The maximal entropy measure of Fatou boundaries

Jane Hawkins; Michael Taylor

doi:10.3934/dcds.2018192

Article Contents

2018, Volume 38, Issue 9: 4421-4431. Doi: 10.3934/dcds.2018192

This issue Previous Article Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models) Next Article Rescaled expansivity and separating flows

The maximal entropy measure of Fatou boundaries

Jane Hawkins^, and
Michael Taylor

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

* Corresponding author
* Corresponding author

Received: July 31, 2017

Published: August 2018

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

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Abstract

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.

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

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References

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