Anna Zdunik

Parabolic orbifolds and the dimension of the maximal measure for rational maps

Let f : G--*G be a rational map of the Riemann sphere, d e g ( f ) > 2 . A natural invariant measure m the measure of maximal entropy was constructed by Ljubich [Lju] and independently by Freire, Lopes and Mafi6 [FLM]. The aim of this paper is to compare this measure with some Hausdorff measures. First recall the following definition. For a probabili ty measure v on (or, more generally, on a smooth manifold) the Hausdorff dimension of v is defined by a formula HD(v) = inf HD(Y) Y : v ( Y ) = l

Hyperbolic dimension of Julia sets of meromorphic maps with logarithmic tracts

We prove that for meromorphic maps with logarithmic tracts (in particular, for transcendental maps in the class , which are entire or meromorphic with a finite number of poles), the Julia set contains a compact invariant hyperbolic Cantor set of Hausdorff dimension greater than 1. Hence, the hyperbolic dimension of the Julia set is greater than 1.

Geometry and ergodic theory of non-hyperbolic exponential maps

We deal with all the maps from the exponential family {λe z } such that the orbit of zero escapes to infinity sufficiently fast. In particular all the parameters A ∈ (1/e, +∞) are included. We introduce as our main technical devices the projection F λ of the map f λ to the infinite cylinder Q = C/2πiZ and an appropriate conformal measure m. We prove that J r (F λ ), essentially the set of points in Q returning infinitely often to a compact region of Q disjoint from the orbit of 0 ∈ Q, has the Hausdorff dimension h λ ∈ (1,2), that the h λ -dimensional Hausdorff measure of J r (F λ ) is positive and finite, and that the h λ -dimensional packing measure is locally infinite at each point of J r (F λ ). We also prove the existence and uniqueness of a Borel probability F λ -invariant ergodic measure equivalent to the conformal measure m. As a byproduct of the main course of our considerations, we reprove the result obtained independently by Lyubich and Rees that the ω-limit set (under f λ ) of Lebesgue almost every point in C, coincides with the orbit of zero under the map f λ . Finally we show that the the function A → h λ , λ ∈ (1/e, +∞), is continuous.

Harmonic measure versus Hausdorff measures on repellers for holomorphic maps

This paper is a continuation of a joint paper of the author with F. Przytycki and M. Urbanski. We study a harmonic measure on a boundary of so-called repelling boundary domain; an important example is a basin of a sink for a rational map. Using the results of the above-mentioned paper we prove that either the boundary of the domain is an analytically embedded circle or interval, or else the harmonic measure is singular with respect to the HausdorS measure corresponding to the function Xc(t) = t exp(c/log t logloglog 1 ) for some c > 0. 0. INTRODUCTION AND STATEMENT OF RESULTS Let Q be a simply connected domain in the Riemann sphere ¢:, card(¢: Q) > 2. Let R: D Q be the Riemann map from the unit disc D onto Q. Since R has nontangential limits almost everywhere on S1 = AD [D], the image R*l of the length measure I on S1 can be considered. This measure (denoted usually by co and called a harmonic measure on AQ ) has been studied a long time (see [Mk2 and PUZ] for the corresponding references), with the most interesting results obtained recently in two remarkable papers, [Mkl and Mk2]. For an increasing function 0: Di+ > Di+, 0(0) = O we define an outer measure for A c C: A¢,(A) = lim inf { (diamBj) } , where the infimum is taken over all coverings of A with balls of a diameter smaller than a (the diameter is computed with respect to the spherical metrics on ¢: ). In particular, for 0(t) = ta, we obtain the ol-dimensional Hausdorff measure. Ata will be denoted by Aa . Let ,u be a probability measure on Borel subsets of ¢: . Hausdorff dimension of ,u is defined as

The parabolic map f1/e(z)=(1/e)ez

Abstract We consider the exponential maps ƒ λ : ℂ → ℂ defined by the formula ƒ λ ( z ) = λ e z , λ(0,1/ e ]. Let J r (ƒ λ ) be the subset of the Julia set consisting of points that do not escape to infinity under forward iterates of ƒ. Our main result is that the function λ h λ :=HD( J r (ƒ λ ),)), λ(0, 1/ e ], is continuons at the point 1/ e . As a preparation for this result we deal with the map ƒ 1 / e itself. We prove that the h 1 / e -dimensional Hausdorff measure of J r (ƒ 1 / e ) is positive and finite on each horizontal strip, and that the h 1 / e -dimensional packing measure of J r (ƒ λ ) is locally infinite at each point of J r (ƒ λ ). Our main technical devices are formed by the, associated with ƒ λ , maps F λ defined on some strip P of height 2π and also associated with them tonformal measures.

Bowen’s formula for meromorphic functions

Let

MAXIMIZING MEASURES ON METRIZABLE NON-COMPACT SPACES

f

Real analyticity for random dynamics of transcendental functions

be an arbitrary transcendental entire or meromorphic function in the class

Dimension properties of the boundaries of exponential basins

\mathcal S

Conformal measures for meromorphic maps

(i.e. with finitely many singularities). We show that the topological pressure