Related papers: Structure of partially hyperbolic H\'enon maps

The Fatou-Julia decomposition is significant in the study of iterations of holomorphic mappings. Such a decomposition can be also considered for foliations in a unified manner. Although the decomposition will be fundamental in the study, it…

Let f be a transcendental map, and let U be an attracting or parabolic basin, or a doubly parabolic Baker domain. Assume U is simply connected. Then, we prove that periodic points are dense in the boundary of U, under certain hypothesis on…

We study the asymptotic expansion of smooth one-dimensional maps. We give an example of an interval map for which the optimal shrinking of components exponential rate is not attained for any neighborhood of a certain fixed point in the…

We study rational functions satisfying summability conditions - a family of weak conditions on the expansion along the critical orbits. Assuming their appropriate versions, we derive many nice properties: There exists a unique, ergodic, and…

We define iterated monodromy groups of more general structures than partial self-covering. This generalization makes it possible to define a natural notion of a combinatorial model of an expanding dynamical system. We prove that a naturally…

We show here the separability of Hamilton-Jacobi equation for a hierarchy of integrable Hamiltonian systems obtained from the constrained flows of the Jaulent-Miodek hierarchy. The classical Poisson structure for these Hamiltonian systems…

The goal of this paper is to study some basic properties of the Fatou and Julia sets for a family of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2$. We are particularly interested in studying these sets for semigroups generated by…

We consider accumulation of periodic points in local uniformly quasiregular dynamics. Given a local uniformly quasiregular mapping $f$ with a countable and closed set of isolated essential singularities and their accumulation points on a…

For infinite-dimensional quasi-compact cocycles over a map satisfying a certain closing condition, we show that periodic orbits carry enough information to guarantee the existence of a dominated splitting. More precisely, we establish that…

Consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally polynomial maps from a finite union of copies of $\C$ to itself. In the space of suitably normalized maps of this type, the hyperbolic maps form an open set called…

Let $f:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ be a hyperbolic rational map of degree $d \geq 2$, and let $J \subset \mathbb{C}$ be its Julia set. We prove that $J$ always has positive Fourier dimension. The case where $J$ is…

Markov partitions persisting in a neighbourhood of hyperbolic components of rational maps were constructed under the condition that closures of Fatou components are disjoint in \cite{R1}. Given such a partition, we characterize all nearby…

Let $f$ and $g$ be commuting meromorphic functions with finitely many poles. By studying the behaviour of Fatou components under this commuting relation, we prove that $f$ and $g$ have the same Julia set whenever $f$ and $g$ have no simply…

We prove that the characterization of the critical locus for complex H\'enon maps that are small perturbations of quadratic polynomials with disconnected Julia sets given by Firsova holds in a much larger HOV-like region from the complex…

We construct a quasiregular mapping in $\mathbb{R}^3$ that is the first to illustrate several important dynamical properties: the quasi-Fatou set contains wandering components; these quasi-Fatou components are bounded and hollow; and the…

We show that the Julia set of quadratic maps with parameters in hyperbolic components of the Mandelbrot set is given by a transseries formula, rapidly convergent at any repelling periodic point. Up to conformal transformations, we obtain…

We give an example of a transcendental entire function with a simply connected fast escaping Fatou component, but with no multiply connected Fatou components. We also give a new criterion for points to be in the fast escaping set.

We show that if a meromorphic function has two completely invariant Fatou components and only finitely many critical and asymptotic values, then its Julia set is a Jordan curve. However, even if both domains are attracting basins, the Julia…

In this paper we study dynamical properties of the area preserving Henon map, as a discrete version of open Hamiltonian systems, that can exhibit chaotic scattering. Exploiting its geometric properties we locate the exit and entry sets,…

We consider fiber-preserving complex dynamics on fiber bundles whose fibers are Riemann spheres and whose base spaces are compact metric spaces. In this context, without any assumption on (semi-)hyperbolicity, we show that the fiberwise…