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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.

We investigate the dynamics of semigroups of rational maps on the Riemann sphere. To establish a fractal theory of the Julia sets of infinitely generated semigroups of rational maps, we introduce a new class of semigroups which we call nicely expanding rational semigroups. More precisely, we prove Bowen’s formula for the Hausdorff dimension of the pre-Julia sets, which we also introduce in this paper. We apply our results to the study of the Julia sets of non-hyperbolic rational semigroups. For these results, we do not assume the cone condition, which has been assumed in the study of infinite contracting iterated function systems. Similarly, we show that Bowen’s formula holds for the limit set of a contracting conformal iterated function system without the cone condition.

We study non-autonomous conformal iterated function systems, with finite or countably infinite alphabet alike. These differ from the usual (autonomous) iterated function systems in that the contractions applied at each step in time are allowed to vary. (In the case where all maps are affine similarities, the resulting system is also called a "Moran set construction".) We shall show that, given a suitable restriction on the growth of the number of contractions used at each step, the Hausdorff dimension of the limit set of such a system is determined by an equation known as Bowen's formula. We also give examples that show the optimality of our results. In addition, we prove Bowen's formula for a class of infinite-alphabet-systems and deal with Hausdorff and packing measures for finite systems, as well as continuity of topological pressure and Hausdorff dimension for both finite and infinite systems. In particular, we strengthen the existing continuity results for infinite autonomous systems. As a simple application of our results, we show that, for a transcendental meromorphic function, the Hausdorff dimension of the set of transitive points (i.e., those points whose orbits are dense in the Julia set) is bounded from below by the hyperbolic dimension (in the sense of Shishikura)., 40 pages. To appear in Transactions of the AMS. V4: Final accepted version. Minor changes and clarifications from V3. V3: A number of corrections and clarifications were made. Also the concept of "Bowen's constant" was renamed to "Bowen dimension". V2: additional references, some revisions

We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $G$ of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any finite critical value of any map $g \in G$. In general, the Julia set of such a semigroup $G$ may be disconnected, and each Fatou component of such $G$ is either simply connected or doubly connected (\cite{Su01,Su9}). In this paper, we show that for any two distinct Fatou components of certain types (e.g., two doubly connected components of the Fatou set), the boundaries are separated by a Cantor set of quasicircles (with uniform dilatation) inside the Julia set of $G.$ Important in this theory is the understanding of various situations which can and cannot occur with respect to how the Julia sets of the maps $g \in G$ are distributed within the Julia set of the entire semigroup $G$. We give several results in this direction and show how such results are used to generate (semi) hyperbolic semigroups possessing this postcritically boundedness condition., To appear in Trans. Amer. Math. Soc

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 study topological dynamics on unshielded planar continua with weak expanding properties at cycles for which we prove that the absence of wandering continua implies backward stability. Then we deduce from this that a polynomial f f with a locally connected Julia set is backward stable outside any neighborhood of its attracting and neutral cycles. For a conformal measure μ \mu this easily implies that one of the following holds: 1. for μ \mu -a.e. x ∈ J ( f ) x\in J(f) , ω ( x ) = J ( f ) \omega (x)=J(f) ; 2. for μ \mu -a.e. x ∈ J ( f ) x\in J(f) , ω ( x ) = ω ( c ( x ) ) \omega (x)=\omega (c(x)) for a critical point c ( x ) c(x) depending on x x .

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

A parabolic rational map of the Riemann sphere admits a non atomic h h -conformal measure on its Julia set where h = h = the Hausdorff dimension of the Julia set and satisfies 1 / 2 > h > 2 1/2 > h > 2 . With respect to this measure the rational map is conservative, exact and there is an equivalent σ \sigma -finite invariant measure. Finiteness of the measure is characterised. Central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one. A theory of Markov fibred systems is developed, and parabolic rational maps are considered within this framework.

The long term analysis of dynamical systems inspired the study of the dynamics of families of mappings. Many of these investigations led to the study of the dynamics of mappings on Cantor sets and on intervals. Julia sets play a critical role in the understanding of the dynamics of families of mappings. In this paper we introduce another class of objects (called hairy objects) which share many properties with the Cantor set and the interval: they are topologically unique and admit only one embedding in the plane. These uniqueness properties explain the regular occurrence of hairy objects in pictures of Julia sets—hairy objects are ubiquitous. Hairy arcs will be used to give a complete topological description of the Julia sets of many members of the exponential family.

We show that for any transcendental meromorphic function f there is a point z in the Julia set of f such that the iterates f n (z) escape, that is, tend to ∞ , arbitrarily slowly. The proof uses new covering results for analytic functions. We also introduce several slow escaping sets, in each of which f n (z) tends to ∞ at a bounded rate, and establish the connections between these sets and the Julia set of f . To do this, we show that the iterates of f satisfy a strong distortion estimate in all types of escaping Fatou components except one, which we call a quasi-nested wandering domain. We give examples to show how varied the structures of these slow escaping sets can be.

By a theorem of S. Demko there exists a balanced measure on the Julia set of an arbitrary nonlinear rational transformation on the Riemann sphere. It is proved here that if the transformation admits an attractive or indifferent cycle, then there is a point with respect to which all the moments of a balanced measure exist; moreover, these moments can be calculated exactly. An explicit balanced measure is exhibited in an example where the Julia set is the whole sphere and for which the moments, with respect to any point, do not all exist. 1. Introduction. The Julia set for a nonlinear rational transformation on the Riemann sphere is the closure of the set of all repulsive cycles of the transformation (Br, Fa, J). The "location" or "distribution" of the set is of interest in the theory of iterated maps, and it may be characterized by an associated invariant measure. In this paper, we extend previous results by showing that when there is an attractive or indifferent cycle, certain moments of such a measure exist and can be calculated exactly from the rational transformation. This means, for example, that when the Julia set is contained strictly in the real line, all of the orthogonal polynomials on the set can be calculated as can the entries of the corresponding infinite dimensional Jacobi matrices. The latter are of the type which have recently been shown to furnish model Schrodinger equations for crystals with almost periodic potentials (BBM). This leads us to the following speculation, which provides one reason for our interest in the present topic. We note first the frequent occurrence, in the physical universe, of objects ("steady-state" systems), which are fractals from some point of

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