Related papers: On the geometry of Julia sets

For the complex quadratic family $q_c:z\mapsto z^2+c$, it is known that every point in the Julia set $J(q_c)$ moves holomorphically on $c$ except at the boundary points of the Mandelbrot set. In this note, we present short proofs of the…

Let $f:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ be a hyperbolic rational map of degree $d \geq 2$, and let $J \subset \mathbb{C}$ be its Julia set. We prove that $J$ always has positive Fourier dimension. The case where $J$ is…

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 the quadratic family $f_{c}(z) = z^2+c$ with $c$ in the exterior of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. Let $\hat{c}$ be a semi-hyperbolic parameter in the boundary of the Mandelbrot…

In this paper we study the dynamics of regular polynomial automorphisms of C^n. These maps provide a natural generalization of complex Henon maps in C^2 to higher dimensions. For a given regular polynomial automorphism f we construct a…

Following the ideas of A.~Douady, we give an alternative proof of the authors' result: for any boundary point $c_0$ of the Mandelbrot set $M$, we can find small quasiconformal copies of $M$ in $M$ that are encaged in nested quasiconformal…

In this article, we consider hyperbolic rational maps restricted on thier Julia sets and study about the recurrence rate of typical orbits in arbitrarily small neighbourhoods around them and their relationship to the Hausdorff dimension of…

We study complex one-dimensional parameter slices in a three-parameter family of rational maps with two free critical points, obtained by imposing the existence of periodic orbits with prescribed multipliers. Using explicit…

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"…

Local similarity between the Mandelbrot set and quadratic Julia sets manifests itself in a variety of ways. We discuss a combinatorial one, in the language of geodesic laminations. More precisely, we compare quadratic invariant laminations…

We present an algorithm for a rigorous computation of lower bounds on the Hausdorff dimensions of Julia sets for a wide class of holomorphic maps. We apply this algorithm to obtain lower bounds on the Hausdorff dimension of the Julia sets…

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.

We will show the Mandelbrot set $M$ is locally conformally inhomogeneous: the only conformal map $f$ defined in an open set $U$ intersecting $\partial M$ and satisfying $f(U\cap\partial M)\subset \partial M$ is the identity map. The proof…

Consider the parameter space $\mathcal{P}_{\lambda}\subset \mathbb{C}^{2}$ of complex H\'enon maps $$ H_{c,a}(x,y)=(x^{2}+c+ay,ax),\ \ a\neq 0 $$ which have a semi-parabolic fixed point with one eigenvalue $\lambda=e^{2\pi i p/q}$. We give…

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.

For a hyperbolic rational map $f$ with connected Julia set, we give upper and lower bounds on the Ahlfors-regular conformal dimension of its Julia set $J_f$ from a family of energies of associated graph maps. Concretely, the dynamics of $f$…

For certain typical perturbations $(f_n)_n$ of a rational map $f$ with parabolic cycles, we investigate the relations between the Hausdorff convergence of Julia sets and invariant rays, and the horocyclic convergence of multipliers of…

We are interested in studying sets of the form \[ \mathcal{U}(\alpha) := \left\{ x\in X: \ \exists M=M(x) \geq 1 \text{ such that } \forall N\geq M, \ \exists n\leq N \text{ such that } d(T^nx, x) \leq |\lambda|^{-\alpha N} \right\} \]…

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 continue the description of Mandelbrot and Multibrot sets and of Julia sets in terms of fibers which was begun in IMS preprints 1998/12 and 1998/13a. The question of local connectivity of these sets is discussed in terms of fibers and…