We address the inverse problem for holomorphic germs of a mapping of the complex line near a fixed point which is tangent to the identity. We provide a preferred parabolic map $\Delta $ realizing a given Birkhoff–Écalle–Voronin modulus $\psi $ and prove its uniqueness in the functional class we introduce. The germ is the time-$1$ map of a Gevrey formal vector field admitting meromorphic sums on a pair of infinite sectors covering the Riemann sphere. For that reason, the analytic continuation of $\Delta $ is a multivalued map admitting finitely many branch points with finite monodromy. In particular, $\Delta $ is holomorphic and injective on an open slit sphere containing $0$ (the initial fixed point) and $\infty $, where the companion parabolic point is situated under the involution ${-1}/{\mathrm {Id}}$. One finds that the Birkhoff–Écalle–Voronin modulus of the parabolic germ at $\infty $ is the inverse $\psi ^{\circ -1}$ of that at $0$.

We describe the structure of regular codimension $1$ foliations with numerically projectively flat tangent bundle on complex projective manifolds of dimension at least $4$. Along the way, we prove that either the normal bundle of a regular codimension $1$ foliation is pseudo-effective, or its conormal bundle is nef.

We study the global behavior of the renormalization operator on a specially constructed Banach manifold that has cubic critical circle maps on its boundary and circle diffeomorphisms in its interior. As an application, we prove results on smoothness of irrational Arnold tongues.

Given a number field $\mathbb {K} \subset \mathbb {C}$ that is not contained in $\mathbb {R}$, we prove the existence of a dense set (with respect to the topology of local uniform convergence) of entire maps $f \colon \mathbb {C} \rightarrow \mathbb {C}$ whose preperiodic points and multipliers all lie in $\mathbb {K}$. This contrasts with the case of rational maps. In addition, we show that there exists an escaping quadratic-like map that is not conjugate to an affine escaping quadratic-like map and whose multipliers all lie in $\mathbb {Q}$.

As discovered by W. Thurston, the action of a complex one-variable polynomial on its Julia set can be modeled by a geodesic lamination in the disk, provided that the Julia set is connected. It also turned out that the parameter space of such dynamical laminations of degree two gives a model for the bifurcation locus in the space of quadratic polynomials. This model is itself a geodesic lamination, the so called quadratic minor lamination of Thurston. In the same spirit, we consider the space of all cubic symmetric polynomials $f_\unicode{x3bb} (z)=z^3+\unicode{x3bb} ^2 z$ in three articles. In the first one, we construct the cubic symmetric comajor lamination together with the corresponding quotient space of the unit circle. As is verified in the third paper, this yields a monotone model of the cubic symmetric connectedness locus, that is, the space of all cubic symmetric polynomials with connected Julia sets. In the present paper, the second in the series, we develop an algorithm for generating the cubic symmetric comajor lamination analogous to the Lavaurs algorithm for constructing the quadratic minor lamination.

We show the existence of transcendental entire functions $f: \mathbb {C} \rightarrow \mathbb {C}$ with Hausdorff-dimension $1$ Julia sets, such that every Fatou component of f has infinite inner connectivity. We also show that there exist singleton complementary components of any Fatou component of f, answering a question of Rippon and Stallard [Eremenko points and the structure of the escaping set. Trans. Amer. Math. Soc. 372(5) (2019), 3083–3111]. Our proof relies on a quasiconformal-surgery approach developed by Burkart and Lazebnik [Interpolation of power mappings. Rev. Mat. Iberoam. 39(3) (2023), 1181–1200].

We present a one-parameter family Fλ of transcendental entire functions with zeros, whose Newton’s method yields wandering domains, coexisting with the basins of the roots of Fλ. Wandering domains for Newton maps of zero-free functions have been built before by, e.g. Buff and Rückert [23] based on the lifting method. This procedure is suited to our Newton maps as members of the class of projectable functions (or maps of the cylinder), i.e. transcendental meromorphic functions f(z) in the complex plane that are semiconjugate, via the exponential, to some map g(w), which may have at most a countable number of essential singularities. In this paper, we make a systematic study of the general relation (dynamical and otherwise) between f and g, and inspect the extension of the logarithmic lifting method of periodic Fatou components to our context, especially for those g of finite-type. We apply these results to characterize the entire functions with zeros whose Newton’s method projects to some map g which is defined at both 0 and $\infty$. The family Fλ is the simplest in this class, and its parameter space shows open sets of λ-values in which the Newton map exhibits wandering or Baker domains, in both cases regions of initial conditions where Newton’s root-findingmethod fails.

We study the exact Hausdorff and packing dimensions of the prime Cantor set, $\Lambda _P$, which comprises the irrationals whose continued fraction entries are prime numbers. We prove that the Hausdorff measure of the prime Cantor set cannot be finite and positive with respect to any sufficiently regular dimension function, thus negatively answering a question of Mauldin and Urbański (1999) and Mauldin (2013) for this class of dimension functions. By contrast, under a reasonable number-theoretic conjecture we prove that the packing measure of the conformal measure on the prime Cantor set is in fact positive and finite with respect to the dimension function $\psi (r) = r^\delta \log ^{-2\delta }\log (1/r)$, where $\delta $ is the dimension (conformal, Hausdorff, and packing) of the prime Cantor set.

Let $f(z)=z^2+c$ be an infinitely renormalizable quadratic polynomial and $J_\infty $ be the intersection of forward orbits of ‘small’ Julia sets of its simple renormalizations. We prove that if f admits an infinite sequence of satellite renormalizations, then every invariant measure of $f: J_\infty \to J_\infty $ is supported on the postcritical set and has zero Lyapunov exponent. Coupled with [13], this implies that the Lyapunov exponent of such f at c is equal to zero, which partly answers a question posed by Weixiao Shen.

The possibilities for limit functions on a Fatou component for the iteration of a single polynomial or rational function are well understood and quite restricted. In non-autonomous iteration, where one considers compositions of arbitrary polynomials with suitably bounded degrees and coefficients, one should observe a far greater range of behavior. We show this is indeed the case and we exhibit a bounded sequence of quadratic polynomials which has a bounded Fatou component on which one obtains as limit functions every member of the classical Schlicht family of normalized univalent functions on the unit disc. The proof is based on quasiconformal surgery and the use of high iterates of a quadratic polynomial with a Siegel disc which closely approximate the identity on compact subsets. Careful bookkeeping using the hyperbolic metric is required to control the errors in approximating the desired limit functions and ensure that these errors ultimately tend to zero.

We prove a multidimensional conformal version of the scale recurrence lemma of Moreira and Yoccoz [Stable intersections of regular Cantor sets with large Hausdorff dimensions. Ann. of Math. (2) 154(1) (2001), 45–96] for Cantor sets in the complex plane. We then use this new recurrence lemma, together with Moreira’s ideas in [Geometric properties of images of Cartesian products of regular Cantor sets by differentiable real maps. Math. Z. 303 (2023), 3], to prove that under the right hypothesis for the Cantor sets $K_1,\ldots ,K_n$ and the function $h:\mathbb {C}^{n}\to \mathbb {R}^{l}$, the following formula holds:

$$ \begin{align*}HD(h(K_1\times K_2 \times \cdots\times K_n))=\min \{l,HD(K_1)+\cdots+HD(K_n)\}.\end{align*} $$

In this paper, we consider random iterations of polynomial maps $z^{2} + c_{n}$, where $c_{n}$ are complex-valued independent random variables following the uniform distribution on the closed disk with center c and radius r. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter $c = -1$, almost every random Julia set is totally disconnected with much smaller radial parameters r than expected. We also introduce several open questions worth discussing.

This paper consists of two parts. The first is to study the existence of a point a at the intersection of the Julia set and the escaping set such that a goes to infinity under iterates along Julia directions or Borel directions. Additionally, we find such points that approximate all Borel directions to escape if the meromorphic functions have positive lower order. We confirm the existence of such slowly escaping points under a weaker growth condition. The second is to study the connection between the Fatou set and argument distribution. In view of the filling disks, we show nonexistence of multiply connected Fatou components if an entire function satisfies a weaker growth condition. We prove that the absence of singular directions implies the nonexistence of large annuli in the Fatou set.

Every Thurston map $f\colon S^2\rightarrow S^2$ on a $2$-sphere $S^2$ induces a pull-back operation on Jordan curves $\alpha \subset S^2\smallsetminus {P_f}$, where ${P_f}$ is the postcritical set of f. Here the isotopy class $[f^{-1}(\alpha )]$ (relative to ${P_f}$) only depends on the isotopy class $[\alpha ]$. We study this operation for Thurston maps with four postcritical points. In this case, a Thurston obstruction for the map f can be seen as a fixed point of the pull-back operation. We show that if a Thurston map f with a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can ‘blow up’ suitable arcs in the underlying $2$-sphere and construct a new Thurston map $\widehat f$ for which this obstruction is eliminated. We prove that no other obstruction arises and so $\widehat f$ is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points. We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem.

DeMarco, Krieger, and Ye conjectured that there is a uniform bound B, depending only on the degree d, so that any pair of holomorphic maps $f, g :{\mathbb {P}}^1\to {\mathbb {P}}^1$ with degree $d$ will either share all of their preperiodic points or have at most $B$ in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, $\mathrm {Rat}_d \times \mathrm {Rat}_d$, for each degree $d\geq 2$. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier and Vigny, Yuan and Zhang, and Mavraki and Schmidt. In addition, we present alternate proofs of the main results of DeMarco, Krieger, and Ye [Uniform Manin-Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), 949–1001; Common preperiodic points for quadratic polynomials, J. Mod. Dyn. 18 (2022), 363–413] and of Poineau [Dynamique analytique sur $\mathbb {Z}$ II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel, Preprint (2022), arXiv:2207.01574 [math.NT]]. In fact, we prove a generalization of a conjecture of Bogomolov, Fu, and Tschinkel in a mixed setting of dynamical systems and elliptic curves.

For a post-critically finite branched covering of the sphere that is a subdivision map of a finite subdivision rule, we define non-expanding spines which determine the existence of a Levy cycle in a non-exhaustive semi-decidable algorithm. Especially when a finite subdivision rule has polynomial growth of edge subdivisions, the algorithm terminates very quickly, and the existence of a Levy cycle is equivalent to the existence of a Thurston obstruction. To show the equivalence between Levy and Thurston obstructions, we generalize the arcs intersecting obstruction theorem by Pilgrim and Tan [Combining rational maps and controlling obstructions. Ergod. Th. & Dynam. Sys. 18(1) (1998), 221–245] to a graph intersecting obstruction theorem. As a corollary, we prove that for a pair of post-critically finite polynomials, if at least one polynomial has core entropy zero, then their mating has a Levy cycle if and only if the mating has a Thurston obstruction.

The independence polynomial originates in statistical physics as the partition function of the hard-core model. The location of the complex zeros of the polynomial is related to phase transitions, and plays an important role in the design of efficient algorithms to approximately compute evaluations of the polynomial.

In this paper we directly relate the location of the complex zeros of the independence polynomial to computational hardness of approximating evaluations of the independence polynomial. We do this by moreover relating the location of zeros to chaotic behaviour of a naturally associated family of rational functions; the occupation ratios.

We study the asymptotic behavior of the family of holomorphic correspondences $\lbrace \mathcal {F}_a\rbrace _{a\in \mathcal {K}}$, given by

$$ \begin{align*}\bigg(\frac{az+1}{z+1}\bigg)^2+\bigg(\frac{az+1}{z+1}\bigg)\bigg(\frac{aw-1}{w-1}\bigg)+\bigg(\frac{aw-1}{w-1}\bigg)^2=3.\end{align*} $$

$\mathcal {F}_a$

$\operatorname {PSL}_2(\mathbb {Z})$

$a\in \mathcal {K}$

$\mathcal {F}_a$

$\mathcal {F}_a$

Recently, Benini et al showed that, in simply connected wandering domains of entire functions, all pairs of orbits behave in the same way relative to the hyperbolic metric, thus giving us our first insight into the general internal dynamics of such domains. The author proved in a recent paper [G. R. Ferreira. Multiply connected wandering domains of meromorphic functions: internal dynamics andconnectivity. J. Lond. Math. Soc. (2) 106 (2022), 1897–1919] that the same is not true for multiply connected wandering domains, a natural question is how inhomogeneous multiply connected wandering domains can be. We give an answer to this question, in that we show that uniform dynamics inside an open subset of the domain generalizes to the whole wandering domain. As an application of this result, we construct the first example of a meromorphic function with a semi-contracting infinitely connected wandering domain.

We study local biholomorphisms with finite orbits in some neighborhood of the origin since they are intimately related to holomorphic foliations with closed leaves. We describe the structure of the set of periodic points in dimension 2. As a consequence we show that given a finite-orbits local biholomorphism F, in dimension 2, there exists an analytic curve passing through the origin and contained in the fixed-point set of some non-trivial iterate of $F.$ As an application we obtain that at least one eigenvalue of the linear part of F at the origin is a root of unity. Moreover, we show that such a result is sharp by exhibiting examples of finite-orbits local biholomorphisms such that exactly one of the eigenvalues is a root of unity. These examples are subtle since we show they cannot be embedded in one-parameter groups.