Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

Ergodic properties of semi-hyperbolic functions with polynomial Schwarzian derivative

Volker Mayera1 and Mariusz Urbańskia2

a1 Université de Lille I, UFR de Mathématiques, UMR 8524 du CNRS, 59655 Villeneuve d'Ascq Cedex, France (volker.mayer@math.univ-lille1.fr)

a2 Department of Mathematics, University of North Texas, Denton, TX 76203-1430, USA (urbanski@unt.edu)

Abstract

The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative are investigated under the condition that the function is semi-hyperbolic, i.e. the asymptotic values of the Fatou set are in attracting components and the asymptotic values in the Julia set are boundedly non-recurrent. We first show the existence, uniqueness, conservativity and ergodicity of a conformal measure m with minimal exponent h; furthermore, we show weak metrical exactness of this measure. Then we prove the existence of a σ-finite invariant measure μ absolutely continuous with respect to m. Our main result states that μ is finite if and only if the order ρ of the function f satisfies the condition h > 3ρ/(ρ+1). When finite, this measure is shown to be metrically exact. We also establish a version of Bowen's Formula, showing that the exponent h equals the Hausdorff dimension of the Julia set of f.

(Received November 03 2007)

Keywords

  • holomorphic dynamics;
  • Hausdorff dimension;
  • meromorphic functions

2010 Mathematics subject classification

  • Primary 30D05