Expanding repellers in limit sets for iterations of holomorphic functions

Feliks Przytycki¹

¹ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland

Fundamenta Mathematicae, Tome 186 (2005) no. 1, pp. 85-96

Résumé

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.

Détail

DOI : 10.4064/fm186-1-7

Keywords: prove mit omega being immediate basin attraction attracting fixed point rational mapping riemann sphere ergodic invariant measure boundary mathop mit omega positive lyapunov exponent there invariant subset mathop mit omega which expanding repeller hausdorff dimension arbitrarily close hausdorff dimension prove generalizations geometric coding tree abstract version paper continuation paper fund math author anna zdunik where density periodic orbits mathop mit omega proved