Ergodic Theory and Dynamical Systems



Geometry and ergodic theory of conformal non-recurrent dynamics


MARIUSZ URBANSKI a1
a1 Department of Mathematics, University of North Texas, Denton, TX 76203-5116, USA (e-mail: urbanski@unt.edu)

Abstract

Let $h$ be the Hausdorff dimension of the Julia set of a rational function $T$ with no non-periodic recurrent critical points and let $m$ be the only $h$-conformal measure for $T$. We prove the existence of a $\sigma$-finite $T$-invariant measure $\mu$ equivalent with $m$. The measure $\mu$ is then proved to be ergodic and conservative and we study the set of those points whose all open neighborhoods have infinite measure $\mu$. Developing the concept of the inverse jump transformation we show that the packing and Hausdorff dimensions of the conformal measure are equal to $h$. We also provide some sufficient conditions for Hausdorff and box dimensions of the Julia set to be equal.

(Received August 12 1995)
(Revised May 3 1996)