Expanding repellers in limit sets for iterations of holomorphic functions

Volume 186 / 2005

Feliks Przytycki 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.

Authors

  • Feliks PrzytyckiInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-956 Warszawa, Poland
    e-mail

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