Related papers: Geometric Limits of Julia Sets with Parameters on …

In this paper, we prove that for any post-critically finite rational map $f$ on the Riemann sphere $\overline{\mathbb{C}}$, and for each sufficiently large integer $n$, there exists a finite and connected graph $G$ in the Julia set of $f$…

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…

For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials $\{P_n\}$ to properties of the support. More precisely we relate the Julia…

Let $f:\bar\bold C\to\bar\bold C$ be a rational map on the Riemann sphere , such that for every $f$-critical point $c\in J$ which forward trajectory does not contain any other critical point, $|(f^n)'(f(c))|$ grows exponentially fast…

According to the Thurston No Wandering Triangle Theorem, a branching point in a locally connected quadratic Julia set is either preperiodic or precritical. Blokh and Oversteegen proved that this theorem does not hold for higher degree Julia…

A. Sannami constructed an example of the differentiable Cantor set embedded in the real line whose difference set has a positive measure. In this paper, we generalize the definition of the difference sets for sets of the two dimensional…

We estimate the upper box and Hausdorff dimensions of the Julia set of an expanding semigroup generated by finitely many rational functions, using the thermodynamic formalism in ergodic theory. Furthermore, we show Bowen's formula, and the…

The goal of this article is to study a rigidity property of Julia sets of certain classes of automorphisms in $\mathbb{C}^k$, $k \ge 3.$ First, we study the relation between two polynomial shift-like maps in $\mathbb{C}^k$, $k \ge 3$, that…

In this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their…

Let $g(z)=\int_0^zp(t)\exp(q(t))\,dt+c$ where $p,q$ are polynomials and $c\in\mathbb{C}$, and let $f$ be the function from Newton's method for $g$. We show that under suitable assumptions the Julia set of $f$ has Lebesgue measure zero.…

We consider a sequence $(p_n)_{n=1}^\infty$ of polynomials with uniformly bounded zeros and $\deg p_1\geq 1$, $\deg p_n\geq 2$ for $n\geq 2$, satisfying certain asymptotic conditions. We prove that the function sequence $\left(\frac{1}{\deg…

We provide the first definition of \emph{Misiurewicz parameter} for the unicritical family of algebraic correspondences $ z^r + c$, with $ r > 1$ rational, and prove that, at every Misiurewicz parameter, the correspondence uniformly expands…

This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by $\exp(C_K |s|^{\delta})$ in strips $|\Re s| \leq K$, where…

In this article, we derive an inequality of {\L}ojasiewicz-Siciak type for certain sets arising in the context of the complex dynamics in dimension 1. More precisely, if we denote by $dist$ the euclidian distance in $\mathbb{C}$, we show…

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. The Julia set of such a semigroup may not be connected in general. We…

Let $f$ be a rational map with degree at least two. We prove that $f$ has at least $2$ disjoint and infinite critical orbits in the Julia set if it has a Herman ring. This result is sharp in the following sense: there exists a cubic…

Let $G_{m,n,k} = \mathbb{Z}_m \ltimes_k \mathbb{Z}_n$ be the split metacyclic group, where $k$ is a unit modulo $n$. We derive an upper bound for the diameter of $G_{m,n,k}$ using an arithmetic parameter called the \textit{weight}, which…

For 0 < c < 1/e the Julia set of f(z) = c exp(z) is an uncountable union of pairwise disjoint simple curves tending to infinity [Devaney and Krych 1984], the Hausdorff dimension of this set is two [McMullen 1987], but the set of curves…

We show that if $\beta>1$ is a rational number and the Julia set $J$ of the holomorphic correspondence $z^{\beta}+c$ is a locally eventually onto hyperbolic repeller, then the Hausdorff dimension of $J$ is bounded from above by the zero of…

We give a description of the group of all quasisymmetric self-maps of the Julia set of $f(z)=z^2-1$ that have orientation preserving homeomorphic extensions to the whole plane. More precisely, we prove that this group is the uniform closure…