ERGODIC PROPERTIES OF SEMI-HYPERBOLIC FUNCTIONS WITH POLYNOMIAL SCHWARZIAN DERIVATIVE

作者：Mayer Volker^*; Urbanski Mariusz

来源：Proceedings of the Edinburgh Mathematical Society, 2010, 53: 471-502.

DOI：10.1017/S0013091507001332

• holomorphic dynamics
• Hausdorff dimension
• meromorphic functions

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 mid 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 sigma-finite invariant measure mu absolutely continuous with respect to in. Our main result states that mu is finite if and only if the order rho of the function f satisfies the condition h > 3 rho/(rho+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.

• 出版日期2010-6