# American Institute of Mathematical Sciences

September  2018, 38(9): 4421-4431. doi: 10.3934/dcds.2018192

## The maximal entropy measure of Fatou boundaries

 Mathematics Dept. CB #3250, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA

* Corresponding author

Received  August 2017 Published  June 2018

Fund Project: The second author was partially supported by NSF grant DMS-1500817.

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.

Citation: Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192
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