Related papers: Computability of the Julia set. Nonrecurrent criti…

We establish necessary and sufficient conditions for the realization of mapping schemata as post-critically finite polynomials, or more generally, as post-critically finite polynomial maps from a finite union of copies of the complex…

A small perturbation of a quadratic polynomial with a non-repelling fixed point gives a polynomial with an attracting fixed point and a Jordan curve Julia set, on which the perturbed polynomial acts like angle doubling. However, there are…

We consider the structure of substantially dissipative complex H\'enon maps admitting a dominated splitting on the Julia set. The dominated splitting assumption corresponds to the one-dimensional assumption that there are no critical points…

Given a continuous dynamical system $f:X\to X$ on a compact metric space $X$ and an $m$-dimensional continuous potential $\Phi:X\to \mathbb R^m$, the (generalized) rotation set ${\rm Rot}(\Phi)$ is defined as the set of all $\mu$-integrals…

Let $k$ be an algebraic closed field of characteristic zero. Let $K$ be the rational function field $K=k(t)$. Let $\phi$ be a non isotrivial rational function in $K(z)$. We prove a bound for the cardinality of the set of $K$--rational…

It is well known that the dynamical behavior of a rational map $f:\widehat{\mathbb C}\to \widehat{\mathbb C}$ is governed by the forward orbits of the critical points of $f$. The map $f$ is said to be postcritically finite if every critical…

We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We provide proof that a random backward iteration algorithm to draw the pictures of the Julia sets, previously proven to work in the…

For the family of complex rational functions of the form R(z)= z^n + a/z^n+b, known as "Generalized McMullen maps", for non-zero a, and integer n fixed and at least 3, we describe the apparent phenomena of baby Julia sets in parameter space…

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…

The bifurcation sets of polynomial functions have been studied by many mathematicians from various points of view. In particular, N\'emethi and Zaharia described them in terms of Newton polytopes. In this paper, we will show analogous…

We present a practical algorithm to compute models of rational functions with minimal resultant under conjugation by fractional linear transformations. We also report on a search for rational functions of degrees 2 and 3 with rational…

For a morphism $f:\P^N \to \P^N$, the points whose forward orbit by $f$ is finite are called preperiodic points for $f$. This article presents an algorithm to effectively determine all the rational preperiodic points for $f$ defined over a…

We establish Bowen's formula for the Julia set of a non-elementary, expanding, irreducible and aperiodic rational graph-directed Markov system satisfying the backward separating condition. Towards this end, we shall prove that the…

We prove an analog of the Tits alternative for rational functions. In particular, we show that if $S$ is a finitely generated semigroup of rational functions over the complex numbers, then either $S$ has polynomially bounded growth or $S$…

Evaluation of relativistic molecular integrals over exponential-type spinor orbitals require using the relativistic auxiliary functions in prolate spheroidal coordinates. They have derived recently by the author [Physical Review E 91,…

Any finite union of disjoint, mutually exterior Jordan curves in the complex plane can be approximated arbitrarily well in the Hausdorff topology by polynomial Julia sets. Furthermore, the proof is constructive.

It is proved that for every complex quadratic polynomial $f$ with Cremer's fixed point $z_0$ (or periodic orbit) for every $\delta>0$, there is at most one periodic orbit of minimal period $n$ for all $n$ large enough, entirely in the disc…

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 the notion of computability of F{\o}lner sets for finitely generated amenable groups. We prove, by an explicit description, that the Kharlampovich group, a finitely presented solvable group with unsolvable word problem, has…

In this article we address the following question, whose interest was recently renewed by problems arising in arithmetic dynamics: under which conditions does there exist a local biholomorphism between the Julia sets of two given…