Related papers: Rational maps whose Julia sets are generalized Sie…

If $X$ is the attractor set of a conformal IFS in dimension two or three, we prove that there exists a quasiregular semigroup $G$ with Julia set equal to $X$. We also show that in dimension two, with a further assumption similar to the open…

A cubic polynomial $P$ with a non-repelling fixed point $b$ is said to be immediately renormalizable if there exists a (connected) QL invariant filled Julia set $K^*$ such that $b\in K^*$. In that case, exactly one critical point of $P$…

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…

This article deals with the question of local connectivity of the Julia set of polynomials and rational maps. It essentially presents conjectures and questions.

In the paper we investigate the behavior of trajectory of rational $p$-adic dynamical system in complex $p$-adic filed $\C_p$. It is studied Siegel disks and attractors of such dynamical systems. We show that Siegel disks may either…

This paper contains a quite detailed description of the C*-algebra arising from the transformation groupoid of a rational map of degree at least two on the Riemann sphere. The algebra is decomposed stepwise via extensions of familiar…

Let $f$ be a rational map with an infinitely-connected fixed parabolic Fatou domain $U$. We prove that there exists a rational map $g$ with a completely invariant parabolic Fatou domain $V$, such that $(f,U)$ and $(g,V)$ are conformally…

We prove that the Julia set $J(f)$ of at most finitely renormalizable unicritical polynomial $f:z\mapsto z^d+c$ with all periodic points repelling is locally connected. (For $d=2$ it was proved by Yoccoz around 1990.) It follows from a…

In this paper we consider the compact plane sets known as Swiss cheese sets, which are a useful source of examples in the theory of uniform algebras and rational approximation. We introduce a notion of 'allocation map' connected with Swiss…

The aim of this work is to describe the equivalence relations in $\Q/\Z$ that arise as the rational lamination of polynomials with all cycles repelling. We also describe where in parameter space one can find a polynomial with all cycles…

We give a purely analytic construction of a self-similar local regular Dirichlet form on the Sierpi\'nski carpet using approximation of stable-like non-local closed forms which gives an answer to an open problem in analysis on fractals.

We introduce and study a non uniform hyperbolicity condition for complex rational maps, that does not involve a growth condition. We call this condition Backward Contraction. We show this condition is weaker than the Collet-Eckmann…

We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials.…

The theory of polynomial-like maps is of fundamental importance in holomorphic dynamics. We study dynamical properties of a larger class of maps. Our main result is that, under some natural conditions, a map of this class has a completely…

Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers. A rational map $\phi\in K(z)$ of degree at least $2$ is subhyperbolic if each critical point in the $\mathbb{C}_p$-Julia set of $\phi$ is eventually periodic. We…

A generalization of the filled-in Julia set is presented using the multicomplex numbers and an algorithm is presented to visualize these sets in the tridimensional space. There are many ways to visualize these higher dimensional fractals…

We give explicit examples of pairs of Julia sets of hyperbolic rational maps which are homeomorphic but not quasisymmetrically homeomorphic.

We study the dynamics of the H\'enon map defined over complete, locally compact non-Archimedean fields of odd residue characteristic. We establish basic properties of its one-sided and two-sided filled Julia sets, and we determine, for each…

We study one of the central open questions in one-dimensional renormalization theory -- the conjectural universality of golden-mean Siegel disks. We present an approach to the problem based on cylinder renormalization proposed by the second…

Some families of graphs, such as the n-cubes and Sierpinski gaskets, are self-similar. In this paper we show how such recursive structure can be used systematically to prove isoperimetric theorems.