Conformal,geometric and invariant measures for transcendental expanding functions

We describe the fractal structure of expanding meromorphic maps of the form , where H and Q are rational functions whose most transparent examples are among the functions of the form with . In particular we show that depending upon whether the Hausdorff dimension of the Julia set is greater or less than 1, the finite non-zero geometric measure is provided by the Hausdorff or packing measure. In order to describe this fractal structure we introduce and explore in detail Walters expanding conformal maps and jump-like conformal maps. Received: 3 May 2001 / Published online: 5 September 2002     

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.     

Gibbs state for some class of meromorphic functions

This paper is a continuation of our earlier works [1,2] on the fractal structure of expanding and subexpanding meromorphic functions of the form F = H o exp o Q, where H and Q are non-constant rational maps. Under some assumptions on the forward trajectories of asymptotic values ofF we define a class of summable potentials for the maps f of the punctured cylinder induced by F. We prove the existence and uniqueness of Gibbs states for these potentials.     

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.     

Hausdorff Dimension and Hausdorff Measures of Julia Sets of Elliptic Functions

It is proved here that if is an elliptic function and q is the maximal multiplicity ofall poles of f, then the Hausdorff dimension of the Julia setof f is greater than 2 q/(q + 1), and the Hausdorff dimensionof the set of points that escape to infinity is less than orequal to 2q/(q + 1). In particular, the area of this latterset is equal to 0. 2000 Mathematics Subject Classification 37F35(primary); 37F10, 30D30 (secondary).     

Existence of invariant measures for transcendental subexpanding functions

We consider the problem of the existence of absolutely continuous invariant measures for transcendental meromorphic functions. We prove sufficient conditions for a subexpanding meromorphic function f to have a C-finite absolutely continuous invariant measure 7 and we find a class of functions satisfying these assumptions.     

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.     

Geometric rigidity of transcendental meromorphic functions

The classes of dynamically and geometrically tame functions meromorphic outside a small set are introduced. The Julia sets of geometrically tame functions are proven to be either geometrical circle (in ) or to have Hausdorff dimension strictly larger than 1. Vast classes of dynamically and geometrically tame functions are identified. The research of both authors was supported in part by the Polish KBN Grant No 2 PO3A 034 25, the Warsaw University of Technology Grant no 504G 11200043000 and by the NSF/PAN grant INT-0306004. The research of the second author was supported in part by the NSF Grant DMS 0400481.