Attractors for graph critical rational maps

Blokh, Alexander; Misiurewicz, Michał

doi:10.1090/S0002-9947-02-02999-9

Attractors for graph critical rational maps
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by Alexander Blokh and Michał Misiurewicz PDF

Trans. Amer. Math. Soc. 354 (2002), 3639-3661 Request permission

Abstract:

We call a rational map $f$ graph critical if any critical point either belongs to an invariant finite graph $G$, or has minimal limit set, or is non-recurrent and has limit set disjoint from $G$. We prove that, for any conformal measure, either for almost every point of the Julia set $J(f)$ its limit set coincides with $J(f)$, or for almost every point of $J(f)$ its limit set coincides with the limit set of a critical point of $f$.

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