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The maximal entropy measure of Fatou boundaries

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The second author was partially supported by NSF grant DMS-1500817

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  • We look at the maximal entropy measure (MME) of the boundaries of connected components of the Fatou set of a rational map of degree $≥ 2$. We show that if there are infinitely many Fatou components, and if either the Julia set is disconnected or the map is hyperbolic, then there can be at most one Fatou component whose boundary has positive MME measure. We also replace hyperbolicity by the more general hypothesis of geometric finiteness.

    Mathematics Subject Classification: Primary: 37F10, 30D05; Secondary: 37A05.

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