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Accessability of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps

Feliks Przytycki
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Abstract

We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, if $A$ is completely invariant (i.e. $f^{-1}(A)=A$), and if $\mu$ is an arbitrary $f$-invariant measure with positive Lyapunov exponents on the boundary of $A$, then $\mu$-almost every point $q$ in the boundary of $A$ is accessible along a curve from $A$. In fact we prove the accessability of every "good" $q$ i.e. such $q$ for which "small neighbourhoods arrive at large scale" under iteration of $f$. This generalizes Douady-Eremenko-Levin-Petersen theorem on the accessability of periodic sources.

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Feliks Przytycki. 1993-03-20. Accessability of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps. https://arxiv.org/abs/math/9303209

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