Hausdorff Dimension of Harmonic Measure for Self-Conformal Sets

Under some technical assumptions it is shown that the Hausdorff dimension of the harmonic measure on the limit set of a conformal infinite iterated function system is strictly less than the Hausdorff dimension of the limit set itself if the limit set is contained in a real-analytic curve, if the iterated function system consists of similarities only, or if this system is irregular. As a consequence of this general result the same statement is proven for hyperbolic and parabolic Julia sets, finite parabolic iterated function systems and generalized polynomial-like mappings. Also sufficient conditions are provided for a limit set to be uniformly perfect and for the harmonic measure to have the Hausdorff dimension less than 1. Some results in the spirit of Przytycki et al. (Ann. of Math.130 (1989), 1-40; Stud. Math.97 (1991), 189-225) are obtained.     

Elliptic Functions With Critical Points Eventually Mapped Onto Infinity

We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto ∞. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) σ-finite f-invariant measure μ equivalent to m. The measure μ is ergodic and conservative.     

Elliptic Functions With Critical Points Eventually Mapped Onto Infinity

We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto ∞. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) σ-finite f-invariant measure μ equivalent to m. The measure μ is ergodic and conservative.     

Asymptotic porosity of planar harmonic measure

We study the distribution of harmonic measure on connected Julia sets of unicritical polynomials. Harmonic measure on a full compact set in ? is always concentrated on a set which is porous for a positive density of scales. We prove that there is a topologically generic set $\mathcal{A}$ in the boundary of the Mandelbrot set such that for every $c\in \mathcal{A}$ , β>0, and λ∈(0,1), the corresponding Julia set is a full compact set with harmonic measure concentrated on a set which is not β-porous in scale λ ⁿ for n from a set with positive density amongst natural numbers.     

Geometry and ergodic theory of non-recurrent elliptic functions

We explore the class of elliptic functions whose critical points all contained in the Julia set are non-recurrent and whose ω-limit sets form compact subsets of the complex plane. In particular, this class comprises hyperbolic, subhyperbolic and parabolic elliptic maps. Leth be the Hausdorff dimension of the Julia set of such an elliptic functionf. We construct an atomlessh-conformal measurem and show that theh-dimensional Hausdorff measure of the Julia set off vanishes unless the Julia set is equal to the entire complex plane ℂ. Theh-dimensional packing measure is positive and is finite if and only if there are no rationally indifferent periodic points. Furthermore, we prove the existence of a (unique up to a multiplicative constant) σ-finitef-invariant measure μ equivalent tom. The measure μ is shown to be ergodic and conservative, and we identify the set of points whose open neighborhoods all have infinite measure μ. In particular, we show that ∞ is not among them. The research of the first author was supported in part by the Foundation for Polish Science, the Polish KBN Grant No 2 PO3A 034 25 and TUW Grant no 503G 112000442200. She also wishes to thank the University of North Texas where this research was conducted. The research of the second author was supported in part by the NSF Grant DMS 0100078. Both authors were supported in part by the NSF/PAN grant INT-0306004.     

Julia sets that are full of holes

Conclusion  Many of the most fundamental properties, such as measure and dimension, remain unknown for most Julia sets. Although there are Julia sets that are the whole Riemann sphere and so have dimension two and positive measure, no other Julia sets of measure bigger than zero have been found. Shishikura’s surprising result (1998) shows that there are other Julia sets of dimension 2, which makes it appear possible that there are other Julia sets of positive measure. Proving that a Julia set is full of holes, or porous, provides a bound on the upper box dimension, but this has so far been possible only for special classes of Julia sets. Mean porosity and mean e-porosity, both found in Koskela and Rohde (1997), provide better dimension bounds; nonuniform porosity (Roth 2006) implies measure zero, but is not known to provide dimension bounds. These notions can be used in some cases when it is not possible to prove porosity. In the end, we do not know in general which Julia sets are porous and which are not. In fact, forJ _R, little is known about its dimension or measure. There is much left to explore.     

On the dynamics of a family of generated renormalization transformations

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?2$d?2$. In particular, we prove that their Julia sets can never be a Sierpiński carpet if the parameter is real. We show that the Julia set is a quasicircle if and only if the parameter lies in the unbounded capture domain of these models. Moreover, the asymptotic formula of the Hausdorff dimension of the Julia set is calculated as the parameter tends to infinity.     

Polynomial diffeomorphisms of C2: V. critical points and Lyapunov exponents

There is an invariant measure μ, which is the pluri-complex version of the harmonic measure of the Julia set for polynomial maps of C.In this paper we give an integral formula for the Lyapunov exponents of a polynomial automorphism with respect to μ, analogous to the Brolin-Manning formula polynomial maps of C.Our formula relates the Lyapunov exponents to the value of a Green function at a type of critical point which we define in this paper. We show that these the critical points have a natural dynamical interpretation.     

Multipliers of periodic orbits of quadratic polynomials and the parameter plane

We prove a result about an extension of the multiplier of an attracting periodic orbit of a quadratic map as a function of the parameter. This has applications to the problem of geometry of the Mandelbrot and Julia sets. In particular, we prove that the size of p/q-limb of a hyperbolic component of the Mandelbrot set of period n is O(4 ⁿ /p), and give an explicit condition on internal arguments under which the Julia set of corresponding (unique) infinitely renormalizable quadratic polynomial is not locally connected. In memory of my grandmother Esfir Garbuz     

Puzzle geometry and rigidity: The Fibonacci cycle is hyperbolic

We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and ``complex bounds', two generalized polynomial-like maps which admit a topological conjugacy, quasiconformal outside the filled-in Julia set, are indeed quasiconformally conjugate. The proof uses a new abstract removability-type result for quasiconformal maps, following ideas of Heinonen and Koskela and of Kallunki and Koskela, optimized for applications in complex dynamics. We prove, as the first application of this new method, that, for even criticalities distinct from two, the period two cycle of the Fibonacci renormalization operator is hyperbolic with -dimensional unstable manifold.

     

Analytic families of semihyperbolic generalized polynomial-like mappings

We show that the Hausdorff dimension of Julia sets in any analytic family of semihyperbolic generalized polynomial-like mappings (GPL) depends in a real-analytic manner on the parameter. For the proof we introduce abstract weakly regular analytic families of conformal graph directed Markov systems. We show that the Hausdorff dimension of limit sets in such families is real-analytic, and we associate to each analytic family of semihyperbolic GPLs a weakly regular analytic family of conformal graph directed Markov systems with the Hausdorff dimension of the limit sets equal to the Hausdorff dimension of the Julia sets of the corresponding semihyperbolic GPLs.     

Analytic families of semihyperbolic generalized polynomial-like mappings

We show that the Hausdorff dimension of Julia sets in any analytic family of semihyperbolic generalized polynomial-like mappings (GPL) depends in a real-analytic manner on the parameter. For the proof we introduce abstract weakly regular analytic families of conformal graph directed Markov systems. We show that the Hausdorff dimension of limit sets in such families is real-analytic, and we associate to each analytic family of semihyperbolic GPLs a weakly regular analytic family of conformal graph directed Markov systems with the Hausdorff dimension of the limit sets equal to the Hausdorff dimension of the Julia sets of the corresponding semihyperbolic GPLs.     

Area of Julia sets of holomorphic self-maps onC *

It is a general problem to study the measure of Julia sets. There are a lot of results for rational and entire functions. In this note, we describe the measure of Julia set for some holomorphic self-maps onC ^*. We'll prove thatJ(f) has positive area, wheref:C ^*→C ^*,f(z)=z ^m c ^P(z)+Q(1/z) ,P(z) andQ(z) are monic polynomials of degreed, andm is an integer.     

Quasi-Homeomorphisms and Measures of Finite Energy Integrals of Generalized Dirichlet Forms

We introduce the quasi-homeomorphisms of generalized Dirichlet forms and prove that any quasi-regular generalized Dirichlet form is quasi-homeomorphic to a semi-regular generalized Dirichlet form. Moreover, we apply this quasi-homeomorphism method to study the measures of finite energy integrals of generalized Dirichlet forms. We show that any 1-coexcessive function which is dominated by a function in is associated with a measure of finite energy integral. Consequently, we prove that a Borel set B is ɛ-exceptional if and only if μ (B) = 0 for any measure μ of finite energy integral. Received May 28, 1999, Revised September 8, 1999, Accepted December 10, 1999     

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 geometry of the Julia sets of McMullen maps

We study the quasisymmetric geometry of the Julia sets of McMullen maps f_λ(z) = z^m + λ/z^?, where λ ∈ ? {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 suffcient 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.     

External rays to periodic points

We prove that for every polynomial-like holomorphic mapP, ifaεK (filled-in Julia set) and the componentK _aofK containinga is either a point ora is accessible along a continuous curve from the complement ofK andK _ais eventually periodic, thena is accessible along an external ray. Ifa is a repelling or parabolic periodic point, then the set of arguments of the external rays converging toa is a nonempty closed “rotation set”, finite (ifK _ais not a one point) or Cantor minimal containing a pair of arguments of external rays of a critical point in ℂ. In the Appendix we discuss constructions via cutting and glueing, fromP to its external map with a “hedgehog”, and backward. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis, sponsored by the Minerva Foundation (Germany). Supported by the Polish KBN Grants 210469101 “Iteracje i Fraktale” and 210909101 “Uklady Dynamiczne”.     

Real analyticity of topological pressure for parabolically semihyperbolic generalized polynomial-like maps

For arbitrary parabolically semihyperbolic generalized polynomial-like maps f, we prove that on a certain interval, which contains the interval (0, HD(J(f))), the pressure function t ? P(−t log |f|) is real-analytic. Our results generalize the work of Makarov and Smirnov in [3] and [7].     

Conservative Exact Rational Maps of the Sphere

In this paper we investigate conditions for conservativity and exactness ofd-to-1 endomorphisms. We prove that all rational maps of degree greater than 1 are either conservative and exact with respect to normalized Lebesgue measure on the sphere, or have an attracting property inside the Julia set. We provide examples of rational maps which can be shown to be conservative and exact by using the previous result which was not formerly known to have this property.     

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.