Expanding repellers in limit sets for iterations of holomorphic functions
Volume 186 / 2005
Fundamenta Mathematicae 186 (2005), 85-96
MSC: Primary 37F15; Secondary 37F35, 37D25.
DOI: 10.4064/fm186-1-7
Abstract
We prove that for ${\mit \Omega }$ being an immediate basin of attraction to an attracting fixed point for a rational mapping of the Riemann sphere, and for an ergodic invariant measure $\mu $ on the boundary $\mathop {\rm {Fr}}{\mit \Omega }$, with positive Lyapunov exponent, there is an invariant subset of $\mathop {\rm {Fr}}{\mit \Omega }$ which is an expanding repeller of Hausdorff dimension arbitrarily close to the Hausdorff dimension of $\mu $. We also prove generalizations and a geometric coding tree abstract version. The paper is a continuation of a paper in Fund. Math. 145 (1994) by the author and Anna Zdunik, where the density of periodic orbits in $\mathop {\rm {Fr}}{\mit \Omega }$ was proved.
