2 Background on the dynamics of exponential maps

In the following, let $f\colon \mathbb {C}\to \mathbb {C}$ denote an entire function. The singular set $S(f)$ of f is defined to be the closure of the set of critical and asymptotic values of f. It is the smallest subset of $\mathbb {C}$ such that the restriction of f to $\mathbb {C}\setminus f^{-1}(S(f))$ is a covering map (see e.g. [Reference Goldberg and KeenGK, Lemma 1.1]). The post-singular set $P(f)$ of f is given by

$$ \begin{align*} P(f):=\overline{\bigcup_{n\geq 0}f^{\circ n}(S(f))}. \end{align*} $$

We call the function f post-singularly finite if $\vert P(f)\vert <\infty $ .

Entire functions have an important forward and backward invariant set, the set of escaping points; it is often more important than Fatou and Julia sets because it is never empty and never all of $\mathbb {C}$ (see [Reference EremenkoE1]), so it provides a non-trivial dynamically invariant decomposition of $\mathbb {C}$ . Here is the formal definition.

Definition 2.1. (Escaping set)

The escaping set of f is given by

$$ \begin{align*} I(f):=\{z\in\mathbb{C}\colon f^{\circ n}(z)\to\infty\}. \end{align*} $$

In [Reference Rottenfusser, Rückert, Rempe and SchleicherRRRS], it was shown that for a large class of transcendental entire functions (finite compositions of functions of bounded type and finite order), the escaping set is organized in the form of disjoint dynamic rays, which are certain arcs consisting of escaping points that terminate at $\infty $ .

In this paper, we focus on the case of post-singularly finite (psf) exponential maps: these are (up to affine conjugation) maps of the form $E_{\lambda }\colon \mathbb {C}\to \mathbb {C}$ , $E_{\lambda }(z)=\lambda \exp (z)$ , where the parameter $\lambda \in \mathbb {C}\setminus \{0\}$ is chosen such that the orbit of the only singular value $0$ is finite and hence strictly pre-periodic (the orbit of $0$ cannot be periodic because $0$ is an omitted value). For these functions, the set of escaping points has been described and classified in [Reference Schleicher and ZimmerSZ1], using the combinatorial concept of external addresses that distinguish dynamic rays.

The shift space $\mathcal {S}$ can be totally ordered using lexicographic order (for $\underline {s},\underline {t}\in \mathcal {S}$ , we write $\underline {s}< \underline {t}$ if and only if $\underline {s}=s_1 s_2 s_3\ldots $ and $\underline {t}=t_1 t_2 t_3\ldots $ have a k so that $s_1=t_1$ , … , $s_k=t_k$ and $s_{k+1}<t_{k+1}$ ). The lexicographic order defines the order topology on $\mathcal {S}$ . Any total order induces a cyclic order on the same set: for distinct $\underline {s},\underline {t},\underline {u}\in \mathcal {S}$ , we write

$$ \begin{align*} \underline{s}\prec\underline{t}\prec\underline{u} :\Leftrightarrow (\underline{s}<\underline{t}<\underline{u}) \lor (\underline{t}<\underline{u}<\underline{s}) \lor (\underline{u}<\underline{s}<\underline{t}). \end{align*} $$

We write $(\underline {s},\underline {t})$ for the open interval between $\underline {s}$ and $\underline {t}$ with respect to the cyclic order on $\mathcal {S}$ , that is, we have $\underline {u}\in (\underline {s},\underline {t})$ if and only if $\underline {s}\prec \underline {u}\prec \underline {t}$ .

The following theorem is a weak version of the classification result proved in [Reference Schleicher and ZimmerSZ1], but it is all we need for this work.

Proposition 2.4. (The escaping set of a post-singularly finite exponential map)

Let $E_{\lambda }$ be a psf exponential map. Then every escaping point either lies on a unique dynamic ray or is the landing point of a unique ray. In particular, distinct dynamic rays are disjoint.

We can assign to each (pre-)periodic external address $\underline {s}$ a dynamic ray $g_{\underline {s}}\colon (0,\infty )\to \mathbb {C}$ in such a way that

$$ \begin{align*} E_{\lambda}(g_{\underline{s}}(t))=g_{\sigma(\underline{s})} (F(t)) \quad \text{for all } t>0, \end{align*} $$

where $F(t)=e^t-1$ , and so that

(1) $$ \begin{align} \lim_{t\to\infty} \operatorname{Re} g_{\underline{s}}(t) = +\infty \quad and \quad \lim_{t\to\infty} \operatorname{Im} g_{\underline{s}}(t) = -\operatorname{Im} \log\lambda+2\pi s_1. \end{align} $$

Here, we chose the branch of the logarithm for which $\operatorname {Im} \log \lambda \in (-\pi ,\pi ]$ .

We partition the complex plane into horizontal strips of the form

$$ \begin{align*} S_k := \{ z\in\mathbb{C}\colon -\operatorname{Im} \log\lambda-\pi+2\pi k < \operatorname{Im} z < -\operatorname{Im} \log\lambda+\pi+2\pi k \} \end{align*} $$

with $k\in \mathbb {Z}$ . This partition is called the static partition for $E_{\lambda }$ .

Equation (1) implies that every dynamic ray is, for all sufficiently large potentials t, contained in a single sector of the static partition, that is, there exists a $k\in \mathbb {Z}$ such that $g(t)\in S_k$ for t large enough.

Definition 2.6. (External address of a dynamic ray)

Let g be a dynamic ray for a psf exponential map $E_{\lambda }$ . The external address $\text {Ad}(g)=s_1 s_2\ldots \in \mathcal {S}$ of the dynamic ray g is the unique external address such that for every $k\in \mathbb {N}$ , we have

$$ \begin{align*} E_{\lambda}^{\circ (k-1)}(g(t))\in S_{s_k} \quad\text{for } t \text{ large enough (depending on } k). \end{align*} $$

Because dynamic rays are disjoint and converge to $\infty $ in a controlled way, as described in (1), they have a well-defined vertical order, defined as follows.

It follows easily from the mapping properties of the exponential map that the vertical order of dynamic rays coincides with the lexicographical order of their external addresses (see [Reference Förster and SchleicherFS, Lemma 3.9]).

As described above for the space of external addresses, the linear order induces a cyclic order on the set of dynamic rays; we write $g\prec g'\prec g''$ if $g'$ lies between g and $g''$ in this cyclic order. The map assigning to each ray its external address preserves the cyclic order (as it preserves the linear order inducing it).

For the purposes of this paper, we are most interested in (pre-)periodic dynamic rays and their landing behavior (compare Definition 2.2). It was shown in [Reference Schleicher and ZimmerSZ2, Theorem 3.2] that every (pre-)periodic dynamic ray of a psf exponential map that is not eventually mapped onto a ray landing at the singular value $0$ lands at a (pre-)periodic point. Conversely, every (pre-)periodic point is the landing point of at least one (pre-)periodic dynamic ray by [Reference Schleicher and ZimmerSZ2, Theorem 5.3]. This result is in analogy to the Douady–Hubbard landing theorem for polynomials.

We construct a partition of the plane that allows us to describe in combinatorial terms which dynamic rays are landing together: Choose a pre-periodic dynamic ray $g_{\underline {s}}$ landing at $0$ . The pre-image $E_{\lambda }^{-1}(\operatorname {\mathrm {Tr}}(g_{\underline {s}}))$ consists of countably many disjoint simple curves which are translations of each other by integer multiples of $2\pi i$ . Furthermore, if g is any lift of $g_{\underline {s}}$ by $E_{\lambda }$ , then we have

$$ \begin{align*} \lim_{t\to 0}\operatorname{Re} (g(t))=-\infty \quad \text{and} \quad \lim_{t\to \infty}\operatorname{Re} (g(t))=+\infty \end{align*} $$

by the mapping properties of exponential maps.

We define a sector to be a connected component of $\mathbb {C}\setminus E_{\lambda }^{-1}(\operatorname {\mathrm {Tr}}(g_{\underline {s}}))$ . These sectors partition $\mathbb {C}$ so that the sector boundaries are exactly the pre-images of the ray $g_{\underline {s}}$ .

Because distinct dynamic rays are disjoint, there is a unique sector $D_0$ containing the ray $g_{\underline {s}}$ and its landing point $0$ . For $k\in \mathbb {Z}$ , the sector obtained by translating $D_0$ by $2k\pi i$ is called $D_k$ . Observe that for all $k\in \mathbb {Z}$ , the restriction $E_{\lambda }\colon D_k\to \mathbb {C}\setminus (\operatorname {\mathrm {Tr}}(g_{\underline {s}})\cup \{0\})$ is biholomorphic. We call $\mathcal {D}:=\bigcup _{k\in \mathbb {Z}}\{D_k\}$ the dynamical partition for $E_{\lambda }$ with respect to $g_{\underline {s}}$ and we call $\partial \mathcal {D}:=E_{\lambda }^{-1}(\operatorname {\mathrm {Tr}}(g_{\underline {s}}))$ the boundary of the partition $\mathcal {D}$ . Note that $\mathcal {D}$ is a collection of subsets of $\mathbb {C}$ , and $\partial \mathcal {D}\subset \mathbb {C}$ .

As dynamic rays are parameterized by external addresses, the dynamical partition can also be constructed on the level of external addresses. Consider again the shift space $\mathcal {S}$ endowed with the lexicographic order. In the following, terms like $t_0\underline {t}\in \mathcal {S}$ with $t_0\in \mathbb {Z}$ and $\underline {t}\in \mathcal {S}$ will denote concatenation of an integer and an external address.

We start with the external address $\underline {s}=s_1 s_2\ldots $ of the dynamic ray $g_{\underline {s}}$ . This address is not constant, so we either have $\underline {s}\in ((s_1-1)\underline {s}, s_1\underline {s})$ or $\underline {s}\in (s_1\underline {s}, (s_1+1)\underline {s})$ . Denote the interval containing $\underline {s}$ by $I_0$ . For all $k\in \mathbb {Z}$ , we define the interval

$$ \begin{align*} I_{k}:=\{ t_1 t_2 t_3\ldots\in\mathcal{S}\colon (t_1-k)t_2 t_3\ldots\in I_0 \}. \end{align*} $$

Observe that $\mathcal {I}:=\bigcup _{k\in \mathbb {Z}} \{I_{k}\}$ is a partition of the shift space $\mathcal {S}$ , which we call the dynamical partition of $\mathcal {S}$ with respect to $\underline {s}$ . We denote by $\partial \mathcal {I}:=\sigma ^{-1}(\underline {s})\subset \mathcal {S}$ the boundary of the partition $\mathcal {I}$ .

It follows from Lemma 2.9 that a dynamic ray g is contained in the sector $D_k$ of the dynamical partition $\mathcal {D}$ of the complex plane if and only if its address $\text {Ad}(g)\in \mathcal {S}$ is contained in the sector $I_k$ of the dynamical partition of the shift space. Recording the sectors in $\mathcal {D}$ to which a point in the plane is mapped under iteration of $E_{\lambda }$ yields so-called itineraries. The same applies to the partition $\mathcal {I}$ of the shift space.

Definition 2.11. (Itineraries of external addresses and the kneading sequence)

Let $\star $ be a formal symbol not contained in $\mathbb {Z}$ and let $\underline {t}\in \mathcal {S}$ be an external address. Then the itinerary $\operatorname {\mathrm {It}}(\underline {t}~\vert ~\underline {s})$ of $\underline {t}$ with respect to $\underline {s}$ is the unique sequence $\operatorname {\mathrm {It}}(\underline {t}~\vert ~\underline {s})=\mathtt {\underline {t}}=\mathtt {t}_1\mathtt {t}_2\ldots \in (\mathbb {Z}\cup \{\star \})^{\mathbb {N}}$ such that

$$ \begin{align*} \mathtt{t}_k:= \begin{cases} i & \text{if } \sigma^{\circ(k-1)}(\underline{t})\in I_i,\\ \star & \text{if } \sigma^{\circ(k-1)}(\underline{t})\in \partial\mathcal{I}. \end{cases} \end{align*} $$

We call the itinerary $\nu :=\operatorname {\mathrm {It}}(\underline {s}~\vert ~\underline {s})$ the kneading sequence of the external address $\underline {s}$ .

Definition 2.12. (Itineraries of points and dynamic rays)

Let $\star $ be a formal symbol not contained in $\mathbb {Z}$ . For a point $z\in \mathbb {C}$ , we define the itinerary of z with respect to $\mathcal {D}$ to be the sequence $\operatorname {\mathrm {It}}(z~\vert ~\mathcal {D})=\mathtt {\underline {t}}=\mathtt {t}_1\mathtt {t}_2\ldots \in (\mathbb {Z}\cup \{\star \})^{\mathbb {N}}$ such that

$$ \begin{align*} \mathtt{t}_k:= \begin{cases} i & \text{if } E_{\lambda}^{\circ (k-1)}(z)\in D_i,\\ \star & \text{if } E_{\lambda}^{\circ (k-1)}(z)\in \partial\mathcal{D}. \end{cases} \end{align*} $$

Let g be a dynamic ray of $E_{\lambda }$ . As distinct dynamic rays do not intersect and the partition boundary $\partial \mathcal {D}$ consists of rays, every iterate of g is either fully contained in a single sector of the dynamical partition or part of the partition boundary. Therefore, all points on g have equal itineraries and we set $\operatorname {\mathrm {It}}(g~\vert ~\mathcal {D}):=\operatorname {\mathrm {It}}(w~\vert ~\mathcal {D})$ for an arbitrary point $w\in \operatorname {\mathrm {Tr}}(g)$ .

Itineraries are written in a different font to distinguish them from external addresses. Observe that the itinerary of a (pre-)periodic point is itself (pre-)periodic. Furthermore, it does not contain the symbol $\star $ because $\partial \mathcal {D}$ consists of escaping points.

The following proposition describes the landing behavior of (pre-)periodic dynamic rays in terms of itineraries. It is crucial for the construction of homotopy Hubbard trees for exponential maps. A proof can be found in [Reference Schleicher and ZimmerSZ2, Lemma 5.2 and Theorem 5.3].

One of the main goals of this paper is to classify post-singularly finite exponential maps in terms of their abstract Hubbard trees. A different classification of psf exponential maps has already been obtained in [Reference Laubner, Schleicher and VicolLSV, Theorems 2.6 and 2.7].

We do not use this result, neither for the construction of homotopy Hubbard trees for exponential maps nor in the proof of our own classification result. Still, it is a nice insight to have in mind during the upcoming constructions.

3 Homotopy Hubbard trees

Hubbard trees were defined for polynomials over thirty years ago in [Reference Douady and HubbardDH1]. We give here a more conceptual but equivalent definition of Hubbard trees, which has a better chance to be generalized to the case of exponential maps than the original one given in [Reference Douady and HubbardDH1] (which uses the concept of filled-in Julia sets). To avoid difficulties arising from unnecessary generality, we do not give the most general definition, but a definition valid for unicritical polynomials (polynomials with only one critical point in the complex plane) with pre-periodic critical value. This makes sense from a conceptual viewpoint as exponential maps are a dynamical limit of unicritical polynomials [Reference Bodelón, Devaney, Hayes, Roberts, Goldberg and HubbardDGH] and for post-singularly finite exponential maps, the unique singular value is pre-periodic.

Let us make the concept of an embedded tree precise. Our definition differs from the standard definition (based on a topological quotient) in the case of infinite trees: infinite trees occur naturally as part of our construction and the topology of these trees at vertices of infinite degree deviates from the usual quotient topology.

Definition 3.1. (Embedded graphs and trees)

A topological space G is called a topological graph if it is homeomorphic to a space

$$ \begin{align*} X=\Big(\dot\bigsqcup_{i\in I} [0^{(i)}, 1^{(i)}] \Big)/_\sim \end{align*} $$

of disjoint copies of the unit interval $[0, 1]$ , where I is some index set (for our purposes, we may restrict to countable index sets) and $\sim $ is an equivalence relation identifying some of the endpoints of the intervals $[0^{(i)}, 1^{(i)}]$ .

Away from equivalence classes of infinite cardinality, we define the topology on X to be the usual quotient topology. In particular, if I is finite, we equip X with the usual quotient topology.

If $x\in X$ is a vertex of infinite degree, that is, an equivalence class of infinite cardinality, there are index sets $I_0, I_1\subset I$ , at least one of which is infinite, such that $0^{(i)}\in x$ if and only if $i\in I_0$ and $1^{(i)}\in x$ if and only if $i\in I_1$ . For all $0 < \epsilon < 1$ , we set

$$ \begin{align*} U_\epsilon := \bigg(\bigg(\bigcup_{i\in I} [0^{(i)}, \epsilon^{(i)}\bigg)\bigg)\cup \bigg(\bigcup_{i\in I} (1-\epsilon^{(i)}, 1^{(i)}] \bigg)\bigg)/_\sim,\end{align*} $$

and define $\{U_\epsilon \}_{0<\epsilon <1}$ to be a neighborhood basis of x in X.

We call G finite if the index set I may be chosen to be finite. The topological graph G is called a topological tree if it is connected and has trivial fundamental group.

The naive way to generalize this definition to the case of exponential maps fails because of the existence of an asymptotic value.

The idea that leads us to a meaningful definition of Hubbard trees for exponential maps is to relax the invariance condition: we do not require a Hubbard tree to be exactly forward invariant, but only invariant up to homotopy relative to the post-singular set. Note that this relaxation is not only necessary (because there is no exactly invariant tree), but also natural from the point of view of Thurston’s theory: an exponential Hubbard tree should determine a psf exponential map up to Thurston equivalence, and the Thurston equivalence class of a map is invariant under homotopies relative to the post-singular set in the domain and co-domain of the map (see Definition 6.4).

Because homotopies rel $P(E_{\lambda })$ cannot be pushed forward by $E_{\lambda }$ (because of the existence of non-post-singular pre-images of post-singular points), the right way to formulate the forward invariance condition is through backwards iteration. We want to say that the pre-image $E_{\lambda }^{-1}(H)$ of a Hubbard tree H contains H up to homotopy rel $P(E_{\lambda })$ . This statement bears a problem. The pre-image $E_{\lambda }^{-1}(H)$ is disconnected because H contains the singular value $0$ . Different connected components of $E_{\lambda }^{-1}(H)$ contain post-singular points which are, by definition, contained in H and are not allowed to move during the homotopy. Hence, by connectedness, H cannot be homotoped into its pre-image $E_{\lambda }^{-1}(H)$ . By adding a pre-image $-\infty $ of $0$ to the complex plane, the pre- image of H becomes connected and in fact becomes an (infinite) embedded tree, so it makes sense to require that H can be homotoped into $E_{\lambda }^{-1}(H)\cup \{-\infty \}$ rel $P(E_{\lambda })$ in the extended plane.

Let us make this idea precise. Let $\mathbb {C}_T$ (as a set) be defined as the disjoint union $\mathbb {C}_T:=\mathbb {C}\mathbin {\dot {\cup }}\{-\infty \}$ , where, for now, the point $-\infty $ is just an abstract point not contained in the complex plane. We turn $\mathbb {C}_T$ into a topological space by choosing a neighborhood basis $(V_n)$ of $0$ and declaring the sets $U_n:=E_{\lambda }^{-1}(V_n)\cup \{-\infty \}$ to be a neighborhood basis of $\{-\infty \}$ . The extension $\widehat {E_{\lambda }}\colon \mathbb {C}_T\to \mathbb {C}$ of $E_{\lambda }$ defined by $\widehat {E_{\lambda }}(-\infty ):=0$ is continuous by definition. The completion $\mathbb {C}_T$ of $\mathbb {C}$ is a special case of a far more general construction of defining a completion of the domain of a holomorphic function by adding all transcendental singularities of its inverse function. See [Reference EremenkoE2] for further information. The extended map $\widehat {E_{\lambda }}$ is not a covering map any more, but we can still lift homeomorphisms and homotopies of the complex plane that fix $0$ under $\widehat {E_{\lambda }}$ .

Lemma 3.5. (Lifting homeomorphisms)

Let $A\subset \mathbb {C}$ be a set containing $0$ and $\varphi \colon \mathbb {C}\to \mathbb {C}$ be a homeomorphism which is homotopic to the identity relative to A. There exists a unique homeomorphism $\Phi \colon \mathbb {C}_T\to \mathbb {C}_T$ which is homotopic to the identity relative to $\widehat {E_{\lambda }}^{-1}(A)$ such that the diagram

commutes. We call $\Phi $ the preferred lift of $\varphi $ . Every homotopy between $\varphi $ and $\text {id}$ rel A lifts to a homotopy between $\Phi $ and $\text {id}$ rel $\widehat {E_{\lambda }}^{-1}(A)$ .

To state our definition of homotopy Hubbard trees, we need a bit more vocabulary.

Assume that $H\subset \mathbb {C}$ is a finite embedded tree spanned by $P(E_{\lambda })$ . By the mapping properties of exponential maps, $\widehat {E_{\lambda }}^{-1}(H)\subset \mathbb {C}_T$ is an infinite embedded tree where the only point of infinite degree is $-\infty $ and every branch of $\widehat {E_{\lambda }}^{-1}(H)$ at $-\infty $ is homeomorphic to H. (This step uses our tailored definition of an infinite tree: the topology near the vertex $-\infty $ is compatible with the extension $\mathbb {C}_T$ .)

As $P(E_{\lambda })$ is forward invariant, we have $P(E_{\lambda })\subset \widehat {E_{\lambda }}^{-1}(H)$ , and therefore it makes sense to talk about the subtree $H':=[P(E_{\lambda })]_{\widehat {E_{\lambda }}^{-1}(H)}$ of the pre-image tree spanned by $P(E_{\lambda })$ . We say that H is invariant up to homotopy rel $P(E_{\lambda })$ if $H'$ is homotopic to H in the extended plane $\mathbb {C}_T$ relative to $P(E_{\lambda })$ . Let us make the last statement precise.

As one might expect, this defines an equivalence relation: if two embedded trees H and $H'$ are homotopic rel A through $I\colon H\times [0,1]\to X$ , then also $H'$ and H are homotopic rel A through the reversed homotopy $\overline {I}\colon H'\times [0,1]\to X$ , $\overline {I}(p,t):=I((I^1)^{-1}(p),1-t)$ . Given a second homotopy $I'\colon H'\times [0,1]\to X$ between $H'$ and an embedded tree $H''\subset X$ rel A, we obtain a homotopy between H and $H''$ rel A by forming the concatenation $I\cdot I':H\times [0,1]\to X$ via

$$ \begin{align*} I\cdot I'(p,t):= \begin{cases} I(p,2t) & \text{for}~t\leq\tfrac{1}{2},\\ I'(I^1(p),2(t-\tfrac{1}{2})) & \text{for}~t\geq\tfrac{1}{2}. \end{cases} \end{align*} $$

The concatenation satisfies $(I\cdot I')^1=(I')^1\circ I^1$ .

Assume from now on that H is invariant up to homotopy rel $P(E_{\lambda })$ . While H is not forward invariant as a set, we still obtain a self-map of H which is well defined up to a certain equivalence relation: the homotopy $I\colon H\times [0,1]\to \mathbb {C}_T$ from Definition 3.7 yields an identification of H and $H'$ via the homeomorphism $\psi :=I_1\colon H\to H'$ and the composition $f:=\widehat {E_{\lambda }}\circ \psi \colon H\to H$ is a self-map of the tree H. We call f the induced self-map of H. There is a distinguished point $v_T:=\psi ^{-1}(-\infty )\in H$ , the singular point of $(H,f)$ , with the property that f is not injective at $v_T$ while it is a local homeomorphism elsewhere and its image $f(v_T)=0$ is the singular value. We call $V_f:=\{f^{\circ n}(v_T)~|~n\in \mathbb {N}_0\}\cup \{p\in H~|~\text {deg}_H(p)\geq 3\}$ the set of marked points of $(H,f)$ . It is the union of the forward orbit of $v_T$ under f and the set of branch points of H, and, as such, contains $P(E_{\lambda })$ .

Different homotopies between H and $H'$ yield different self-maps of H and we want to investigate this ambiguity. Let $\varphi \colon H\to H'$ be another identification of H with $H'$ obtained by a homotopy between H and $H'$ rel $P(E_{\lambda })$ , and let $f':=\widehat {E_{\lambda }}\circ \varphi $ be the corresponding self-map of H and set $v_T':=\varphi ^{-1}(-\infty )$ . As both maps $\psi $ and $\varphi $ are induced by homotopies relative to $P(E_{\lambda })$ , we have $\psi (p)=\varphi (p)=p$ for all $p\in P(E_{\lambda })$ . Therefore the ‘change of identification’ $\theta :=\varphi ^{-1}\circ \psi \colon H\to H$ restricts to the identity on $P(E_{\lambda })$ and, in particular, on the set of endpoints of H. A (graph-theoretic) isomorphism between finite trees is uniquely determined by its values on endpoints; this implies $\theta (b)=b$ for all branch points $b\in H$ . It might happen that $v_T\neq v_T'$ , but we still have $\theta (v_T)=v_T'$ by definition of the singular point. We claim that $\theta (f(v))=f(v)$ for all $v\in V_f$ . Indeed, if $v\neq v_T$ is a branch point of H, then f is a local homeomorphism at v, so $f(v)$ is also a branch point of H, and we have $\theta (f(v))=f(v)$ . If $v\in P(E_{\lambda })$ , then $f(v)\in P(E_{\lambda })$ because $P(E_{\lambda })$ is forward invariant under the dynamics of f, so again we have $\theta (f(v))=f(v)$ . Finally, if $v=v_T$ , then $f(v_T)=0\in P(E_{\lambda })$ , so we have $\theta (f(v))=f(v)$ . It follows that $\theta $ restricts to a conjugation between f and $f'$ on $V_f$ because $\theta (f(v))=f(v)=f'(\theta (v))$ for all $v\in V_f$ .

It turns out that if we just replace the forward invariance condition of Hubbard trees by the weaker condition of being invariant up to homotopy rel $P(E_{\lambda })$ , the graph structure of the resulting tree is not uniquely determined. For example, if there exists a periodic branch point $b\in H$ of degree four, one can split it into two degree three branch points $b_1$ and $b_2$ , changing the tree only in an arbitrary small neighborhood of b as indicated in Figure 1. If we perform this change consistently along the (forward and backward) orbit of b under the self-map f, we again obtain an invariant tree. Note, however, that in the new tree, we have $f^{\circ {n}}([b_1,b_2])=[b_1,b_2]$ , where n is the period of b (and hence also of $b_1$ and $b_2$ ) under f, that is, the in-tree connection of $b_1$ and $b_2$ is forward invariant.

Figure 1

Replacing one degree four branch point b by two degree three branch points $b_1$ and $b_2$ .

To obtain uniqueness of the graph structure, we want f to be expansive in the following sense.

Because different self-maps of H are conjugate on their sets of marked points, the definition of expansivity does not depend on the choice of homotopy between H and $H'$ . Again, the motivation for this definition comes from the polynomial case: the map obtained by restricting a unicritical polynomial with pre-periodic critical value to its Hubbard tree is always expansive in the sense that some iterated image of the in-tree connection of two marked points contains the unique critical point.

Summing up, we obtain the following definition of homotopy Hubbard trees for exponential maps.

Homotopy Hubbard trees for exponential maps are only required to be invariant up to homotopy. To prove a meaningful uniqueness statement, we have to define a suitable equivalence relation on homotopy Hubbard trees that deals with the increased flexibility compared to the polynomial case. Naturally, being a homotopy Hubbard tree is a property of homotopy classes of embedded trees relative to the post-singular set.

Proof By symmetry of Definition 3.7, both directions are equivalent and it suffices to show one of them. Assume that H is a homotopy Hubbard tree for $E_{\lambda }$ , and let $I_0$ be a homotopy between H and $\tilde {H}$ rel $P(E_{\lambda })$ . In analogy to Lemma 3.5, one shows that $I_0$ lifts to a homotopy $\widehat {I}_1\colon \widehat {E_{\lambda }}^{-1}(H)\times [0,1]\to \mathbb {C}_T$ between the pre-images of H and $\tilde {H}$ rel $\widehat {E_{\lambda }}^{-1}(P(E_{\lambda }))$ . As $P(E_{\lambda })\subset \widehat {E_{\lambda }}^{-1}(P(E_{\lambda }))$ , the homotopy $\widehat {I}_1$ fixes $P(E_{\lambda })$ pointwise, so the restriction $I_1:=\widehat {I}_1|_{H'\times [0,1]}$ is a homotopy between $H'$ and $\tilde {H'}$ rel $P(E_{\lambda })$ . By the invariance of H, there exists a homotopy $I\colon H\times [0,1]\to \mathbb {C}_T$ between H and $H'$ . We see that $\tilde {H}$ is homotopic to $\tilde {H}'$ rel $P(E_{\lambda })$ via the concatenation $\overline {I_0}\cdot I\cdot I_1$ . The self-map $\tilde {f}:=\widehat {E_{\lambda }}\circ I_1^1\circ I^1 \circ \overline {I_0}^1$ of $\tilde {H}$ obtained via this homotopy is conjugate to the self-map $f:=\widehat {E_{\lambda }}\circ I^1$ of H. This can be seen using the fact that $\widehat {E_{\lambda }}\circ I_1^1=I_0^1\circ \widehat {E_{\lambda }}$ because $I_1^1$ is a lift of $I_0^1$ . Indeed,

$$ \begin{align*} \tilde{f}=\widehat{E_{\lambda}}\circ I_1^1\circ I^1 \circ\overline{I_0}^1 = I_0^1\circ\widehat{E_{\lambda}}\circ I^1\circ\overline{I_0}^1 =I_0^1\circ f\circ (I_0^1)^{-1}. \end{align*} $$

Therefore, the expansivity of $\tilde {f}$ follows from the expansivity of f.

Let us look more closely at the homotopy involved in the invariance condition of homotopy Hubbard trees. We want to see that after a small modification, this homotopy can be replaced by a stronger kind of homotopy called an ambient isotopy.

The distinction between relative homotopy and isotopy is known to be subtle; in general, $\phi $ being homotopic to the identity $\text {id}$ rel A does not imply $\phi $ being isotopic to $\text {id}$ rel A. In our setting, however, these notions are equivalent; we will use this in §6.

For later use (in the proof of Proposition 4.9), we want to prove the following result regarding the homotopy involved in Definition 3.9.

One might hope to obtain an ambient isotopy between H and $H'$ rel $P(E_{\lambda })$ . Unfortunately, such an isotopy does not exist. The tree $H'$ contains $-\infty $ , while H does not, so the ambient isotopy would have to send $-\infty $ to some point in the complex plane. However, the space $\mathbb {C}_T$ is not locally compact at $-\infty $ , while it is locally compact at every point in the complex plane, so there does not exist a homeomorphism of $\mathbb {C}_T$ sending $-\infty $ to a point in the complex plane.

Therefore, we have to take an intermediate step, homotoping only the tree $H'$ to a tree $H''\subset \mathbb {C}$ . Then, we use a classical result on homotopies between graphs on surfaces to find an ambient isotopy between the modified tree $H''$ and H.

It remains to show that $H''$ is ambient isotopic to H rel $P(E_{\lambda })$ in $\mathbb {C}$ . The question under which conditions homotopic embedded graphs on a surface are ambient isotopic has already been studied extensively. One rather general result can be found in [Reference Farb and MargalitFM, Lemma 2.9]. We state a weaker version of this result that is sufficient for our purposes.

Here, a proper arc is an arc $\gamma \colon [0,1]\to \overline {\mathbb {C}}$ satisfying $\gamma ^{-1}(A)=\{0,1\}$ , and two arcs are called homotopic rel A if they are homotopic rel A in the sense of Definition 3.7. We cannot apply Lemma 3.15 directly to our setting because no post-singular branch points of H are allowed to move during the homotopy, so there is no decomposition of H into proper arcs. The following lemma allows us to deal with this problem.

4 The triod algorithm: determining the graph structure

In this section, we show that for a fixed post-singularly finite exponential map, the structure of a homotopy Hubbard tree as a dynamical tree (see Definition 4.2) is uniquely determined by the kneading sequence (see Definition 2.11) of the external address of a ray landing at the singular value. Together with the fact that a homotopy Hubbard tree does not intersect dynamic rays landing at post-singular points (up to homotopy), this implies uniqueness of homotopy Hubbard trees.

If several dynamic rays land at the singular value, the kneading sequences of their external addresses agree by the following lemma, so we speak of the kneading sequence of a post-singularly finite exponential map. A proof of the result can be found in [Reference Laubner, Schleicher and VicolLSV, Proof of Theorem 2.7, $(1)\Rightarrow (5)$ ].

Let H be a homotopy Hubbard tree and f be an induced self-map of H. We will see that two post-singular points are contained in the same branch of H at the singular point $v_T$ if and only if their itineraries start with the same integer $k_i$ , and we index the different branches $B_{k_i}$ by the corresponding integers. If we accept this fact for the moment, the triple $(H,f,B_{k_i})$ satisfies the following definition.

Let us state some simple properties of exponential dynamical trees first. Their proof is not very hard and works in complete analogy to the proof of [Reference Bruin, Kaffl and SchleicherBKS, Lemma 2.3], so we omit it here.

We have seen in the previous section that different self-maps of a homotopy Hubbard tree H yield equivalent exponential dynamical trees (see the paragraphs after Definition 3.7). Furthermore, if $\tilde {H}$ and H are equivalent homotopy Hubbard trees, we can choose their induced self-maps to be conjugate to each other (see the proof of Lemma 3.10). Therefore, we obtain a well-defined map from the set of equivalence classes of homotopy Hubbard trees to the set of equivalence classes of exponential dynamical trees. The main goal of this section is to give a constructive proof of the following statement.

Note that we do not require H and $\tilde {H}$ to be equivalent. The proof of this theorem works in two steps. To explain the outline of the proof, we also have to define kneading sequences for exponential dynamical trees.

Definition 4.5. (Itineraries and kneading sequence)

Let $(H,f,B_{k_i})$ be an exponential dynamical tree. The itinerary of a point $p\in H$ is the infinite sequence $\operatorname {\mathrm {It}}(p)=\mathtt {\underline {t}}=\mathtt {t_1 t_2\ldots }$ where

$$ \begin{align*} \mathtt{t}_i:= \begin{cases} i & \text{if } f^{\circ (k-1)}(p)\in B_{k_i},\\ \star & \text{if } f^{\circ (k-1)}(p)=v_T. \end{cases} \end{align*} $$

The itinerary $\operatorname {\mathrm {It}}(f(v_T))=:\nu =\nu _1\nu _2\ldots $ of the singular value is called the kneading sequence of $(H,f,B_{k_i})$ .

The proof of Theorem 4.4 works in two steps. First we show that different homotopy Hubbard trees for the same psf exponential map yield exponential dynamical trees with equal kneading sequences. The second step is to show that an exponential dynamical tree is already determined (up to equivalence) by its kneading sequence.

To prove the first statement, we will show that for every homotopy Hubbard tree, there exists a dynamic ray landing at $0$ such that, up to homotopy, the tree does not intersect the ray. Therefore, the pre-image of the tree does not intersect the boundary of the corresponding dynamical partition (up to homotopy). This implies the statement above because itineraries are independent of the choice of partition by Lemma 4.1 and every partition sector contains at most one branch of $H'=[P(E_{\lambda })]_{\widehat {E_{\lambda }}^{-1}(H)}$ at $-\infty $ .

For the proof, we need a topological characterization of dynamic rays landing at periodic post-singular points. Let $p\in P(E_{\lambda })$ be periodic. An arc $\gamma \colon [0,1]\to \overline {\mathbb {C}}$ with $\gamma (0)=p$ , $\gamma (1)=\infty $ , and $\operatorname {\mathrm {Tr}}(\gamma )\cap P(E_{\lambda })=\{p\}$ is called a leg at p. We denote its homotopy class relative to $P(E_{\lambda })\cup \{\infty \}$ by $[\gamma ]$ . By definition, a leg $\tilde {\gamma }$ at p is in the same homotopy class as $\gamma $ if there is a homotopy between $\gamma $ and $\tilde {\gamma }$ in $\overline {\mathbb {C}}\setminus (P(E_{\lambda })\setminus \{p\})$ fixing the endpoints.

For every leg $\gamma $ , we set $\tilde {\mathcal {L}}(\gamma )=\Gamma $ , where $\Gamma \colon [0,1]\to \overline {\mathbb {C}}$ with $\Gamma (0)=p'$ and $\Gamma (1)=\infty $ is the lift of $\gamma $ starting at the unique periodic pre-image $p'$ of p. Lifting is always possible as $0\notin \operatorname {\mathrm {Tr}}(\gamma )$ and $E_{\lambda }\colon \mathbb {C}\to \mathbb {C}\setminus \{0\}$ is a covering map. As $P(E_{\lambda })$ is forward invariant, the arc $\Gamma $ satisfies $\operatorname {\mathrm {Tr}}(\Gamma )\cap P(E_{\lambda })=\{p'\}$ , so $\Gamma $ is a leg at $p'$ .

The map $\tilde {\mathcal {L}}$ defined in this way descends to a well-defined map $\mathcal {L}([\gamma ])=[\Gamma ]$ because homotopies lift under covering maps. We call $\mathcal {L}$ the leg map, using the terminology from [Reference Mihaljević-BrandtB, Definition 4.3].

We prove that for every access to a homotopy Hubbard tree at a post-singular point, there exists a dynamic ray that approaches H through this access and does not intersect H up to homotopy. Before stating the theorem, let us formally define what we mean by an access.

Proof Let $\{0=p_0,p_1=E_{\lambda }(p_0),\ldots ,p_n=E_{\lambda }^{\circ n}(p_0), p_{n+1}=p_{l+1}\}$ be the post-singular orbit, where $p_{l+1}$ is the first periodic point on the forward orbit of $0$ . Denote the accesses to H at $p_m$ by $A_m^{(1)},\ldots , A_m^{(d_m)}$ , where $d_m\in \mathbb {N}$ is the number of branches of H at $p_m$ and the accesses $A_m^{(i)}$ are numbered according to their cyclic order at $p_m$ . For every access $A_m^{(i)}$ , choose a leg $\gamma _m^{(i)}\colon [0,1]\to \overline {\mathbb {C}}$ approaching $p_m$ through $A_m^{(i)}$ . Note that if $\Gamma _m^{(i)}$ is another leg approaching $p_m$ through $A_m^{(i)}$ , then $\gamma _m^{(i)}$ is homotopic to $\Gamma _m^{(i)}$ rel $P(E_{\lambda })$ (which is easy to see using the theory of prime ends). We first consider the periodic part of the forward orbit of $0$ . For $m\in \{l+1,\ldots ,n\}$ , the number of branches of H at $p_m$ is independent of m (say equal to $d\in \mathbb {N}$ ) because the induced self-map $f\colon H\to H$ is a local homeomorphism at $p_m$ and $p_m$ is periodic. Choose a neighborhood V of $0$ such that none of the legs $\gamma _m^{(i)}$ for $m\in \{l+1,\ldots , n\}$ intersect V. We set $U:=\widehat {E_{\lambda }}^{-1}(V)$ and apply Lemma 3.13 to obtain an embedded tree $H''\subset \mathbb {C}$ homotopic to $H'$ rel $\mathbb {C}_T\setminus U$ . The lifts $\tilde {\mathcal {L}}(\gamma _m^{(i)})$ of the $\gamma _m^{(i)}$ along the periodic post-singular orbit do not intersect U and therefore are disjoint from $H''$ (as $H'$ and $H''$ are equal on $\mathbb {C}\setminus U$ ) except for their endpoints. Now we see why we need Lemma 3.13: there exists an ambient isotopy $I\colon \mathbb {C}\times [0,1]\to \mathbb {C}$ between $H''$ and H rel $P(E_{\lambda })$ , and it extends to an ambient isotopy $I\colon \overline {\mathbb {C}}\times [0,1]\to \overline {\mathbb {C}}$ fixing $\infty $ . Hence, $\Gamma _m^{(i)}:= I|_{\overline {\mathbb {C}}\times \{1\}}(\tilde {\mathcal {L}}(\gamma _m^{(i)}))$ approaches $p_{m-1}$ (or $p_n$ for $m=l+1$ ) through an access $A_{m-1}^{(\sigma _m(i))}$ . By the above, $\Gamma _m^{(i)}$ is homotopic to $\gamma _{m-1}^{(\sigma _m(i))}$ , so we have

$$ \begin{align*} \mathcal{L}([\gamma_m^{(i)}])=[\gamma_{m-1}^{(\sigma_m(i))}]\quad\text{for all } m\in\{l+1,\ldots,n\}~\text{and all}~i\in\{1,\ldots,d\}. \end{align*} $$

As $E_{\lambda }$ is locally an orientation-preserving homeomorphism and as ambient isotopies preserve the cyclic order of curves landing at a common point, we see that each $\sigma _m\colon \{1,\ldots ,d\}\to \{1,\ldots ,d\}$ is a power of the d-cycle $(1~2~\ldots ~d)$ . Therefore, there exists an $N\in \mathbb {N}$ such that $\mathcal {L}^{\circ N}([\gamma _m^{(i)}])=[\gamma _m^{(i)}]$ for all m and i. By Proposition 4.6, each $\gamma _m^{(i)}$ is homotopic to a dynamic ray $g_m^{(i)}$ rel $P(E_{\lambda })\cup \{\infty \}$ .

Because the $\gamma _m^{(i)}$ as well as the $g_m^{(i)}$ are pairwise non-homotopic rel $P(E_{\lambda })\cup \{\infty \}$ and intersect each other at most at their endpoints, the two curve systems fulfill the hypotheses of Lemma 3.15. Thus, there exists an ambient isotopy $I\colon \overline {\mathbb {C}}\times [0,1]\to \overline {\mathbb {C}}$ rel $P(E_{\lambda })\cup \{\infty \}$ satisfying $I^1(\gamma _m^{(i)})=g_m^{(i)}$ for all $m\in \{l+1,\ldots ,n\}$ and all $i\in \{1,\ldots ,d\}$ . By Lemma 3.10, the tree $H_{0}:=I^1(H)$ is again a homotopy Hubbard tree and it is equivalent to H.

The last part of the proof works by induction. Assume that $H_j\cong H$ is equivalent to our initial homotopy Hubbard tree and for each post-singular point $p\in P(E_{\lambda })$ , for which $E_{\lambda }^{\circ j}(p)$ is periodic and every access A to p in $H_j$ , there exists a dynamic ray g approaching p through A. Choose a small neighborhood $V_j$ of $0$ such that $V_j$ does not intersect the dynamic rays landing at the points $p_{l+1-j},\ldots ,p_n$ . The pre-image $U_l:=\widehat {E_{\lambda }}^{-1}(V_l)$ does not intersect any of the dynamic rays landing at the points $p_{l-j},\ldots ,p_n$ . By Lemma 3.13, there exists a tree $H_{j}''\subset \mathbb {C}$ which is isotopic to $H_{j}'$ rel $\mathbb {C}_T\setminus U_j$ and isotopic to $H_{j}$ rel $P(E_{\lambda })$ in $\mathbb {C}$ . By Lemma 3.10, the tree $H_{j+1}:=H_{j}''$ is itself a homotopy Hubbard tree and we have $H_{j+1}\cong H_{j}\cong H$ . By construction, for every $p\in \{p_{l-j},\ldots ,p_n\}$ and every access to $H_{j+1}$ at p, there exists a dynamic ray approaching p through this access. Iterating this procedure l times yields a tree $\tilde {H}:=H_l$ satisfying the conditions of the theorem.

We will now show that an exponential dynamical tree is already determined (up to equivalence) by its kneading sequence. The following ideas, leading to the proof of this fact, are inspired by and, in many parts, analogous to results of [Reference Bruin, Kaffl and SchleicherBKS]. Let us start with a simple observation.

Let $(H,f,B_{k_i})$ be an exponential dynamical tree and let $p,q,r\in V_f$ be distinct marked points. The subtree $[p,q,r]$ spanned by p, q, and r is called a triod. It must look in one of two ways: if $[p,q,r]$ is homeomorphic to the letter Y, we call it branched. Else, it is homeomorphic to the letter I and we call it linear. It should now be clear how to define the middle point $b[p,q,r]\in H$ of the triod $[p,q,r]$ . Note that the middle point $b[p,q,r]\in V_f$ is also a marked point.

The shape of the triod $[p,q,r]$ as well as the itinerary of the middle point $b[p,q,r]$ can be determined algorithmically from the itineraries of the marked points p, q, and r. This algorithm is known as the triod algorithm. The triod algorithm is purely combinatorical and could be applied to any triple of pre-periodic sequences (not just the itineraries of p, q, and r). We apply it only to itineraries of vertices of an abstract exponential tree $(H, f, B_{k_i})$ ; let $\nu $ be its kneading sequence.

Definition 4.12. (Formal triods and the formal triod map)

Let $\star $ be a formal symbol not contained in $\mathbb {Z}$ . The space of formal (pre-)periodic points $\mathcal {S}_\nu \subset (\mathbb {Z}\cup \{\star \})^{\mathbb {N}}$ consists of all (pre-)periodic sequences $\mathtt {\underline {t}}\in (\mathbb {Z}\cup \{\star \})^{\mathbb {N}}$ such that either $\mathtt {\underline {t}}\in \mathbb {Z}^{\mathbb {N}}$ and $\sigma ^{\circ n}(\mathtt {\mathtt {\underline {t}}})\neq \nu $ for all $n\in \mathbb {N}$ or $\mathtt {\underline {t}}$ is contained in the backwards orbit

$$ \begin{align*} \Omega^-(\star\nu):= \{k_1\ldots k_n\star\nu~\vert ~n\in\mathbb{N}_0, k_i\in\mathbb{Z} \}\subset (\mathbb{Z}\cup\{\star\})^{\mathbb{N}} \end{align*} $$

of finite integer sequences followed by $\star \nu $ . We call a point $\mathtt {\underline {t}}\in \mathcal {S}_\nu $ a pre-singular point if $\mathtt {\underline {t}}\in \Omega ^-(\star \nu )$ .

We have the shift map $\sigma \colon (\mathbb {Z}\cup \{\star \})^{\mathbb {N}}\to (\mathbb {Z}\cup \{\star \})^{\mathbb {N}}$ acting on the full space of sequences. Note that we are using the same symbol for the shift map on $\mathcal {S}$ . It will always be clear from the context which map we are considering.

Any triple of distinct sequences $\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}\in \mathcal {S}_\nu $ is called a formal triod $\mathtt {[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]}$ . Given a formal triod, we define the formal triod map $\mathcal {T}$ as follows:

$$ \begin{align*} \mathcal{T}[\mathtt{\underline{t}},\mathtt{\underline{u}},\mathtt{\underline{v}}]:= \begin{cases} [\sigma(\mathtt{\underline{t}}), \sigma(\mathtt{\underline{u}}), \sigma(\mathtt{\underline{v}})] &\text{if } \mathtt{t}_1=\mathtt{u}_1=\mathtt{v}_1,\\ [\sigma(\mathtt{\underline{t}}), \sigma(\mathtt{\underline{u}}), \nu] &\text{if } \mathtt{t}_1=\mathtt{u}_1\neq \mathtt{v}_1,\\ [\sigma(\mathtt{\underline{t}}), \nu, \sigma(\mathtt{\underline{v}})] &\text{if } \mathtt{t}_1=\mathtt{v}_1\neq \mathtt{u}_1,\\ [\nu, \sigma(\mathtt{\underline{u}}), \sigma(\mathtt{\underline{v}})] &\text{if } \mathtt{t}_1\neq \mathtt{u}_1=\mathtt{v}_1,\\ \textbf{stop} &\text{if } \mathtt{t}_1, \mathtt{u}_1, \mathtt{v}_1 \text{ distinct}.\end{cases} \end{align*} $$

In all cases other than the stop case, $\mathcal {T}{\kern1.5pt}[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]$ is again a formal triod: because $\mathcal {S}_\nu $ is forward invariant under $\sigma $ and $\nu \in \mathcal {S}_\nu $ , the three image sequences are contained in $\mathcal {S}_\nu $ and they are distinct, as we have $\{\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}\}\cap \{k\nu ~\vert ~k\in \mathbb {Z}\}=\emptyset $ . By construction, the only sequence that starts with $\star $ is $\star \nu $ , so in all cases other than the stop case, at least two of the first entries of the involved sequences are equal integers. If exactly two of the three first entries are equal, we say that the sequence whose first entry differs from the other two gets chopped off under iteration of $\mathcal {T}$ .

The formal triod map can be iterated as long as the stop case is not reached. We write $\mathcal {T}^{{\kern1.5pt}\circ n}[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]$ for the resulting formal triod after n iterations of $\mathcal {T}$ (if iteration is possible). Note that if the triod can be iterated indefinitely, at least two sequences each get chopped off under iteration of $\mathcal {T}$ infinitely many times: otherwise, there would exist an $n\in \mathbb {N}$ such that two of the three sequences of the triod $\mathcal {T}^{{\kern1.5pt}\circ n}[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]$ never get chopped off. However, this implies that these sequences are equal, which contradicts the fact that all three sequences stay distinct under iteration of $\mathcal {T}$ .

Definition 4.13. (Majority vote and middle point of a formal triod)

Let $[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]$ be a formal triod. If $\mathcal {T}{\kern1.5pt}[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]\neq \textbf {stop}$ , then, as noted above, at least two of the three sequences start with the same integer. We denote this integer by $\mathtt {m}(\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}})$ and call it the majority vote of the triod $[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]$ . Let $i_0\in \mathbb {N}_0$ be chosen such that $\mathcal {T}^{{\kern1.5pt}\circ i_0}[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]=\textbf {stop}$ , if the triod eventually reaches the stop case, and let $i_0=\infty $ otherwise. We define a sequence $\mathtt {\underline {b}}(\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}})\in \mathbb {Z}^{\mathbb {N}}\cup \Omega ^-(\star \nu )$ by setting

$$ \begin{align*} (\mathtt{\underline{b}}(\mathtt{\underline{t}},\mathtt{\underline{u}},\mathtt{\underline{v}}))_i:= \begin{cases} \mathtt{m}(\mathcal{T}^{\circ (i-1)}[\mathtt{\underline{t}},\mathtt{\underline{u}},\mathtt{\underline{v}}])&\text{if}~i<i_0,\\ \star&\text{if}~i=i_0,\\ \nu_{i-i_0}&\text{if}~i>i_0, \end{cases} \end{align*} $$

and we call $\mathtt {\underline {b}}(\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}})$ the middle point of the triod. If we have $\mathtt {\underline {b}}(\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}})\notin \{\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}\}$ , we call $[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]$ branched, otherwise we call it linear. Sometimes, we want to be more precise and call a triod pre-singularly branched or pre-singularly linear if $i_0<\infty $ , that is, if it eventually reaches the stop case under iteration of the triod map.

We now prove that the triod algorithm really determines the itineraries of the branch points of an exponential dynamical tree. The proof is an adaption of [Reference Bruin, Kaffl and SchleicherBKS, Proposition 3.5].

Lemma 4.14. (Correctness of the triod algorithm)

Let $(H,f,B_{k_i})$ be an exponential dynamical tree and let $[p_1,p_2,p_3]\subset H$ be a triod. Then $[p_1,p_2,p_3]$ is branched if and only if the formal triod $[\operatorname {\mathrm {It}}(p_1),\operatorname {\mathrm {It}}(p_2),\operatorname {\mathrm {It}}(p_3)]$ is branched. Furthermore, we have

$$ \begin{align*} \operatorname{\mathrm{It}}(b[p_1,p_2,p_3])=\mathtt{\underline{b}}(\operatorname{\mathrm{It}}(p_1),\operatorname{\mathrm{It}}(p_2),\operatorname{\mathrm{It}}(p_3)). \end{align*} $$

Proof If $p_1$ , $p_2$ , and $p_3$ are contained in distinct branches of H at $v_T$ , then $b[p_1,p_2,p_3]=v_T$ , and this is the branch point determined (on the level of itineraries) by the triod algorithm. If one of the $p_i$ equals $v_T$ and the other two points are contained in different branches of H, then $b[p_1,p_2,p_3]=p_i$ , and again this is the output of the triod algorithm.

If both of these cases do not occur, there exist a branch $B_{k_1}$ of H at $v_T$ and distinct indices $i,j\in \{1,2,3\}$ , such that $p_i,p_j\in B_{k_1}\setminus \{v_t\}$ . Hence, we also have $b[p_1,p_2,p_3]\in B_{k_1}\setminus \{v_T\}$ . We see, that the first entry of $\operatorname {\mathrm {It}}(b[p_1,p_2,p_3])$ is calculated correctly by the triod algorithm.

If all of the $p_l$ are contained in $B_{k_1}$ , the spanned subtree $[p_1,p_2,p_3]$ is also entirely contained in $B_{k_1}$ . By Lemma 4.3, the restriction $f|_{[p_1,p_2,p_3]}$ is injective, so $[f(p_1),f(p_2),f(p_3)]$ is also a triod and we have $b[f(p_1),f(p_2),f(p_3)]=f(b[p_1,p_2,p_3])$ . On the level of formal triods, we have $\mathcal {T}{\kern1.5pt}[\operatorname {\mathrm {It}}(p_1),\operatorname {\mathrm {It}}(p_2),\operatorname {\mathrm {It}}(p_3)]=[\operatorname {\mathrm {It}}(f(p_1)),\operatorname {\mathrm {It}}(f(p_2)),\operatorname {\mathrm {It}}(f(p_3))]$ . If instead $p_l\notin B_{k_1}$ , while $p_i,p_j\in B_{k_1}$ , then the chopped-off triod $[v_T, p_i, p_j]$ still gets mapped forward injectively. Hence, we have $b[0,f(p_i),f(p_j)]=f(b[p_1,p_2,p_3])$ . On the level of formal triods, we have $\mathcal {T}{\kern1.5pt}[\operatorname {\mathrm {It}}(p_1),\operatorname {\mathrm {It}}(p_2),\operatorname {\mathrm {It}}(p_3)]=[\operatorname {\mathrm {It}}(0),\operatorname {\mathrm {It}}(f(p_i)),\operatorname {\mathrm {It}}(f(p_j))]$ .

If we are in one of the two preceding cases (we have not reached the stop case in the first iteration step), we apply the same reasoning as before to the image triod. Hence, the triod algorithm correctly calculates $\operatorname {\mathrm {It}}(b[p_1,p_2,p_3])$ . It remains to show that $[p_1,p_2,p_3]$ is branched if and only if the formal triod $[\operatorname {\mathrm {It}}(p_1),\operatorname {\mathrm {It}}(p_2),\operatorname {\mathrm {It}}(p_3)]$ is branched. If $[p_1,p_2,p_3]$ is branched, then

$$ \begin{align*} \operatorname{\mathrm{It}}(b[p_1,p_2,p_3])=\mathtt{\underline{b}}(\operatorname{\mathrm{It}}(p_1),\operatorname{\mathrm{It}}(p_2),\operatorname{\mathrm{It}}(p_3))\neq\operatorname{\mathrm{It}}(p_i)\quad\text{for all } i\in\{1,2,3\} \end{align*} $$

by Proposition 4.11, so the formal triod $[\operatorname {\mathrm {It}}(p_1),\operatorname {\mathrm {It}}(p_2),\operatorname {\mathrm {It}}(p_3)]$ is also branched. If $[p_1,p_2,p_3]$ is linear, then

$$ \begin{align*} \operatorname{\mathrm{It}}(b[p_1,p_2,p_3])=\mathtt{\underline{b}}(\operatorname{\mathrm{It}}(p_1),\operatorname{\mathrm{It}}(p_2),\operatorname{\mathrm{It}}(p_3))=\operatorname{\mathrm{It}}(p_i)\quad\text{for some } i\in\{1,2,3\}, \end{align*} $$

so the formal triod $[\operatorname {\mathrm {It}}(p_1),\operatorname {\mathrm {It}}(p_2),\operatorname {\mathrm {It}}(p_3)]$ is also linear.

Theorem 4.4 is an important step in proving uniqueness of homotopy Hubbard trees, but we still have to see that the embedding of the tree into the plane is also unique. Let H be a homotopy Hubbard tree and let $[p,q,r]$ be a triod of post-singular points. If $[p,q,r]$ is linear with middle point p, Proposition 4.9 implies that there are two dynamic rays $g_{\underline {p}},g_{\underline {p}'}$ landing at p and separating q from r in the sense that q and r are contained in different connected components of $\mathbb {C}\setminus (\operatorname {\mathrm {Tr}}(g_{\underline {p}})\cup \operatorname {\mathrm {Tr}}(g_{\underline {p}'})\cup \{p\}$ . Conversely, if such separating rays exist, the triod $[p,q,r]$ is linear with middle point p.

We are now in the position to prove uniqueness of homotopy Hubbard trees.

5 Separating dynamic rays: embedding the tree into the plane

In this section, we show the existence of homotopy Hubbard trees through an explicit construction. We use the triod algorithm from the preceding section to determine the middle points of post-singular triods, and thereby the set of marked points of the homotopy Hubbard tree on the level of itineraries. Using Proposition 2.13, we pick (pre-)periodic points in the complex plane realizing the itineraries in the formal set of marked points (for pre-singular itineraries, the construction is a bit more involved because there is no actual (pre-)periodic point in the plane realizing this itinerary).

It remains to find the right way to embed the edges of the tree into the complex plane. Our main idea is to find an embedded tree which does not intersect any dynamic rays landing at a marked point. (This statement is only approximately true; some rays landing at pre-singular marked points need to be intersected, but we do so in a controlled way.)

This is partially motivated by Proposition 4.9: if there exists a homotopy Hubbard tree, then, up to homotopy rel $P(E_{\lambda })$ , every access to the tree at every post-singular point contains a dynamic ray. In particular, this ray does not intersect the homotopy Hubbard tree. Another reason is the analogy to polynomials. A polynomial Hubbard tree does not intersect any dynamic rays. If a polynomial does not have bounded Fatou components, every access to every point on its Hubbard tree contains a dynamic ray.

The key observation is that the union of the rays landing at marked points partitions the plane in a meaningful way: there exists an embedded tree spanned by $P(E_{\lambda })$ that does not intersect these rays (except for some rays landing at pre-singular points, as noted above), and this tree is unique up to homotopy rel $P(E_{\lambda })$ . As the rays landing at marked points form a forward invariant set, the pre-image tree also does not intersect them, so the embedded tree is invariant up to homotopy. Expansivity of the induced self-map follows because different marked points have different itineraries, so the embedded tree is a homotopy Hubbard tree for $E_{\lambda }$ .

Let us begin with the construction of homotopy Hubbard trees. At first, we determine the set of marked points on the level of itineraries by using the triod algorithm. Throughout this section, $\nu $ denotes the kneading sequence of the exponential map $E_{\lambda }$ (recall Lemma 4.1). Let $\Omega ^+(\star \nu ):=\{\sigma ^{\circ k}(\star \nu )~\colon ~k\geq 0\}$ denote the forward orbit of $\star \nu $ under the shift map $\sigma $ . This is a finite set because $\nu $ is pre-periodic. We set

$$ \begin{align*} V_\nu:=\Omega^+(\star\nu)\cup\bigcup_{[\mathtt{\underline{t}},\mathtt{\underline{u}},\mathtt{\underline{v}}]} \{\mathtt{\underline{b}}(\mathtt{\underline{t}},\mathtt{\underline{u}},\mathtt{\underline{v}})\}, \end{align*} $$

where the union runs over all formal triods consisting of sequences in $\Omega ^+(\star \nu )$ : we are adding all branch points of triods formed by post-singular points and $\star \nu $ to the set of post-singular points. We call $V_\nu $ the formal vertex set. The following properties of $V_\nu $ will be proved later in this section.

Next, we embed the formal vertex set into the complex plane. The resulting set will become the set of marked points of the yet to be constructed homotopy Hubbard tree. First, assume that $\mathtt {\underline {t}}\in V_\nu $ is not a pre-singular point, that is, the itinerary $\mathtt {\underline {t}}\in \mathbb {Z}^{\mathbb {N}}$ is (pre-)periodic and $\sigma ^{\circ n}(\mathtt {\underline {t}})\neq \nu $ for all $n\in \mathbb {N}$ . By Proposition 2.13, there is exactly one (pre-)periodic point $v_{\mathtt {\underline {t}}}\in \mathbb {C}$ with $\operatorname {\mathrm {It}}(v_{\mathtt {\underline {t}}}~\vert ~\underline {s})=\mathtt {\underline {t}}$ . This is the point we assign to our formal vertex $\mathtt {\underline {t}}$ .

For a pre-singular itinerary $\mathtt {\underline {t}}\in V_\nu $ , there is no (pre-)periodic point in $\mathbb {C}$ of itinerary $\mathtt {\underline {t}}$ . This issue could be addressed by adding further points at infinity corresponding to iterated pre-images of $-\infty $ to the plane. For brevity, we take a more hands-on approach: we choose surrogate points in the plane which are sufficiently close to these iterated pre-images. In our terms, this means that the itinerary of the vertex $v_{\mathtt {\underline {t}}}\in \mathbb {C}$ shares sufficiently many entries with $\mathtt {\underline {t}}$ . To make this precise, let

$$ \begin{align*} N:=\max\{n\in\mathbb{N}_0~\vert~\text{ there exists }~\mathtt{\underline{t}}\in V_\nu\colon\mathtt{\underline{t}}=k_1\ldots k_n\star\nu\} \end{align*} $$

and pick a closed neighborhood U of the singular value $0$ with the following properties.

1. (1) $E_{\lambda }^{-n}(U)\cap E_{\lambda }^{-m}(U)=\emptyset $ for all distinct $n,m\in \{0,\ldots , N+1\}.$

2. (2) For every $n\in \{1,\ldots , N+1\}$ , the pre-image $E_{\lambda }^{-n}(U)$ does not intersect the union of the set of non-pre-singular vertices and the dynamic rays landing at these points.

3. (3) U is bounded by a Jordan curve such that the dynamic ray $g_{\underline {s}}$ landing at the singular value intersects the boundary of U exactly once, that is, there is a unique potential $t_0$ such that $g_{\underline {s}}(t_0)\in \partial U$ , whereas $g_{\underline {s}}(t)\in U$ for $t<t_0$ and $g_{\underline {s}}(t)\in \mathbb {C}\setminus \overline {U}$ for $t>t_0$ . This is only to simplify topological considerations involving U.

The first condition is equivalent to $U\cap E_{\lambda }^{\circ n}(U)=\emptyset $ for all $n\in \{1,\ldots , N+1\}$ and this is obviously fulfilled for U small enough because $0$ is not periodic. The second condition is equivalent to U not intersecting the union of the set of non-pre-singular vertices and the dynamic rays landing at those except for the singular value $0$ . Again, this is true for U small enough. Finally, condition (3) can always be ensured by shrinking a neighborhood that fulfills conditions (1) and (2).

Let $\log _{\underline {s},k}\colon \mathbb {C}\setminus (\operatorname {\mathrm {Tr}}(g_{\underline {s}})\cup \{0\})\to D_k$ be the branch of the inverse of $E_{\lambda }$ with the stated domain and co-domain. For every itinerary $\mathtt {\underline {t}}\in \mathcal {S}_\nu $ of the form $\mathtt {\underline {t}}=k_1\ldots k_n\star \nu $ with $n\in \{0,\ldots , N\}$ , we define

$$ \begin{align*} U_{\mathtt{\underline{t}}}:=\log_{\underline{s},k_1}\ldots \log_{\underline{s},k_n}(E_{\lambda}^{-1}(U)), \end{align*} $$

that is, $U_{\mathtt {\underline {t}}}$ is the iterated pre-image under $E_{\lambda }$ of the domain U constructed above by the branches of the logarithm prescribed by the entries of $\mathtt {\underline {t}}$ . Note that by property (1) of U, distinct $U_{\mathtt {\underline {t}}}$ and $U_{\mathtt {\underline {u}}}$ are disjoint. We assign to $\star \nu \in V_\nu $ an arbitrary point $v_{\star \nu }\in U_{\star \nu }$ which will later become the singular point (see the paragraph after Definition 3.7) of the homotopy Hubbard tree. If $\mathtt {\underline {t}}=k_1\ldots k_n\star \nu $ is pre-singular, we associate to $\mathtt {\underline {t}}$ the point

$$ \begin{align*} v_{\mathtt{\underline{t}}}:=\log_{\underline{s},k_1}\ldots \log_{\underline{s},k_n}(v_{\star\nu})\in U_{\mathtt{\underline{t}}}. \end{align*} $$

To every formal vertex $\mathtt {\underline {t}}\in V_\nu $ , we have thus associated a point $v_{\mathtt {\underline {t}}}\in \mathbb {C}$ .

As triods of vertices are in natural bijection to triods of formal vertices, we use the terminology introduced for formal triods for vertex triods too. In particular, we call a triod of vertices (pre-singularly) branched or (pre-singularly) linear if the corresponding formal triod is of this type.

We now turn our attention to constructing the edges of a homotopy Hubbard tree. The partitioning property mentioned at the beginning of this section rests on a result about dynamic rays separating the vertex set V: if $p, q, r\in V$ are distinct vertices and the corresponding formal triod $[p, q, r]$ is branched, then there exist (pre-)periodic dynamic rays $g_1,g_2,g_3$ of itinerary $\mathtt {\underline {b}}[p, q, r]$ , such that p, q, and r are separated by the rays $g_i$ , that is, such that these three points are contained in different connected components of $\mathbb {C}\setminus \bigcup \operatorname {\mathrm {Tr}}(g_i)$ . If $[p,q,r]$ is linear, there are two (pre-)periodic rays landing at the middle point and separating the other two points from each other. To prove the existence of such separating dynamic rays, we use the language of external addresses. This requires a variant of the triod algorithm operating on the level of external addresses.

Definition 5.3. (Formal triods and the formal triod map)

Recall our notation $\mathcal {I}=\{I_k\}_{k\in \mathbb {Z}}$ for the dynamical partition of the shift space $\mathcal {S}$ with respect to $\underline {s}$ (see §2). In particular, $\underline {s}$ is the external address of a dynamic ray $g_{\underline {s}}$ landing at the singular value $0$ .

A formal triod of external addresses is a triple $[\underline {t}, \underline {u}, \underline {v}]$ of external addresses $\underline {t},\underline {u}, \underline {v}\in \mathcal {S}$ such that $\underline {t}\prec \underline {u}\prec \underline {v}$ , and $\operatorname {\mathrm {It}}(\underline {t}~\vert ~\underline {s}),\operatorname {\mathrm {It}}(\underline {u}~\vert ~\underline {s}),\operatorname {\mathrm {It}}(\underline {v}~\vert ~\underline {s})\in \mathcal {S}_\nu $ are distinct itineraries. We define the formal triod map $\mathcal {T}_{\mathcal {S}}$ on the level of external addresses as follows:

$$ \begin{align*} \mathcal{T}_{\mathcal{S}}[\underline{t}, \underline{u}, \underline{v}]:= \begin{cases} [\sigma(\underline{t}), \sigma(\underline{u}), \sigma(\underline{v})]&\text{if}~\underline{t},\underline{u},\underline{v}\in I_k~\text{for some}~k\in\mathbb{Z},\\ [\sigma(\underline{t}), \sigma(\underline{u}), \underline{s}]&\text{if}~\underline{t},\underline{u}\in I_k~\text{for some}~k\in\mathbb{Z},~\underline{v}\notin I_k,\\ [\sigma(\underline{t}), \underline{s}, \sigma(\underline{t})]&\text{if}~\underline{t},\underline{v}\in I_k~\text{for some}~k\in\mathbb{Z},~\underline{u}\notin I_k,\\ [\underline{s}, \sigma(\underline{u}), \sigma(\underline{v})]&\text{if}~\underline{u},\underline{v}\in I_k~\text{for some}~k\in\mathbb{Z},~\underline{t}\notin I_k,\\ \textbf{stop}&\text{otherwise}. \end{cases} \end{align*} $$

We claim that if the stop case is not reached, the image is again a formal triod. The following result will help us to show this fact.

Lemma 5.4. (Order-preserving restrictions)

Let $I=(k\underline {s},(k+1)\underline {s})$ be a partition sector and let $I^-:=(k\underline {s},(k+1)\underline {s}]$ . The restriction

$$ \begin{align*} \sigma|_{I^-}\colon I^-\to\mathcal{S} \end{align*} $$

is a bijection and preserves the cyclic order.

Figure 2

Sketch illustrating the mapping behavior of the shift map. The left sketch shows how the intervals $I_l$ and $I_u$ are exchanged globally, while the restriction of the shift to each of them is strictly monotonically increasing. The right figure shows that the linear order of the image addresses might change, but the cyclic order stays the same. It also illustrates that an interval splits if it contains the address $ \underline {s}$ .

Let us write $[\underline {\tilde {t}},\underline {\tilde {u}},\underline {\tilde {v}}]:=\mathcal {T}_{\mathcal {S}}[\underline {t}, \underline {u}, \underline {v}]$ . Lemma 5.4 shows that $\underline {\tilde {t}}\prec \underline {\tilde {u}}\prec \underline {\tilde {v}}$ . It remains to show that $\tilde {\underline {\mathtt {x}}}:=\operatorname {\mathrm {It}}(\underline {\tilde {x}}~\vert ~\underline {s})$ are distinct for $x\in \{t,u,v\}$ . The map $[\underline {t},\underline {u},\underline {v}]\mapsto [\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]$ (where $\mathtt {\underline {x}}:=\operatorname {\mathrm {It}}(\underline {{x}}~\vert ~\underline {s})$ as before) is a semi-conjugation between $\mathcal {T}_{\mathcal {S}}$ and $\mathcal {T}$ . The sequences $\tilde {\underline {\mathtt {x}}}$ are distinct because $[\underline {\tilde { \mathtt{t}}},\tilde {\underline {\mathtt {u}}},\tilde {\underline {\mathtt {v}}}]=\mathcal {T}{\kern1.5pt}[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]$ , and triods of formal (pre-)periodic points are mapped to triods of formal (pre-)periodic points under iteration of $\mathcal {T}$ by the considerations after Definition 4.12. We call the triods $[\underline {t}, \underline {u}, \underline {v}]$ and $[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]$ associated to each other. We define the middle point $\mathtt {\underline {b}}[\underline {t},\underline {u},\underline {v}]$ to be the middle point of $[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]$ , and the majority vote $\mathtt {\underline {m}}[\underline {t}, \underline {u}, \underline {v}]$ to be the majority vote of $[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]$ . We call $[\underline {v},\underline {t},\underline {u}]$ branched if $[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]$ is branched, and linear otherwise. If $\mathcal {T}^{{\kern1.5pt}\circ i_0}[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]=\textbf {stop}$ , then $[\underline {t}, \underline {u}, \underline {v}]$ reaches the stop case at the same iteration step and vice versa. We have seen in §4 that if $[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]$ can be iterated indefinitely, all of the three sequences will eventually be contained in $\mathcal {S}_\nu \setminus \Omega ^-(\star \nu )$ , and $[\mathtt {\underline {t}},\mathtt {\underline {u}},\mathtt {\underline {v}}]$ is (pre-)periodic under iteration of $\mathcal {T}$ . By Theorem 2.10 and Proposition 2.13, there are only finitely many external addresses associated to a formal (pre-)periodic point $\mathtt {\underline {t}}\in \mathcal {S}_\nu \setminus \Omega ^-(\star \nu )$ . Therefore, $[\underline {t}, \underline {u}, \underline {v}]$ is eventually periodic under iteration of $\mathcal {T}_{\mathcal {S}}$ .

For the proof of the existence of separating rays, which works on the combinatorial level, the following results will be useful.

Lemma 5.5. (Pullbacks of intervals)

Let $[\underline {t}^1, \underline {t}^2, \underline {t}^3]$ be a formal triod of external addresses which can be iterated at least once before reaching the stop case, let $[\underline {u}^1, \underline {u}^2, \underline {u}^3]:=\mathcal {T}_{\mathcal {S}}[\underline {t}^1, \underline {t}^2, \underline {t}^3]$ be its image triod, and let $k:=\mathtt {m}(\underline {t}^1, \underline {t}^2, \underline {t}^3)$ be its majority vote. Write $J_n:=(\underline {t}^n,\underline {t}^{n+1})$ and $J_n':=(\underline {u}^n, \underline {u}^{n+1})$ for $n\in \{1,2,3\}$ (with indices labeled mod 3) for the intervals of the partition of the shift space by the initial triod and the image triod, respectively. Then we have

$$ \begin{align*} \sigma^{-1}(J_n')\cap I_k\subseteq J_n. \end{align*} $$