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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 Cantor circle Julia sets of hyperbolic rational maps. By analyzing the combinatorial information of the rational maps whose Julia sets are Cantor circles, we give a computational formula of the number of the Cantor circle hyperbolic components in the moduli space of rational maps for any fixed degree. We calculate the Hausdorff dimensions of the Julia sets which are Cantor circles, and prove that for any Cantor circle hyperbolic component $\mathcal{H}$ in the space of rational maps, the infimum of the Hausdorff dimensions of the Julia sets of the maps in $\mathcal{H}$ is equal to the conformal dimension of the Julia set of any representative $f_0\in\mathcal{H}$, and that the supremum of the Hausdorff dimensions is equal to $2$.

We give a criterion to determine when two external rays land at the same point for polynomials with locally connected Julia sets. As an application, we provide an elementary proof of the monotonicity of the core entropy along arbitrary veins of the Mandelbrot set., Comment: 22 pages, 2 figures, writing polished, results unchanged. To appear in Transactions of the American Mathematical Society

We prove that a nonempty, proper subset $S$ of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if $S$ is bounded and $\hat{\mathbb{C}} \setminus \textrm{int}(S)$ is connected. The proof that such a set is approximable by filled Julia sets is constructive and relies on Fekete polynomials. Illustrative examples are presented. We also prove an estimate for the rate of approximation in terms of geometric and potential theoretic quantities.

In this paper we show that there are infinitely many rings S k , k ≥ 1 {\mathcal S}^k, k \geq 1 , around the McMullen domain in the parameter plane for the family of complex rational maps of the form z n + λ / z n z^n + \lambda /z^n where λ ∈ C \lambda \in \mathbb {C} and n ≥ 3 n \geq 3 . These rings converge to the boundary of the McMullen domain as k → ∞ k \rightarrow \infty . The rings S k {\mathcal S}^k contain ( n − 2 ) n k − 1 + 1 (n-2)n^{k-1} + 1 parameter values that lie at the center of Sierpinski holes. That is, these parameters lie at the center of an open set in the parameter plane in which all of the corresponding maps have Julia sets that are Sierpinski curves. The rings also contain the same number of superstable parameter values, i.e., parameter values for which one of the critical points is periodic of period either k k or 2 k 2k .

We show that there exist two cubic polynomials with connected Julia sets which are combinatorially equivalent but not topologically conjugate on their Julia sets. This disproves a conjecture by McMullen from 1995.

We call a rational map f graph critical if any critical point either belongs to an invariant finite graph G, or has minimal limit set, or is non-recurrent and has limit set disjoint from G. We prove that, for any conformal measure, either for almost every point of the Julia set J(f) its limit set coincides with J(f), or for almost every point of J(f) its limit set coincides with the limit set of a critical point of f.

We prove that the Julia set of a quadratic polynomial which admits an infinite sequence of unbranched, simple renormalizations with complex bounds is locally connected. The method in this study is three-dimensional puzzles.

We obtain results on the structure of the Julia set of a quadratic polynomial P with an irrationally indifferent fixed point zo in the iterative dynamics of P. In the Cremer point case, under the assumption that the Julia set is a decomposable continuum, we obtain a building block structure theorem for the corresponding Julia set J = J(P): there exists a nowhere dense subcontinuum B C J such that P(B) = B, B is the union of the impressions of a minimally invariant Cantor set A of external rays, B contains the critical point, and B contains both the Cremer point zo and its preimage. In the Siegel disk case, under the assumption that no impression of an external ray contains the boundary of the Siegel disk, we obtain a similar result. In this case B contains the boundary of the Siegel disk, properly if the critical point is not in the boundary, and B contains no periodic points. In both cases, the Julia set J is the closure of a skeleton S which is the increasing union of countably many copies of the building block B joined along preimages of copies of a critical continuum C containing the critical point. In addition, we prove that if P is any polynomial of degree d > 2 with a Siegel disk which contains no critical point on its boundary, then the Julia set J(P) is not locally connected. We also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected Julia set have homeomorphic Julia sets.

In this paper we study how certain loops in the parameter space of quadratic complex polynomials give rise to shift-automorphisms of quotients of the set Σ 2 of sequences on two symbols. The Mandelbrot set M is the set of parameter values for which the Julia set of the corresponding polynomial is connected. Blanchard, Devaney, and Keen have shown that closed loops in the complement of the Mandelbrot set give rise to shift-automorphisms of Σ 2 , i.e., homeomorphisms of Σ 2 that commute with the shift map. We study what happens when the loops are not entirely in the complement of the Mandelbrot set. We consider closed loops that cross the Mandelbrot set at a single main bifurcation point, surrounding a component of M attached to the main cardioid

On considere l'ensemble de Julia pour l'application rationnelle complexe Z→Z 2 −λ+e/Z, ou λ et e sont des parametres complexes dans la limite e→0