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 (

a2 Department of Mathematics, University of North Texas, Denton, TX 76203-1430, USA (


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)


  • holomorphic dynamics;
  • Hausdorff dimension;
  • meromorphic functions

2010 Mathematics subject classification

  • Primary 30D05