相关论文: Splitting hairs with transcendental entire functio…

We prove several results concerning the relative position of points in the postsingular set $P(f)$ of a meromorphic map $f$ and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer…

It is obtained necessary and sufficient conditions of dependence on $\aleph$ coordinates for functions of several variables, each of which is a product of metrizable factors. The set of discontinuity points of such functions is…

A function is boundedly finite-to-one if there is a natural number $k$ such that each point has at most $k$ inverse images. In this paper, we prove in $\mathsf{ZF}$ (i.e., the Zermelo--Fraenkel set theory without the axiom of choice)…

This article concerns the iteration of quasiregular mappings on $\mathbb{R}^d$ and entire functions on $\mathbb{C}$. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate.…

We study the behaviour of a transcendental entire map $ f\colon \mathbb{C}\to\mathbb{C} $ on an unbounded invariant Fatou component $ U $, assuming that infinity is accessible from $ U $. It is well-known that $ U $ is simply connected.…

Adopting the approach of [7] we study rational function carrying invariant line fields on the Julia set. In particular, we show that under certain weak conditions all possible measurable invariant line fields of a rational function on its…

If M is a Drinfeld module over a local function field L, we may view M as a dynamical system, and consider its filled Julia set J. If J^0 is the connected component of the identity, relative to the Berkovich topology, we give a…

In this paper, we study the exceptional sets $S_f$ of $p$-adic transcendental analytic functions $f$ with rational and algebraic coefficients. We establish a necessary condition for a subset $S \subseteq \overline{\mathbb{Q}} \cap B(0,…

The purpose of this paper is to present several new, sometimes surprising, results concerning a class of hyperholomorphic functions over quaternions, the so-called slice regular functions. The concept of slice regular function is a…

A completely stable multicurve of a post-critically finite rational map induces a combinatorial decomposition. The projections of the small Julia sets are immersed within the original Julia set. We prove that two small Julia sets are…

The set of points that escape to infinity under iteration of a cosine map, that is, of the form $C_{a,b} \colon z \mapsto ae^z+be^{-z}$ for $a,b\in \mathbb{C}^\ast$, consists of a collection of injective curves, called dynamic rays. If a…

This note concerns non-autonomous dynamics of rational functions and, more precisely, the fractal behavior of the Julia sets under perturbation of non-autonomous systems. We provide a necessary and sufficient condition for holomorphic…

We give an example of two rational functions with non-equal Julia sets that generate a rational semigroup whose completely invariant Julia set is a closed line segment. We also give an example of polynomials with unequal Julia sets that…

We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which…

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 show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the ``hairiness phenomenon'', there exist many Feigenbaum Julia sets $J(f)$ whose Hausdorff dimension is strictly smaller…

We construct an entire function in the Eremenko-Lyubich class $\B$ whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative. On the other hand, we show that for many functions in $\B$,…

Let $f$ be a post-critically finite endomorphism (PCF map for short) on $\mathbb{P}^2$, let $J_1$ denote the Julia set and let $J_2$ denote the support of the measure of maximal entropy. In this paper we show that: 1. $J_1\setminus J_2$ is…

For any infinite subset $X$ of the rationals and a subset $F \subseteq X$ which has no isolated points in $X$ we construct a function $f: X \to X$ such that $f(f(x))=x$ for each $x\in X$ and $F $ is the set of discontinuity points of $f$.

Structural stability of holomorphic functions has been the subject of much research in the last fifty years. Due to various technicalities, however, most of that work has focused on so-called finite-type functions (functions whose set of…