Related papers: On uniformly disconnected Julia sets

For Cantor circle Julia sets of hyperbolic rational maps, we prove that they are quasisymmetrically equivalent to standard Cantor circles (i.e., connected components are round circles). This gives a quasisymmetric uniformization of all…

We give three families of parabolic rational maps and show that every Cantor set of circles as the Julia set of a non-hyperbolic rational map must be quasisymmetrically equivalent to the Julia set of one map in these families for suitable…

It is well-known that the Julia set J(f) of a rational map is uniformly perfect; that is, every ring domain which separates J(f) has bounded modulus, with the bound depending only on f. In this article we prove that an analogous result is…

In this note we provide a quasisymmetric taming of uniformly perfect and uniformly disconnected sets that generalizes a result of MacManus from 2 to higher dimensions. In particular, we show that a compact subset of $\mathbb{R}^n$ is…

Let $f:\hat{\mathbb C}\to\hat{\mathbb C}$ be a hyperbolic rational map of degree $d\ge2$ on the Riemann sphere. We give several conditions which are equivalent to the condition for the Julia set $J_f$ to be a Cantor set. It has been known…

There are two natural definitions of the Julia set for complex H\'enon maps: the sets $J$ and $J^\star$. Whether these two sets are always equal is one of the main open questions in the field. We prove equality when the map acts…

We prove that every transcendental meromorphic map f with a disconnected Julia set has a weakly repelling fixed point. This implies that the Julia set of Newton's method for finding zeroes of an entire map is connected. Moreover, extending…

The object of the paper is to characterize gasket Julia sets of rational maps that can be uniformized by round gaskets. We restrict to rational maps without critical points on the Julia set. Under these conditions, we prove that a Julia set…

For a hyperbolic polynomial automorphism of C^2 with a disconnected Julia set, and under a mild dissipativity condition, we give a topological description of the components of the Julia set. Namely, there are finitely many "quasi-solenoids"…

For a sequence of complex parameters $\{c_n\}$ we consider the compositions of functions $f_{c_n} (z) = z^2 + c_n$, which is the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are…

There exist uniformly quasiregular maps $f:\mathbb{R}^3 \to \mathbb{R}^3$ whose Julia sets are wild Cantor sets.

We prove some new continuity results for the Julia sets $J$ and $J^{+}$ of the complex H\'enon map $H_{c,a}(x,y)=(x^{2}+c+ay, ax)$, where $a$ and $c$ are complex parameters. We look at the parameter space of dissipative H\'enon maps which…

For any integers $d\ge 3$ and $n\ge 1$, we construct a hyperbolic rational map of degree $d$ such that it has $n$ cycles of the connected components of its Julia set except single points and Jordan curves.

It is known that the disconnected Julia set of any polynomial map does not contain buried Julia components. But such Julia components may arise for rational maps. The first example is due to Curtis T. McMullen who provided a family of…

We explore the connected/disconnected dichotomy for the Julia set of polynomial automorphisms of C^2. We develop several aspects of the question, which was first studied by Bedford-Smillie. We introduce a new sufficient condition for the…

We study the quasisymmetric geometry of the Julia sets of McMullen maps $f_\lambda(z)=z^m+\lambda/z^\ell$, where $\ell$, $m\geq 2$ are integers satisfying $1/\ell+1/m<1$ and $\lambda\in\mathbb{C}\setminus\{0\}$. If the free critical points…

We show that Fatou components of a semi-hyperbolic rational map are John domains and that the converse does not hold. This generalizes a famous result of Carleson, Jones and Yoccoz. We show that a connected Julia set is locally connected…

For a rational function $R$, let $N_R(z)=z-\frac{R(z)}{R'(z)}.$ Any such $N_R$ is referred to as a Newton map. We determine all the rational functions $R$ for which $N_R$ has exactly two attracting fixed points, one of which is an…

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…

In this paper, we prove that a rational map with a Cantor Julia set carries no invariant line fields on its Julia set. It follows that a structurally stable rational map with a Cantor Julia set is hyperbolic.