Related papers: Julia sets with Ahlfors-regular conformal dimensio…

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 present a number of rigidity results concerning holomorphic dynamical systems admitting rotation quasicircles. Firstly, we show the absence of line fields on the Julia set of any rational map that is geometrically finite away from a…

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…

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…

We prove that the meandering set for $f_a(z)=e^z+a$ is homeomorphic to the space of irrational numbers whenever $a$ belongs to the Fatou set of $f_a$. This extends recent results by Vasiliki Evdoridou and Lasse Rempe. It implies that the…

The dynamical classification of rational maps is a central concern of holomorphic dynamics. Much progress has been made, especially on the classification of polynomials and some approachable one-parameter families of rational maps; the goal…

In this paper, we study the existence and properties of conformal measures on limit sets of (anti)holomorphic correspondences. We show that if the critical exponent satisfies $1\leq \delta_{\operatorname{crit}}(x) <+\infty,$ the…

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…

Contents: 1. Quasiconformal Surgery and Deformations: Ben Bielefeld, Questions in quasiconformal surgery; Curt McMullen, Rational maps and Teichm\"uller space; John Milnor, Thurston's algorithm without critical finiteness; Mary Rees, A…

Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers, and $\phi\in K(z)$ be a rational map of degree at least $2$. We prove that the $K$-Julia set of $\phi$ is the natural restriction of $\mathbb{C}_p$-Julia set,…

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…

Using Ahlfors functions, Grunsky maps and the Bell representation theorem, we show that a certain subset of the rational maps of degree $n$ forms a trivial bundle over the moduli space of non-degenerate $n$-connected domains with one marked…

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…

Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are those for which the postsingular set is a compact subset of the Fatou set. Equivalenty, they are characterized as being expanding.…

In this work we consider a class of endomorphisms of $\mathbb{R}^2$ defined by $f(x,y)=(xy+c,x)$, where $c\in\mathbb{R}$ is a real number and we prove that when $-1<c<0$, the forward filled Julia set of $f$ is the union of stable manifolds…

In this paper, we prove that a postcritically finite rational map with non-empty Fatou set is Thurstion equivalent to an expanding Thurston map if and only if its Julia set is homeomorphic to the standard Sierpinski carpet

We show that an invariant Fatou component of a hyperbolic transcendental entire function is a bounded Jordan domain (in fact, a quasidisc) if and only if it contains only finitely many critical points and no asymptotic curves. We use this…

We study the family of renormalization transformations of the generalized $d$--dimensional diamond hierarchical Potts model in statistical mechanic and prove that their Julia sets and non-escaping loci are always connected, where $d\geq 2$.…

We consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic.…

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…