure
$\mu _{\mathbf {g}}$
, continuously depending on
$\mathbf {g}$
, such that
$$ \begin{align*}\dfrac{1}{d^n}(g_n)^* \mu \to \mu_{\mathbf{g}},\end{align*} $$
for a generic probability measure
$\mu $
. To this end, we consider a sequence
$\{f_k\}\subset {\operatorname {Rat}_d}$
converging to
$\mathbf {g}$
. First we make some assumptions on the sequence
$\{f_k\}$
and prove Theorem A under these assumptions. Later, using compactness of
$\widehat {{\operatorname {Rat}}}_d$
and
$M^1(\overline {\mathbb {C}})$
, we deduce Theorem A in full generality.
We may simultaneously consider the limits of
$f_k^{\circ n}$
and of 
; to record these limits, let us introduce the following definition:
Definition 4.1. A sequence
$\{f_k\}\subset {\operatorname {Rat}_d}$
is totally convergent if for any
$n\ge 1$
, there exist
$g_n \in \overline {\operatorname {Rat}}_{d^n}$
and
$\varphi _n\in {\operatorname {Rat}}_{d^n}^*$
such that the following hold:
-
1.
$f_k^{\circ n}$
converges to
$g_n \in \overline {\operatorname {Rat}}_{d^n}$
(i.e.,
$\{ f_k \}$
converges to
$(g_n)$
in
$\widehat {{\operatorname {Rat}}}_d$
), and -
2.
converges to .
converges to 
.We say that
$\{f_k\}$
totally converges to
$(g_n,\varphi _n)$
.
To prove Theorem A, we assume that
$\{f_k\} \subset {\operatorname {Rat}_d}$
totally converges to
$(g_n, \varphi _n)$
, as
$k \to \infty $
(see Definition 4.1). We also assume that the conjugacy classes
$\{ [f_k] \} \subset \overline {\operatorname {rat}}_d$
converge in moduli space to some
$[g] \in \overline {\operatorname {rat}}_d$
. Then we consider the following cases:
-
1.
$\deg \tilde {\varphi }_n = o(d^n)$
. -
2.
$\deg \tilde {\varphi }_n \neq o(d^n)$
and-
(a)
$[g] \in {\operatorname {rat}_d}$
or, -
(b)
$[g] \in \partial {\operatorname {rat}_d}$
.
-
In each case, we show that a measure
$\mu _{\mathbf {g}}$
with the desired properties exists and that
$\mu _{f_k}$
converges to
$\mu _{\mathbf {g}}$
. After discussing some preliminary facts in §4.1, case (1) is considered in §4.2, case (2a) in §4.3, and case (2b) in §4.4. Finally, in §4.5, we assemble these results to produce a proof of Theorem A.
4.1 Convergence of depth measures
Recall from Definition 2.7 that the depth measure of a map
$g \in \partial {\operatorname {Rat}_d}$
is
$$ \begin{align*}\eta_g := \sum_{h \in \operatorname{Hole}(g)} d_h(g) \delta_h.\end{align*} $$
Lemma 4.2. Let
$\{f_k\} \subset {\operatorname {Rat}_d}$
be a sequence such that 
in 
for each
$n\ge 1$
, as
$k \to \infty $
. Then there exists a purely atomic measure
$\eta $
such that, as
$n\to \infty $
,
$$ \begin{align*}\frac{1}{d^n} \eta_{\varphi_n}\to \eta.\end{align*} $$
Proof. Apply Lemma 2.4 to write
$\varphi _{n+1} = \varphi _{n,n+1} \circ \varphi _n$
where
$\deg \tilde {\varphi }_{n,n+1} \ge 1$
. For all
$h \in \overline {\mathbb {C}}$
, in view of (2.2), we have
$$ \begin{align*}d_h (\varphi_{n+1}) = d_h (\varphi_n) \cdot d + \deg_{h} \tilde\varphi_n \cdot d_{\tilde\varphi_n(h)}(\varphi_{n,n+1}).\end{align*} $$
Division by
$d^{n+1}$
yields that
$$ \begin{align*}\frac{\eta_{\varphi_n} (\{h\})}{d^n} = \dfrac{d_h(\varphi_n)}{d^n}\end{align*} $$
is nondecreasing. Let
$\eta $
be the purely atomic measure defined by
$$ \begin{align*}\eta(\{h\}) := \lim_n\frac{\eta_{\varphi_n} (\{h\})}{d^n}.\end{align*} $$
It follows that
$\eta _{\varphi _n}/d^n \to \eta $
.
4.2 Small degree growth
Let
$\{ f_k\}\subset {\operatorname {Rat}_d}$
be a sequence totally converging to
$(g_n,\varphi _n)$
. The growth of
$\deg \tilde {\varphi }_n$
plays a crucial role in our argument. In the case of small degree growth, that is,
$\deg \tilde {\varphi }_n = o(d^n)$
, we have the following convergence of measures.
Proposition 4.3. Consider a sequence
$\{f_k\} \subset {\operatorname {Rat}_d}$
totally converging to
$(g_n, \varphi _n)$
. Suppose
$\mu $
is a nonexceptional probability measure for
$(g_n)$
. Let
$\eta := \lim (\eta _{\varphi _n}/d^n)$
. Assume that
$$ \begin{align*}\deg \tilde{\varphi}_n = o(d^n).\end{align*} $$
Then
$$ \begin{align*}\eta = \lim_{n\to\infty} \dfrac{1}{d^n} (g_n)^* \mu = \lim_{k\to\infty} \mu_{f_k}.\end{align*} $$
Proof. Let
$\nu $
be an accumulation point of
$\mu _{f_k}$
. Consider
$A_{n,k} \in {\operatorname {Rat}}_1$
such that
$A_{n,k} \circ f_k^{\circ n} \to \varphi _n$
, as
$k \to \infty $
, for all n. Passing to a subsequence of
$\{f_k\}$
, we may assume that
$\mu _{f_k}$
converges to
$ \nu $
and
$(A_{n,k})_* \mu _{f_k}$
converges to some measure
$\nu _n$
for all n. By Proposition 2.10,
$$ \begin{align*}\nu = \dfrac{1}{d^n} (\varphi_n)^* \nu_n = \frac{1}{d^n}\eta_{\varphi_n} + \dfrac{1}{d^n}(\tilde{\varphi}_n)^*\nu_n \to \eta,\end{align*} $$
since
$(\tilde {\varphi }_n)^* \nu _n (\overline {\mathbb {C}}) = \deg \tilde {\varphi }_n = o(d^n)$
. Therefore, the measures
$\mu _{f_k}$
converge to
$\eta $
.
To show the measure
$(g_n)^* \mu /d^n$
converges to
$\eta $
, we now prove that the measures
$(g_n)^*\mu $
and
$\eta _{\varphi _n}$
differ in at most
$o(d^n)$
. Although the conclusion is the same, there are two cases, according to whether
$\tilde {g}_n$
is constant or not.
For all n such that
$\tilde {g}_n$
is constant, from Lemma 2.5, the sequence
$\{A_{n,k} \}$
converges to some
${A_n \in \partial {\operatorname {Rat}}_1}$
, as
$k \to \infty $
, and
$$ \begin{align*} (g_n)^*\mu = \eta_{g_n} = \eta_{\varphi_n} + \sum_{z \in \tilde{\varphi}_n^{-1}(\tilde{A}_n)} \deg_z \tilde{\varphi}_n \cdot \delta_z = \eta_{\varphi_n} + \varepsilon_n, \end{align*} $$
where
$\varepsilon _n (\overline {\mathbb {C}}) = \deg \tilde {\varphi }_n$
.
If
$\tilde {g}_n$
is not constant, then
$$ \begin{align*} (g_n)^* \mu = \eta_{g_n} + (\tilde{g}_n)^* \mu = \eta_{\varphi_n} + (\tilde{g}_n)^* \mu = \eta_{\varphi_n} + \varepsilon^{\prime}_n, \end{align*} $$
where
$\varepsilon ^{\prime }_n (\overline {\mathbb {C}}) = \deg \tilde {g}_n$
.
For all
$n \ge 1$
, we have that
$\deg \tilde {g}_n \le \deg \tilde {\varphi }_n = o (d^n)$
. Therefore, both
$\varepsilon _n(\overline {\mathbb {C}})$
and
$\varepsilon ^{\prime }_n(\overline {\mathbb {C}})$
are
$o(d^n)$
, and hence
$$ \begin{align*}\lim_{n\to\infty} \dfrac{1}{d^n}(g_n)^* \mu = \lim_{n\to\infty} \dfrac{1}{d^n} \eta_{\varphi_n} =\eta.\end{align*} $$
This completes the proof.
We will deal with sequences such that
$\deg \tilde {\varphi }_n \neq o(d^n)$
in the next two subsections, but before, let us record the following observation:
Lemma 4.4. Let
$\{ f_k\}$
be a sequence totally converging to
$(g_n, \varphi _n)$
and
$g_1 \in \partial {\operatorname {Rat}_d}$
. Suppose that
$\deg \tilde {\varphi }_n \neq o(d^n)$
. Then
$$ \begin{align*}0<\dfrac{1}{d^n} \deg \tilde{\varphi}_n\le 1\end{align*} $$
is eventually constant (independent of n). Moreover, for all n
$$ \begin{align*}\deg \tilde{g}_n =0.\end{align*} $$
Proof. Let
$\varphi _{n,n+1} \in {\operatorname {Rat}}^*_d$
be such that
$\varphi _{n+1}=\varphi _{n,n+1} \circ \varphi _n$
, as in Lemma 2.4. Since
$\deg \tilde {\varphi }_{n+1} = \deg \tilde {\varphi }_{n,n+1} \cdot \deg \tilde {\varphi }_{n}$
and
$\deg \tilde {\varphi }_n \neq o(d^n)$
, it follows that
$\deg \tilde \varphi _{n,n+1} = d$
for all n sufficiently large. Therefore,
$\deg \tilde {\varphi }_n/d^n$
is eventually constant in
$(0,1]$
.
We now prove that
$\deg \tilde {g}_n =0$
, by contradiction. First observe that
$g_n \in \partial {\operatorname {Rat}}_{d^n}$
because
$g_1 \in \partial {\operatorname {Rat}_d}$
and iteration is a proper map. Suppose that, for some
$n_0$
, we have
$\deg \tilde {g}_{n_0} \ge 1$
. Then
$\deg \tilde {\varphi }_{\ell n_0}=\deg \tilde {g}_{\ell n_0} = \deg \tilde {g}_{n_0}^{\circ \ell } = o(d^{n_0 \ell })$
, as
$\ell \to \infty $
, since
$\deg \tilde {g}_{n_0} < d^{n_0}$
. Therefore,
$\deg \tilde {\varphi }_{\ell n}/d^{n\ell }$
is not eventually constant, as
$\ell \to \infty $
, which is a contradiction.
4.3 Potential good reduction
Here we consider the case in which
$\{ f_k\}$
totally converges to
$(g_n,\varphi _n)$
with
$\deg \tilde {\varphi }_n \neq o(d^n)$
and
$\{ [f_k]\} \subset {\operatorname {rat}_d}$
converges in
${\operatorname {rat}_d}$
. In order to state the hypothesis that are really used, we consider a slightly more general case:
Proposition 4.5. Let
$\{ f_k\}\subset {\operatorname {Rat}_d}$
be a sequence converging to
$g \in \partial {\operatorname {Rat}_d}$
. Assume that
$\{[f_k]\}$
converges in moduli space
${\operatorname {rat}_d}$
. Then g has a unique hole, say h. Moreover, if
$f_k^{\circ n} \to g_n$
, as
$k \to \infty $
and
$\tilde {g}_n$
is constant, for all n, then
$$ \begin{align*}\lim_{k\to\infty} \mu_{f_k}=\delta_h = \lim_{n\to\infty}\dfrac{1}{d^n}(g_n)^* \mu\end{align*} $$
for all probability measures
$\mu $
that are nonexceptional for
$(g_n)$
.
Proof. Let
$\{ L_k \} \subset {\operatorname {Rat}}_1$
be such that
$\psi _k:=L_k \circ f_k \circ L^{-1}_k$
converges (uniformly) to some
$ \psi \in {\operatorname {Rat}_d}$
, as
$k\to \infty $
. We may assume that
$\{L_k\}$
converges to a degenerate Möbius transformation with a hole at h and with constant reduction a. By contradiction we prove that h is the unique hole of g. Suppose that
$h' \neq h$
is a hole of g. Consider a small neighborhood U of
$h'$
. Pick a generic point
$z_0\in \overline {\mathbb {C}}$
such that
$z_0\neq \psi (a)$
. For sufficiently large k, from Lemma 2.6 (1), we deduce that
$z_0 \in L_k \circ f_k(U)$
; therefore,
$z_0 \in \psi _k \circ L_k(U)$
. The uniform convergence, in U, of
$\{L_k\}$
to the constant a implies that
$z_0 = \psi (a)$
, which contradicts the choice of
$z_0$
. Thus the first assertion holds.
We now prove the second assertion. Since the delta measure
$\mu _\psi $
is nonexceptional for
$L=\lim L_k$
and
$\mu _{\psi _k} \to \mu _\psi $
, we conclude that
$\mu _{f_k} = (L_k)^* \mu _{\psi _k} \to \delta _h$
, as
$k\to \infty $
, by Proposition 2.9. To establish
$\delta _h = (g_n)^* \mu /d^n$
, note that
$L_k\circ f_k^{\circ n}\circ L_k^{-1}$
converges to
$\psi ^{\circ n}\in \mathrm {Rat}_{d^n}$
. From the previous paragraph, we conclude that h is the unique hole of
$g_n$
, for all
$n \ge 1$
. Moreover,
$d_h(g_n)=d^n$
because
$\deg \tilde {g}_n=0$
. Hence
$(g_n)^* \mu /d^n = \delta _h$
, for all probability measures
$\mu $
that are nonexceptional for
$(g_n)$
.
4.4 Degeneration with large degree growth
The aim of this section is to apply the Structure Theorem 3.1 and its corollaries, to prove the following:
Proposition 4.6. Let
$\{f_k\}$
be a sequence in
${\operatorname {Rat}_d}$
totally converging to
$(g_n,\varphi _n)$
such that
$[f_k]$
converges to
$[g] \in \partial {\operatorname {rat}_d}$
. Assume that
$\deg \tilde {\varphi }_{n} \neq o(d^n)$
. Then there exists a probability measure
$\nu $
such that
$$ \begin{align*}\nu = \lim_{k\to\infty} \mu_{f_k} = \lim_{n\to\infty} \dfrac{1}{d^n} (g_n)^* \mu ,\end{align*} $$
for any probability measure
$\mu $
that is nonexceptional for
$(g_n)$
.
Proof. Consider an accumulation point
$\nu $
of
$\{\mu _{f_k}\}$
. After passing to a subsequence, we may assume
$\mu _{f_k} \to \nu $
. We may also assume that there exists a collection of scalings
$\{A_{n,k} : n,k \ge 1\}$
for
$\{f_k\}$
with convergent changes of coordinates. Since
$\deg \tilde {\varphi }_n \neq o(d^n)$
, any sufficiently large n is a fully ramified time. Thus, we may apply Corollary 3.15 (2) to conclude that
$(A_{n,k})_*\mu _{f_k} \to \delta _{a_n}$
, for all sufficiently large n. By Proposition 2.10, we have that
$$ \begin{align*}\nu=\frac{1}{d^n}(\varphi_n)^\ast\delta_{a_n},\end{align*} $$
and, therefore,
$$ \begin{align*} \nu&=\frac{1}{d^n}\left(\sum_{h\in\mathrm{Hole}(\varphi_n)}d_h (\varphi_n) \cdot \delta_h+(\tilde{\varphi}_n)^\ast\delta_{a_n}\right)\\ &=\frac{1}{d^n}\left(\sum_{h\in\mathrm{Hole}(\varphi_n)}d_h (\varphi_n) \cdot \delta_h+\sum_{\tilde{\varphi}_n(h)=a_n}\deg_h \tilde{\varphi}_n \cdot \delta_{h}\right). \end{align*} $$
By Lemma 4.4, we may assume that
$g_n$
has constant reduction and apply Lemma 2.5 to conclude that
$\operatorname {Hole}(g_n) = \operatorname {Hole}(\varphi _n) \cup \tilde {\varphi }^{-1}_n(a_n)$
. Then
$$ \begin{align*}\nu=\frac{1}{d^n}\sum_{h\in\mathrm{Hole}(g_n)} d_h(g_n) \cdot \delta_h = \dfrac{1}{d^n} g_n^\ast \mu,\end{align*} $$
for any probability measure
$\mu $
that is nonexceptional for
$g_n$
and, as a consequence,
$\mu _{f_k}$
converges to
$\nu $
.
4.5 Proof of Theorem A
Given any sequence
$\{f_k\} \subset {\operatorname {Rat}_d}$
converging to
$\mathbf {g} = (g_n) \in \widehat {{\operatorname {Rat}}}_d$
, we simultaneously show that there exists a probability measure
$\mu _{\mathbf {g}}$
such that
$$ \begin{align*}\dfrac{1}{d^n}(g_n)^* \mu \to \mu_{\mathbf{g}},\end{align*} $$
and that if along a subsequence of
$\{f_k \}$
we have convergence of the measures
$\mu _{f_k}$
to a measure
$\nu $
, then
$\nu = \mu _{\mathbf {g}}$
.
So given a subsequence such that the measures of maximal entropy converge, there exists a further subsequence
$\{f_{k_i}\}$
such that
$\{f_{k_i}\}$
totally converges to
$(g_n,\varphi _n)$
and
$[f_{k_i}] \to [g] \in \overline {\operatorname {rat}}_d$
. If
$\deg \tilde {\varphi }_n = o(d^n)$
or
$[g] \in {\operatorname {rat}_d}$
, by Propositions 4.3 and 4.5, there exists a probability measure
$\mu _{\mathbf {g}}$
such that
$$ \begin{align*}\dfrac{1}{d^n}(g_n)^* \mu \to \mu_{\mathbf{g}}=\nu.\end{align*} $$
If
$[g] \in \partial {\operatorname {rat}_d}$
and
$\deg \tilde {\varphi }_n \neq o(d^n)$
, then we apply Proposition 4.6 to obtain the same conclusion.
5 Continuous extension of barycentered measures
The moduli space
${\operatorname {rat}_d}$
is a complex orbifold of dimension
$2d- 2$
. Its compactification
$\overline {\operatorname {rat}}_d$
is the categorical quotient of semistable points in
$\overline {\operatorname {Rat}}_d$
by the
$\mathrm {SL}(2,\mathbb {C})$
conjugation action, see [Reference Silverman17]. According to [Reference DeMarco3, Section 3], the (semi)stable points can be characterized by the depths; more precisely,
-
• a point
$\varphi \in \overline {\operatorname {Rat}}_d$
is semistable if
$d_z(\varphi )\le (d+1)/2$
for any
$z\in \overline {\mathbb {C}}$
and
$\tilde \varphi (h)\not =h$
for any
$h\in \overline {\mathbb {C}}$
with
$d_h(\varphi )\ge d/2$
; -
• a point
$\varphi \in \overline {\operatorname {Rat}}_d$
is stable if
$d_z(\varphi )\le d/2$
for any
$z\in \overline {\mathbb {C}}$
and
$\tilde \varphi (h)\not =h$
for any
$h\in \overline {\mathbb {C}}$
with
$d_h(\varphi )\ge (d-1)/2$
.
In §5.1, we review basic facts from [Reference DeMarco3]. In particular, to each element in
${\operatorname {rat}_d}$
one may continuously assign a well-defined barycentered measure of maximal entropy, modulo push-forward by a rotation in
$\operatorname {SO}(3)$
. In §5.2, we establish Theorem B, which asserts continuous extension of this assignment to the resolution space
$\widehat {{\operatorname {rat}}}_d \subset \Pi _{n\ge 1} \overline {\operatorname {rat}}_{d^n}$
of the iterate map 
discussed in the introduction §1.
5.1 Barycentered measures
Recall that
$M^1(\overline {\mathbb {C}})$
is the space of probability measures endowed with the weak*-topology. Let
$M^1_o(\overline {\mathbb {C}})\subset M^1(\overline {\mathbb {C}})$
be the subset of probability measures with no atom of weight
$\ge 1/2$
. Then, under the identifications of
$\overline {\mathbb {C}}$
with
$S^2\subset \mathbb {R}^3$
by the stereographic projection and of hyperbolic space
${\mathbb {H}}^3$
with the unit ball in
$\mathbb {R}^3$
, the conformal barycenter
$C(\mu ) \in {\mathbb {H}}^3$
of
$\mu \in M^1_o(\overline {\mathbb {C}})$
is well-defined and continuous on
$M^1_o(\overline {\mathbb {C}})$
. More precisely, given
$\mu \in M^1_o(\overline {\mathbb {C}})$
, the barycenter
$C(\mu )$
is uniquely determined by the following two properties:
-
1.
$C(\mu )=0\in {\mathbb {H}}^3$
if and only if
$\int _{S^2}xd\mu (x)=0$
; and -
2.
$C(A_*\mu )=A(C(\mu ))$
for any
$A\in {\operatorname {Rat}}_1\cong \text {Isom}^+{\mathbb {H}}^3$
.
The conformal barycenter
$C(\mu )$
of any
$\mu \in M^1(\overline {\mathbb {C}})\setminus M^1_o(\overline {\mathbb {C}})$
is undefined. For more details, see [Reference DeMarco3, Section 8] and [Reference Douady and Earle5].
A measure
$\mu \in M^1(\overline {\mathbb {C}})$
is barycentered if
$C(\mu )$
is well-defined and equals
$0$
. Denote by
$M_{bc}^1(\overline {\mathbb {C}})\subset M^1(\overline {\mathbb {C}})$
the subset of barycentered measures, and let
$\overline {M}^1_{bc}(\overline {\mathbb {C}})$
be the closure of
$M_{bc}^1(\overline {\mathbb {C}})$
in
$M^1(\overline {\mathbb {C}})$
. Each element in
$\overline {M}^1_{bc}(\overline {\mathbb {C}})$
is either barycentered or of the form
$(\delta _a+\delta _{-1/\bar a})/2$
. for some
$a\in \overline {\mathbb {C}}$
, see [Reference DeMarco3, Lemma 8.3]. Then both
$M^1_{bc}(\overline {\mathbb {C}})$
and
$\overline {M}^1_{bc}(\overline {\mathbb {C}})$
are invariant under the push-forward action of the compact group of rotations
$\operatorname {SO}(3)\subset \operatorname{PSL}(2,\mathbb{C})$
. Consider the quotient topological spaces
$$ \begin{align*}M^1_{dm}(\overline{\mathbb{C}}):=M^1_{bc}(\overline{\mathbb{C}})/\operatorname{SO}(3)\ \ \text{and}\ \ \overline{M}^1_{dm}(\overline{\mathbb{C}}):=\overline{M}^1_{bc}(\overline{\mathbb{C}})/\operatorname{SO}(3).\end{align*} $$
The space
$M^1_{dm}(\overline {\mathbb {C}})$
is locally compact, Hausdorff and its one-point compactification is
$\overline {M}^1_{dm}(\overline {\mathbb {C}})$
, see [Reference DeMarco3, Theorem 8.1]. Denote by
$[\infty ]$
the unique point in
$\overline {M}^1_{dm}(\overline {\mathbb {C}})\setminus M^1_{dm}(\overline {\mathbb {C}})$
.
For each
$\mu \in M^1_o(\overline {\mathbb {C}})$
, there exists
$A\in {\operatorname {Rat}}_1$
such that
$A_*\mu \in M^1_{bc}(\overline {\mathbb {C}})$
. Writing
$[\mu ]:=[A_*\mu ]\in M^1_{dm}(\overline {\mathbb {C}})$
, one obtains a continuous map
$$ \begin{align*}\begin{array}{cccc} &M^1_o(\overline{\mathbb{C}})&\rightarrow& M^1_{dm}(\overline{\mathbb{C}})\\ & \mu & \mapsto & [\mu ], \end{array} \end{align*} $$
which induces a continuous map
${\operatorname {rat}}_d\to M^1_{dm}(\overline {\mathbb {C}})$
, sending
$[f]$
to
$[\mu _f]$
. We remark here that the above continuous map can not extend continuously to
$M^1(\overline {\mathbb {C}})$
; for instance, considering
$\mu _n\in M^1(\overline {\mathbb {C}})$
such that
$\mu _n(\{0\})=\mu _n(\{1/n\})=\mu _n(\{1\})=1/3$
, we have that
$[\lim _{n\to \infty }\mu _n]=[\infty ]\not =\lim _{n\to \infty }[\mu _n]$
.
5.2 Proof of Theorem B
Consider a sequence
$\{[f_k]\}\subset {\operatorname {Rat}_d}$
converging to
$([g_n])$
in
$\widehat {{\operatorname {rat}}}_d$
. Let
$\mu \in M^1(\overline {\mathbb {C}})$
be a nonexceptional measure for
$(g_n)$
. For notational simplicity,
$$ \begin{align*} \mu_k &:= \mu_{f_k}\\ \nu_n &:= \dfrac{(g_n)^*\mu}{d^n}. \end{align*} $$
Given accumulation points
$[\mu _\infty ]$
of
$\{[\mu _k]\}$
and
$[\nu ]$
of
$\{[\nu _n]\}$
, after some work, we will show that
${[\nu ]=[\mu _\infty ]}$
, that is, Theorem B holds. By continuity of the measure of maximal entropy for nondegenerate rational maps [Reference Mañé13], if
$[g_1]\in {\operatorname {rat}}_d$
, equivalently,
$[f_k]$
converges in
${\operatorname {rat}}_d$
, then
$[\nu ]=[\mu _\infty ]$
. Thus we work under the assumption that
$$ \begin{align*}[g_1]\in\partial{\operatorname{rat}}_d.\end{align*} $$
Passing to a subsequence in k, we assume that
$[\mu _k]$
converges to
$[\mu _\infty ]$
. Passing to a subsequence
$n_i$
, we assume that
$[\nu _{n_i}]$
converges to
$[\nu ]$
. It is sufficient to establish that
$[\mu _\infty ]$
and
$[\nu ]$
coincide when
$[\mu _\infty ] \neq [\infty ]$
or
$[\nu ] \neq [\infty ]$
.
Lemma 5.1. Assume that
$[\mu _\infty ]\not =[\infty ]$
. Then
$$ \begin{align*}[\nu]=\lim_{n\to\infty}[\nu_n]=[\mu_\infty].\end{align*} $$
Proof. After changing the representative of
$[f_k]$
, we may assume that
$\mu _{f_k}$
is barycentered, for all k. Passing to a subsequence,
$\{f_k\}$
converges in
$\widehat {{\operatorname {Rat}}}_d$
, say to
$\mathbf {h}=(h_n)$
. By Theorem A, we have that
$\mu _{f_k}$
converges to some measure
$\mu _{\mathbf {h}}$
. Since
$[\mu _\infty ] \neq [\infty ]$
, we have
$[\mu _{\mathbf {h}}]=[\mu _\infty ] \neq [\infty ]$
. Therefore,
$\mu _{\mathbf {h}}(z) < 1/2$
for all
$z \in \overline {\mathbb {C}}$
. Hence, for n sufficiently large,
$h_n$
is GIT-stable. Therefore,
$[f_k^{\circ n}]$
converges to
$[h_n]$
and
$[h_n]=[g_n]$
. In particular,
$[(h_n)^* \mu /d^n]= [\nu _n]$
provided that
$\mu $
is not exceptional for
$h_n$
, for all n. The sequence
$(h_n)$
is uniquely determined up to
$\operatorname {SO}(3)$
conjugacy (the same for all n). For any given n, there are at most countably many elements of
$\operatorname {SO}(3)$
so that
$\mu $
is exceptional for the conjugacy of
$h_n$
by that element. Hence we may find a conjugate so that
$\mu $
is not exceptional for all
$h_n$
. Thus the conclusion holds.
Now we assume that
$[\nu ] \neq [\infty ]$
. Passing to a further subsequences of
$\{f_k\}$
and
$\{g_{n_i}\}$
the following hold:
-
1. There exists
$\varepsilon>0$
such that, for all
$n_i$
and all
$z\in \overline {\mathbb {C}}$
,
$$ \begin{align*}d_z(g_{n_i}) < \left(\dfrac{1}{2} - 2 \varepsilon\right) \cdot d^{n_i}.\end{align*} $$
-
2. For each
$n_i$
, there exists
$B_{n_i,k} \in {\operatorname {Rat}}_1$
such that, as
$k \to \infty $
,
$$ \begin{align*}B_{n_i,k} \circ f_k^{\circ n_i} \circ B_{n_i,k}^{-1} \to g_{n_i}.\end{align*} $$
-
3. Let
$$ \begin{align*}f_{n_i,k}:=B_{n_i,k} \circ f_k \circ B_{n_i,k}^{-1}.\end{align*} $$
Then, for all
$n_i$
and m there exists
$g_{n_i,m} \in \overline {\operatorname {Rat}}_{d^{m}}$
such that
$$ \begin{align*}f_{n_i,k}^{\circ m} \to g_{n_i,m}.\end{align*} $$
-
4. For each
$n_i$
and each
$m\ge 1$
, there exist a collection of scalings
$\{A_{n_i,m,k}\}$
and maps
$\psi _{n_i,m} \in {\operatorname {Rat}}^*_{d^{m}}$
such that, as
$k\to \infty $
,
$$ \begin{align*}A_{n_i,m,k} \circ f_{n_i,k}^{\circ m} \to \psi_{n_i,m}.\end{align*} $$
-
5. There exists
$c \in [0,1]$
such that, as
$n_i \to \infty $
,
$$ \begin{align*}\dfrac{\deg \tilde\psi_{n_i,n_i}}{d^{n_i}} \to c \in [0,1].\end{align*} $$
Note that
$g_{n_i}$
is GIT-stable for each
$n_i$
by statement (1). Thus we may choose
$B_{n_i, k}$
as in statement (2). Statements (3) and (4) say that
$\{f_{n_i,k}\}$
totally converges to
$(g_{n_i,m},\psi _{n_i,m})$
with associated scalings
$\{A_{n_i,m,k}\}$
.
Observe that
$g_{n_i,n_i} = g_{n_i}$
. If
$\tilde {g}_{n_i,m}$
is nonconstant, then in (4) we choose the scaling
$A_{n_i,m,k}$
to be the identity and, therefore,
$\psi _{n_i,m}={g}_{n_i,m}$
.
For brevity, write
$\psi _{n_i}:=\psi _{n_i,n_i}$
. We establish the following two lemmas according to the growth of
$\deg \tilde \psi _{n_i}$
; more precisely, according to whether
$c>0$
or
$c =0$
.
Lemma 5.2. Suppose that
$c>0$
, that is,
$\deg \tilde \psi _{n_i}\not =o(d^{n_i})$
. Given a sufficiently large
$n_i$
, for any
$\ell \ge n_i$
and any
$z\in \overline {\mathbb {C}}$
,
$$ \begin{align*}\frac{d_z(g_{n_i,\ell})}{d^\ell}=\frac{d_z(g_{n_i})}{d^{n_i}}.\end{align*} $$
Moreover,
$$ \begin{align*}[\nu] = [\nu_{n_i}] = [\mu_\infty].\end{align*} $$
Proof. Since
$c>0$
, for sufficiently large
$n_i$
, the number of times
$m<n_i$
such that
$\deg \tilde \psi _{n_i,m+1} < d \cdot \deg \tilde \psi _{n_i,m}$
is bounded independently of
$n_i$
. Thus, we can pick N sufficiently large so that for any
$n_i\ge N$
, there is
$1\le m_i\le n_i-3$
satisfying that
$A_{n_i,m_i+3,k}\circ f_{n_i,k}^{\circ 3}\circ A_{n_i,m_i,k}^{-1}$
converges in
${\operatorname {Rat}}_{d^3}$
. Since we are working under the assumption that
$[g_1]\in \partial {\operatorname {rat}}_d$
, by Corollary 3.16, we conclude that for any
$\ell \ge n_i$
and any
$z\in \overline {\mathbb {C}}$
, the reduction
$\tilde g_{n_i,\ell }$
is a constant independent of
$\ell $
and
$$ \begin{align*}\frac{d_z(g_{n_i,\ell})}{d^\ell}=\frac{d_z(g_{n_i})}{d^{n_i}}.\end{align*} $$
Hence
$$ \begin{align*}\frac{(g_{n_i,\ell})^*\mu}{d^\ell}=\frac{(g_{n_i})^*\mu}{d^{n_i}}=\nu_{n_i}.\end{align*} $$
Since by Theorem A
$$ \begin{align*}\lim_{\ell\to\infty}\frac{(g_{n_i,\ell})^*\mu}{d^\ell}=\lim\limits_{k\to\infty}\mu_{f_{n_i,k}},\end{align*} $$
we have that
$$ \begin{align*}\nu_{n_i}=\lim_{k\to\infty}\mu_{f_{n_i,k}}.\end{align*} $$
Observing that
$\nu _{n_i}\in M^1_o(\overline {\mathbb {C}})$
since
$g_{n_i}$
is stable, by the continuity of the map
$M^1_o(\overline {\mathbb {C}})\rightarrow M^1_{dm}(\overline {\mathbb {C}})$
, we obtain that
$$ \begin{align*}[\nu_{n_i}]=\lim\limits_{k\to\infty}[\mu_{f_{n_i,k}}]=\lim_{n\to\infty}[\mu_{f_k}]=[\mu_\infty].\end{align*} $$
Thus the conclusion follows.
Lemma 5.3. Suppose that
$c=0$
, that is,
$\deg \tilde \psi _{n_i}=o(d^{n_i})$
. Then
$$ \begin{align*}[\nu]=[\mu_\infty].\end{align*} $$
Proof. By Lemma 5.1, we may assume that
$[\mu _\infty ]=[\infty ]$
. Passing to a subsequence in k we can assume that
$B_{n_i,k}\circ B_{n_j,k}^{-1}$
converges to some
$B_{n_j,n_i}\in \overline {\operatorname {Rat}}_1$
for any
$n_i\not =n_j$
. Then passing to a further subsequence in
$n_i$
if necessary, we can assume that
$B_{n_j,n_i}\in {\operatorname {Rat}}_1$
for all
$n_i,n_j$
or
$B_{n_j,n_i}\in \partial {\operatorname {Rat}}_1$
for all
$n_i,n_j$
.
Case 1:
$B_{n_j,n_i}\in {\operatorname {Rat}}_1$
for all
$n_i\not =n_j$
. Since
$\{f_{n_1,k}\}$
totally converges to
$(g_{n_1,m}, \psi _{n_1,m})$
, passing to a subsequence in k, we may choose
$\{A_{n_1,n_i,k}\}$
so that
$\psi _{n_i} = \psi _{n_1,n_i} \circ C_{n_i}$
for some
$C_{n_i} \in {\operatorname {Rat}}_1$
.
By Lemma 4.2, let
$$ \begin{align*}\nu':=\lim_{i\to\infty}{\eta_{\psi_{n_1,n_i}}} \in M^1(\overline{\mathbb{C}}),\end{align*} $$
where
$\eta _{\psi _{n_1,n_i}}$
is the corresponding depth measure from Definition 2.7. We claim that
$\nu '\in M^1_o(\overline {\mathbb {C}})$
and hence
$[\nu ']\not =[\infty ]$
. Indeed, given
$z \in \overline {\mathbb {C}}$
, let
$w_i = C_{n_i}^{-1}(z)$
, then
$$ \begin{align*}\nu'(\{z\}) =\lim \dfrac{d_z (\psi_{n_1,n_i})}{d^{n_i}} = \lim \dfrac{d_{w_i} (\psi_{n_i})}{d^{n_i}} \le \lim \dfrac{d_{w_i}(g_{n_i})}{d^{n_i}} \le \dfrac12 - 2 \varepsilon.\end{align*} $$
The first equality follows from the proof of Lemma 4.2 since
$\nu '(\{z\})$
is the limit of the nondecreasing sequence
$\{\eta _{\psi _{n_1,n_i}}(\{z\})=d^{-{n_i}} \cdot d_z (\psi _{n_1,n_i})\}$
. The second equality follows from
$\psi _{n_i} = \psi _{n_1,n_i} \circ C_{n_i}$
. The first inequality is Lemma 2.5 applied to
$z = w_i$
and
$\{f_{n_i,k}^{\circ n_i}\}$
. The second inequality is statement (1).
Moreover, since
$\deg \tilde \psi _{n_i}=o(d^{n_i})$
, we have that
$\deg \tilde {\psi }_{n_1, n_i} = o (d^{n_i})$
, as
$i\to \infty $
, and hence
$\deg \tilde {\psi }_{n_1,n} = o(d^n)$
, as
$n\to \infty $
. Thus, applying Proposition 4.3 to
$\{f_{n_1,k}\}$
, we conclude that
$$ \begin{align*}\lim_{k\to\infty} \mu_{f_{n_1,k}} = \lim_{i\to\infty}\frac{(g_{n_1,n_i})^* \mu'}{d^{n_i} } =\nu',\end{align*} $$
whenever
$\mu '\in M^1(\overline {\mathbb {C}})$
is any nonexceptional measure for
$(g_{n_1,n})$
. Continuity of the map
$M^1_o(\overline {\mathbb {C}})\rightarrow M^1_{dm}(\overline {\mathbb {C}})$
yields
$$ \begin{align*}[\mu_\infty]= \lim_{n\to\infty}[\mu_{f_k}]=\lim_{k\to\infty}[\mu_{f_{n_1,k}}]=[\nu'].\end{align*} $$
Hence
$[\mu _\infty ]\neq [\infty ]$
, which is a contradiction.
Case 2:
$B_{n_j,n_i}\in \partial {\operatorname {Rat}}_1$
for any
$n_i\not =n_j$
. We show that this case cannot occur under our assumptions; more precisely, by counting preimages, we will obtain a contradiction with statement (1).
Since
$\deg \tilde \psi _{n_i}=o(d^{n_i})$
, we consider
$n_{i_0}$
such that for
$n_i \ge n_{i_0}$
,
$$ \begin{align*}\deg\tilde\psi_{n_i}<\varepsilon\cdot d^{n_i}.\end{align*} $$
Let
$\ell :=n_{i_0}$
and
$s:=n_{i_0+1}$
. For brevity write
$$ \begin{align*} \gamma_k := B_{\ell,k} \circ B_{s,k}^{-1} \end{align*} $$
with
$\gamma _k \to B_{(s,\ell )}$
and
$\gamma _k^{-1} \to B_{(\ell ,s)}$
in
$\overline {\operatorname {Rat}}_1$
. Let
$b_{\ell }:=\tilde B_{s,\ell }$
and
$b_{s}:=\tilde B_{\ell ,s}$
.
Intuitively, our argument is as follows. We restrict our attention to the collection of scalings
$\{B_{\ell ,k}, B_{s,k} \}$
with associated space formed by two spheres
$\overline {\mathbb {C}}_\ell $
and
$\overline {\mathbb {C}}_s$
. We consider the set of preimages S of a generic point in
$\overline {\mathbb {C}}_s$
, under an appropriate iterate
$f_{s,k}^{\circ s}$
. Then statement (1) implies that less than half of the elements of S map, under
$f_{s,k}^{\circ s}\circ \gamma _k^{-1}$
, close to
$b_\ell $
, the hole of
$B_{(\ell ,s)}$
. So more than half of the elements are bounded away from
$b_\ell $
. Applying the change of coordinates
$\gamma _k^{-1}$
, one concludes that more than half of the elements of S are close to
$b_s$
, which is a contradiction with statement (1).
More precisely, consider
$$ \begin{align*} \alpha_k &:= B_{s,k} \circ f_k^{\circ s-\ell} \circ B_{\ell,k}^{-1},\\ \beta_k &:= B_{\ell,k} \circ f_k^{\circ \ell} \circ B_{\ell,k}^{-1}. \end{align*} $$
Note that
$$ \begin{align*}f_{s,k}^{\circ s} =\alpha_k \circ \beta_k \circ \gamma_k \to g_s.\end{align*} $$
Let V be a small neighborhood of
$b_{s}$
. For
$w\in \overline {\mathbb {C}}$
, we now estimate
$\#(f_{s,k}^{-s}(w)\cap V)$
. First observe that
$\#(\alpha _k)^{-1}(w)=d^{s-\ell }$
. Given a small neighborhood D of
$b_{\ell }$
, since
$d_{b_{\ell }}(g_\ell ) < (1/2-2\varepsilon )\cdot d^\ell $
from the statement (1) in the assumptions on
$\{f_k\}$
and
$\{g_{n_i}\}$
, by Lemma 2.6 (2), we have that
$$ \begin{align*}\#((\alpha_k\circ\beta_k)^{-1}(w)\cap D)< (1/2-2\varepsilon)\cdot d^s+\varepsilon\cdot d^s=(1/2-\varepsilon)\cdot d^s.\end{align*} $$
Therefore,
$$ \begin{align*}\#((\alpha_k\circ\beta_k)^{-1}(w)\cap(\overline{\mathbb{C}}\setminus D))\ge(1/2+\epsilon)\cdot d^s.\end{align*} $$
In
$ \overline {\mathbb {C}} \setminus D$
, we have uniform convergence of
$\gamma ^{-1}_k$
to the constant
$b_s$
. We conclude that
$$ \begin{align*}\#(f_{s,k}^{\circ -s}(w)\cap V)\ge(1/2+\varepsilon)\cdot d^s.\end{align*} $$
By Lemma 2.6 (2), we have that
$d_{b_{s}}(g_s)\ge d^s/2$
, which contradicts statement (1).
This completes the proof of Theorem B.
Acknowledgments
Part of this work was done when the authors were in the Institute for Mathematical Sciences (IMS) at Stony Brook University. The authors are grateful to the IMS for its hospitality. We thank the anonymous referees for their careful reading and constructive comments, which helped improve the paper.
Competing interests
The authors have no competing interests to declare.
Financial support
The first author acknowledges the support of ANID/FONDECYT Regular 1240508.
A Trees of bouquets of spheres
A.1 Hausdorff limits of spheres
Let J be a finite set of indices. Consider a collection of scalings
$$ \begin{align*}\mathbf{A}:=\{ A_{n,k}: n \in J,k \ge 1 \} \subset {\operatorname{Rat}}_1,\end{align*} $$
indexed by J. Assume that for all
$n,n' \in J$
with
$n \neq n'$
, there exists
$A_{(n,n')} \in \partial {\operatorname {Rat}}_1$
such that
$$ \begin{align*}A_{n',k} \circ A_{n,k}^{-1} \to A_{(n,n')}\ \ \text{as}\ \ k \to \infty.\end{align*} $$
We say that
$\mathbf {A}$
is a collection of independent scalings indexed by J.
For
$ n \neq n'$
, let
$$ \begin{align*}a_{n,n'} := \widetilde{A}_{(n,n')}.\end{align*} $$
Note that
$$ \begin{align*}a_{n',n} = \operatorname{Hole}(A_{(n,n')}).\end{align*} $$
For each
$n \in J$
, consider a copy of
$\overline {\mathbb {C}}$
inside
$\overline {\mathbb {C}}^{J}$
:
$$ \begin{align*}\overline{\mathbb{C}}_n:= \left\{ (w_j) \in \overline{\mathbb{C}}^{J} : w_j = a_{n,j} \text{ for all } j \neq n \right\}.\end{align*} $$
Denote by
$\rho _n:\overline {\mathbb {C}}^{J} \to \overline {\mathbb {C}}$
the projection to the n-th coordinate. This projection provides a standard coordinate for
$\overline {\mathbb {C}}_n$
. That is,
$\rho _n : \overline {\mathbb {C}}_n \to \overline {\mathbb {C}}$
is a conformal isomorphism.
Consider the union of Riemann spheres
We will show that 
is a tree of bouquets of spheres that arises as the Hausdorff limit of Riemann spheres. We say that 
is the space associated to
$\mathbf {A}$
.
For all
$n \neq n' \in J$
, note that
$\rho _n (\overline {\mathbb {C}}_{n'}) = \{a_{n',n}\}$
. Moreover, the intersection
$\overline {\mathbb {C}}_{n'} \cap \overline {\mathbb {C}}_{n}$
is not empty if and only if
$$ \begin{align*}a_{n',n"} = a_{n,n"}\end{align*} $$
for all
$ J \ni n" \neq n,n'$
. In this case, the intersection point is
$a_{n',n} \in \overline {\mathbb {C}}_{n}$
and
$a_{n,n'} \in \overline {\mathbb {C}}_{n'}$
in the respective standard coordinates.
An alternative description of 
can be given as the quotient of the disjoint union of abstract Riemann spheres
$\sqcup _{n \in J}\overline {\mathbb {C}}_{n}$
by an equivalence relation
$\sim $
. In fact, let
$\sim $
be the equivalence relation such that
$a_{n',n} \in \overline {\mathbb {C}}_{n}$
is identified with
$a_{n,n'} \in \overline {\mathbb {C}}_{n'}$
if and only if
$$ \begin{align*}a_{n,n"} = a_{n',n"}\end{align*} $$
for all
$ n" \neq n, n'$
. All other
$\sim $
-classes are trivial. Then
Associated to
$\mathbf {A}$
we also have a sequence of embedding of
$\overline {\mathbb {C}}$
in
$\overline {\mathbb {C}}^{J}$
:
Proposition A.1. Consider a collection
$\mathbf {A}$
of independent scalings indexed by J. Let 
be the space associated to
$\mathbf {A}$
. If 
is the associated sequence of embeddings, then
Moreover, 
is simply connected.
Proof. Consider points 
such that
$w_{\ell } \neq a_{j,\ell }$
and
$w_j=a_{\ell ,j}$
for all
$j \neq \ell $
. Such elements are dense in
$\overline {\mathbb {C}}_\ell $
. Denoting by
$z_k := A^{-1}_{\ell ,k}(w_\ell )$
, we have that 
. So 
is contained in the Hausdorff limit of 
.
We prove that the Hausdorff limit of 
is contained in 
. Suppose on the contrary that
$\{z_k\}$
is a sequence such that 
converges to some 
. Then for all
$j\in J$
, we may choose
$J \ni s(j) \neq j$
such that
$u_{s(j)} \neq a_{j,s(j)}$
. Therefore,
$u_j = a_{s(j),j}$
since the change of coordinates
$A_{j,k} \circ A_{s(j),k}^{-1}$
converges uniformly in compact subsets of
$\overline {\mathbb {C}}\setminus \{a_{j,s(j)}\}$
to
$a_{s(j),j}$
. Now consider
$j_0\in J$
and let
$j_i =s^{\circ i}(j_0)$
for
$i\ge 1$
. We may assume that
$j_p=j_0$
for a minimal
$p \ge 2$
. Since
$a_{j_0,j_1} \neq u_{j_1} = a_{j_2,j_1}$
, it follows that
$a_{j_2, j_0} = a_{j_1,j_0}$
. Recursively, we obtain that
$a_{j_{p-1},j_0}=a_{j_1,j_0}=u_{j_0}$
. However,
$u_{j_0}=u_{s(j_{p-1})} \neq a_{j_{p-1},j_0}$
. It is a contradiction.
Any Hausdorff limit of connected sets is connected, so 
is connected. We now show that 
is simply connected. By contradiction, suppose that for some
$q \ge 2$
, there exists a cycle of spheres of length q. That is, there exist a chain of pairwise distinct spheres
$\overline {\mathbb {C}}_{\ell _0},\dots ,\overline {\mathbb {C}}_{\ell _{q-1}}$
that intersect at pairwise distinct points
$x_0, \cdots , x_{q-1}$
; that is,
$\{ x_i \}= \overline {\mathbb {C}}_{\ell _i} \cap \overline {\mathbb {C}}_{\ell _{i+1}} $
for all
$0 \le i <q$
, subscripts mod q. Then
$x_{j+1}-x_j$
is a vector in
$\mathbb {C}^J$
with unique nonvanishing component in its
$\ell _j$
-coordinate for all j. But
$\sum x_{j+1}-x_j =x_0-x_0=0$
, which is a contradiction.
A.2 Maps between trees of bouquets of spheres
Let
$J, J'$
be finite sets. Consider collections
$$ \begin{align*}\mathbf{A}:=\{A_{n,k} : n \in J, k \ge 1\} \subset {\operatorname{Rat}}_1,\end{align*} $$
$$ \begin{align*}\mathbf{B}:=\{B_{n,k} : n \in J', k \ge 1\} \subset {\operatorname{Rat}}_1.\end{align*} $$
Assume that
$\mathbf {A}$
and
$\mathbf {B}$
are collections of independent scalings.
Consider a sequence
$\{f_k\} \subset {\operatorname {Rat}_d}$
. We say that
$\{f_k\}$
maps
$\mathbf {A}$
into
$\mathbf {B}$
by
$\tau : J \to J'$
if for all
$n \in J$
,
$$ \begin{align*}B_{\tau(n),k} \circ f_k \circ A^{-1}_{n,k}\end{align*} $$
converges to some element
$\varphi _{n,\tau (n)}\in {\operatorname {Rat}}^*_d$
. When
$\varphi _{n,\tau (n)}\in {\operatorname {Rat}}_d$
, for all
$n \in J$
, we say that
$\{f_k\}$
is fully ramified at
$\mathbf {A}$
.
In coordinates, given by the associated embeddings 
and 
of
$\overline {\mathbb {C}}$
into
$\overline {\mathbb {C}}^J$
and
$\overline {\mathbb {C}}^{J'}$
, the maps
$f_k$
become
In the limit,
$F_k$
is, in a certain sense, semiconjugate to the action of
${\varphi }_{n,\tau (n)}$
on
$\overline {\mathbb {C}}_n$
. More precisely, provided that z is not a hole of
${\varphi }_{n,\tau (n)}$
,
$$ \begin{align*}\rho_{\tau (n)} \circ F_k (z) \to \tilde{\varphi}_{n,\tau(n)} (\rho_n(z)).\end{align*} $$
In general, we have the following:
Proposition A.2. Consider a sequence
$\{f_k\} \subset {\operatorname {Rat}_d}$
and collections of independent scalings
$\mathbf {A}$
and
$\mathbf {B}$
indexed by J and
$J'$
with associated spaces 
and 
, respectively, such that
$\{f_k\}$
maps
$\mathbf {A}$
to
$\mathbf {B}$
by
$\tau :J \to J'$
. Let 
and 
be the associated embeddings of
$\overline {\mathbb {C}}$
into
$\overline {\mathbb {C}}^{J}$
and
$\overline {\mathbb {C}}^{J'}$
, respectively. Let
Consider 
and 
. Then, for every small neighborhood
$U_0 \subset \overline {\mathbb {C}}^{J} $
of
$z_0$
, there exists a neighborhood
$W_0 \subset \overline {\mathbb {C}}^{J'}$
of
$w_0$
such that for all 
,
$$ \begin{align*}\#F_k^{-1} (w) \cap U_0 = \begin{cases} d_{z_0}(\varphi_{j,\tau(j)}), & \text{ if } \tilde{\varphi}_{j,\tau(j)}(z_0) \neq w_0,\\ d_{z_0}(\varphi_{j,\tau(j)}) + \deg_{z_0}\tilde{\varphi}_{j,\tau(j)}, & \text{ if } \tilde{\varphi}_{j,\tau(j)}(z_0) = w_0. \end{cases} \end{align*} $$
Proof. For simplicity, let 
and 
. Note that 
for any
$U \subset \overline {\mathbb {C}}$
. A similar statement holds of
$S_k'$
.
Given a small neighborhood
$U_0$
of
$z_0$
, by compactness, it contains
$\rho _j^{-1}(U_0')$
for some neighborhood
$U_0'$
of
$\rho _j(z_0)$
in
$\overline {\mathbb {C}}$
. Let
$h_k := A_{\tau (j),k} \circ f_k \circ A_{j,k}^{-1}$
. Then
$h_k \to \varphi _{j,\tau (j)}$
. By Lemma 2.6 (1), there exists a neighborhood
$W_0'$
of
$w^{\prime }_0:=\rho _j(w_0)$
in
$\overline {\mathbb {C}}$
such that
$$ \begin{align*}\# h_k^{-1} (w') \cap U_0' = d_{z_0} (\varphi_{j,\tau(j)}) + \deg_{z_0}\varphi_{j,\tau(j)},\end{align*} $$
for all
$w' \in W_0'$
. Let
$W_0: = \rho ^{-1}_{\tau (j)} (W_0')$
. Given
$w \in S_k \cap W_0$
, consider
$w' \in W_0'$
such that 
. Now,
$z' \in U^{\prime }_0$
is such that
$h_k(z') = w'$
if and only if 
is such that
$F_k (z) =w$
. That is, 
is a bijection between
$F_k$
-preimages of w in
$U_0$
and
$h_k$
-preimages of
$w'$
in
$U_0'$
.
A.3 Fully ramified maps
Let us now consider the special case in which the limiting degree at each sphere is maximal.
Proposition A.3. Consider a sequence
$\{f_k\} \subset {\operatorname {Rat}_d}$
that maps
$\mathbf {A}$
to
$\mathbf {B}$
by a bijection
$\tau : J \to J'$
, where
$\mathbf {A}$
and
$\mathbf {B}$
are independent collections of scalings with associated spaces 
and 
. If
$\{f_k\}$
is fully ramified at
$\mathbf {A}$
, then there is a well defined and continuous map 
given by:
for
$z \in \overline {\mathbb {C}}_n$
and
$n \in J$
. Moreover, there exist a finite set 
and a (multiplicity) function 
such that the following hold:
-
1.
-
2. The multiplicity of a critical point x of
$\varphi _{n,\tau (n)}$
is
We say that 
is a critical point of multiplicity
$m(\omega )$
of 
.
Proof. Recall that to obtain 
, we identify
$a_{n,n'}$
with
$a_{n',n}$
if and only if
$$ \begin{align*}a_{n',n"} = a_{n,n"}\end{align*} $$
for all
$ J \ni n" \neq n,n'$
. Similarly, we identify points of different spheres in 
. Since
$\tau : J \to J'$
is a bijection, for the continuity of 
, it suffices to check that
$\varphi _{n,\tau (n)}(a_{n',n}) = a_{\tau (n'),\tau (n)}$
for all
$n \neq n' \in J$
. Choose
$z \in \overline {\mathbb {C}}_{n'}$
such that
$z \neq a_{n,n'}$
and
$\varphi _{n',\tau (n')} (z) \neq a_{\tau (n),\tau (n')}$
. Then, uniformly in a neighborhood of z, the change of coordinates
$A_{n,k} \circ A^{-1}_{n',k}$
converges to
$a_{n',n}$
. Thus
$$ \begin{align*}(A_{\tau(n),k} \circ f_k \circ A_{n,k}^{-1} ) \circ (A_{n,k} \circ A^{-1}_{n',k}) (z) \to \varphi_{n,\tau(n)}(a_{n',n}).\end{align*} $$
Reorganizing the above expression yields
$$ \begin{align*}(A_{\tau(n),k} \circ A^{-1}_{\tau(n'),k} )\circ (A_{\tau(n'),k} \circ f_k \circ A_{n',k}^{-1} ) (z) \to a_{\tau(n'),\tau(n)}.\end{align*} $$
Hence
$\varphi _{n,\tau (n)}(a_{n',n}) = a_{\tau (n'),\tau (n)}$
, as claimed.
Passing to a subsequence, we may label the critical points of
$f_k$
by
$c_k(1), \dots , c_k(2d-2)$
with repetitions according to multiplicity. We may also assume that 
converges to 
, as
$k\to \infty $
, for all i. We say that 
is a critical point of 
of multiplicity m if there exist exactly m indices i such that
$c(i) =\omega $
. Let 
be the set formed by the critical points. Note that, a priori, 
depends on the choice of the subsequence. It can be shown that it is independent, but we will not use nor prove this fact.
Given
$x \in \overline {\mathbb {C}}$
, since
$A_{n+1,k} \circ f_k \circ A_{n,k}^{-1}$
converges locally uniformly to
$\tilde \varphi _{n,n+1}$
away from the holes of
$\varphi _{n,n+1}$
, the number of critical points of
$A_{n+1,k} \circ f_k \circ A_{n,k}^{-1}$
in a neighborhood W of x, counted with multiplicity, is
$m: = \deg _x\tilde \varphi _{n, n+1} -1$
. That is, the number of critical point of
$f_k$
in
$A_{n,k}^{-1}(W)$
is m. Therefore, 
contains m critical points of 
for all k sufficiently large. That is,
$\rho _n^{-1}(x)$
contains m critical points 
, counted with multiplicities.
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