## Institute for Mathematical Sciences

## Preprint ims93-3

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

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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