Quasisymmetric geometry of the Julia sets of McMullen maps Dedicated to the Memory of Professor Lei Tan

We study the quasisymmetric geometry of the Julia sets of McMullen maps f_λ(z) = z~m+ λ/z~?,where λ∈ C \ {0} and ? and m are positive integers satisfying 1/? + 1/m 1. If the free critical points of f_λ are escaped to the infinity, we prove that the Julia set J_λ of f_λ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpiński carpet(which is also standard in some sense).If the free critical points are not escaped, we give a sufficient condition on λ such that J_λ is a Sierpiński carpet and prove that most of them are quasisymmetrically equivalent to some round carpets. In particular, there exist infinitely renormalizable rational maps whose Julia sets are quasisymmetrically equivalent to the round carpets.     

Julia sets of permutable Holomorphic maps

Let f and g be two permutable transcendental holomorphic maps in the plane. We shall discuss the dynamical properties of f, g and f o g and prove, among other things, that if either f has no wandering domains or f is of bounded type, then the Julia sets of f and f(g) coincide. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Rigidity for rational maps with Cantor Julia sets

In this paper, we prove that two rational maps with the Cantor Julia sets are quasicon- formally conjugate if they are topologically conjugate.     

Dynamics of McMullen maps

In this article, we develop the Yoccoz puzzle technique to study a family of rational maps termed McMullen maps. We show that the boundary of the immediate basin of infinity is always a Jordan curve if it is connected. This gives a positive answer to the question of Devaney. Higher regularity of this boundary is obtained in almost all cases. We show that the boundary is a quasi-circle if it contains neither a parabolic point nor a recurrent critical point. For the whole Julia set, we show that the McMullen maps have locally connected Julia sets except in some special cases.     

Extending external rays throughout the Julia sets of rational maps

For polynomial maps in the complex plane, the notion of external rays plays an important role in determining the structure of and the dynamics on the Julia set. In this paper we consider an extension of these rays in the case of rational maps of the form Fλ(z) = z ⁿ + λ/z ⁿ where n > 1. As in the case of polynomials, there is an immediate basin of ∞, so we have similar external rays. We show how to extend these rays throughout the Julia set in three specific examples. Our extended rays are simple closed curves in the Riemann sphere that meet the Julia set in a Cantor set of points and also pass through countably many Fatou components. Unlike the external rays, these extended rays cross infinitely many other extended rays in a manner that helps determine the topology of the Julia set.     

Hausdorff measures on Julia sets of subexpanding rational maps

Leth be the Hausdorff dimension of the Julia setJ(R) of a Misiurewicz’s rational mapR : (subexpanding case). We prove that theh-dimensional Hausdorff measure H _h onJ(R) is finite, positive and the onlyh-conformal measure forR : up to a multiplicative constant. Moreover, we show that there exists a uniqueR-invariant measure onJ(R) equivalent to H _h .     

Hausdorff dimensions of the Julia sets of reluctantly recurrent rational maps

In this paper, we consider a rational map f of degree at least two acting on Riemman sphere that is expanding away from critical points. Assuming that all critical points of f in the Julia set J(f) are reluctantly recurrent, we prove that the Hausdorff dimension of the Julia set J(f) is equal to the hyperbolic dimension, and the Lebesgue measure of Julia set is zero when the Julia set J(f) ≠ .     

Quasisymmetric equivalence of self-similar sets

We prove that the self-similar sets with the strong separation condition are all quasisymmetrically equivalent.     

Metric diophantine approximation in Julia sets of expanding rational maps

Let T : J → J be an expanding rational map of the Riemann sphere acting on its Julia set J andf : J →R denote a Hölder continuous function satisfyingf(x)?log | T′(x vb for allx in J. Then for any pointz ₀ in J define the set D_{z 0}(f) of “well-approximable” points to be the set of points in J which lie in the Euclidean ball $B(\gamma ,{\text{ exp(}} - \sum {_{i - 0}^{\mathfrak{n} - 1} } f(T^\ell x)))$ for infinitely many pairs (y, n) satisfying T ⁿ (y)=z₀. We prove that the Hausdorff dimension of D_{z 0}(f) is the unique positive numbers(f) satisfying the equation P(T,?s(f).f)=0, where P is the pressure on the Julia set. This result is then shown to have consequences for the limsups of ergodic averages of Hölder continuous functions. We also obtain local counting results which are analogous to the orbital counting results in the theory of Kleinian groups.     

Coding and tiling of Julia sets for subhyperbolic rational maps

Let be a subhyperbolic rational map of degree d. We construct a set of “proper” coding maps Cod^°(f)={π_r:Σ→J}_r of the Julia set J by geometric coding trees, where the parameter r ranges over mappings from a certain tree to the Riemann sphere. Using the universal covering space for the corresponding orbifold, we lift the inverse of f to an iterated function system I=(g_i)_i=1,2,…,d. For the purpose of studying the structure of Cod^°(f), we generalize Kenyon and Lagarias-Wang's results : If the attractor K of I has positive measure, then K tiles φ^-1(J), and the multiplicity of π_r is well-defined. Moreover, we see that the equivalence relation induced by π_r is described by a finite directed graph, and give a necessary and sufficient condition for two coding maps π_r and π_r^′ to be equal.     

Sierpinski and non-Sierpinski curve Julia sets in families of rational maps

We discuss the dynamics as well as the structure of the parameterplane of certain families of rational maps with few criticalorbits. Our paradigm is the family R_t(z) = (1 + (4/27)z³/(1– z)), with dynamics governed by the behaviour of thepostcritical orbit (Rⁿ())_n. In particular, it is shown thatif escapes (that is, Rⁿ() tends to infinity), then the Juliaset of R is a Cantor set, or a Sierpiski curve, or a curve withone or else infinitely many cut-points; each of these casesactually occurs.     

An explicit bound for uniform perfectness of the Julia sets of rational maps

A compact set C in the Riemann sphere is called uniformly perfect if there is a uniform upper bound on the moduli of annuli which separate C. Julia sets of rational maps of degree at least two are uniformly perfect. Proofs have been given independently by Ma né and da Rocha and by Hinkkanen, but no explicit bounds are given. In this note, we shall provide such an explicit bound and, as a result, give another proof of uniform perfectness of Julia sets of rational maps of degree at least two. As an application, we provide a lower estimate of the Hausdorff dimension of the Julia sets. We also give a concrete bound for the family of quadratic polynomials in terms of the parameter c. Received: 7 June 1999; in final form: 9 November 1999 / Published online: 17 May 2001     

Continuity of Julia sets

A sufficient and necessary condition is given for the continuity of Julia sets in the space of all rational maps with degreek>1. Project supported by the National Natural Science Foundation of China (Grant No. 19871002).     

Dynamics, topology and bifurcations of McMullen maps

We give a survey of many diff erent topological structure arise in the dynamical and parameter planes of McMullen maps.     

Some cubic Julia sets

We figure out geometric properties of the Julia setJ _a of cubic complex polynomialC _a(z) =z ³ +az(a ∈ ?) and the smallest ellipse which surroundsJ _a.     

Alternate superior Julia sets

Alternate Julia sets have been studied in Picard iterative procedures. The purpose of this paper is to study the quadratic and cubic maps using superior iterates to obtain Julia sets with different alternate structures. Analytically, graphically and computationally it has been shown that alternate superior Julia sets can be connected, disconnected and totally disconnected, and also fattier than the corresponding alternate Julia sets. A few examples have been studied by applying different type of alternate structures.