# 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 and Continuous Dynamical Systems, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192
##### References:
 [1] A. Beardon, Iteration of Rational Functions, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4422-6. [2] C. Curry, J. Mayer, J. Meddaugh and J. Rogers, Any counterexample to Makienko's conjecture is an indecomposable continuum, Ergodic Theory and Dynam. Sys., 29 (2009), 875-883.  doi: 10.1017/S014338570800059X. [3] C. Curry, J. Mayer and E. Tymchatyn, Topology and measure of buried points in Julia sets, Fund. Math., 222 (2013), 1-17.  doi: 10.4064/fm222-1-1. [4] A. Freire, A. Lopes and R. Mañé, An invariant measure for rational maps, Bol. Soc. Brasil. Mat., 14 (1983), 45-62.  doi: 10.1007/BF02584744. [5] J. Hawkins, Lebesgue ergodic rational maps in parameter space, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1423-1447.  doi: 10.1142/S021812740300731X. [6] M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory and Dynam. Sys., 3 (1983), 351-385.  doi: 10.1017/S0143385700002030. [7] C. McMullen, Complex Dynamics and Renormalization, Princeton Univ. Press, 1994. [8] J. Milnor, Dynamics in One Complex Variable (3rd ed.), Princeton Univ. Press, 2006. [9] S. Morosawa, On the residual Julia sets of rational functions, Ergodic Theory and Dynam. Sys., 17 (1997), 205-210.  doi: 10.1017/S0143385797069848. [10] S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge Univ. Press, 2000. [11] J. Qiao, Topological complexity of Julia sets, Sci. China Ser. A, 40 (1997), 1158-1165.  doi: 10.1007/BF02931834. [12] L. Tan and Y. Yin, Local connectivity of the Julia set for geometrically finite rational maps, Sci. China Ser. A, 39 (1996), 39-47.

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##### References:
 [1] A. Beardon, Iteration of Rational Functions, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4422-6. [2] C. Curry, J. Mayer, J. Meddaugh and J. Rogers, Any counterexample to Makienko's conjecture is an indecomposable continuum, Ergodic Theory and Dynam. Sys., 29 (2009), 875-883.  doi: 10.1017/S014338570800059X. [3] C. Curry, J. Mayer and E. Tymchatyn, Topology and measure of buried points in Julia sets, Fund. Math., 222 (2013), 1-17.  doi: 10.4064/fm222-1-1. [4] A. Freire, A. Lopes and R. Mañé, An invariant measure for rational maps, Bol. Soc. Brasil. Mat., 14 (1983), 45-62.  doi: 10.1007/BF02584744. [5] J. Hawkins, Lebesgue ergodic rational maps in parameter space, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1423-1447.  doi: 10.1142/S021812740300731X. [6] M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory and Dynam. Sys., 3 (1983), 351-385.  doi: 10.1017/S0143385700002030. [7] C. McMullen, Complex Dynamics and Renormalization, Princeton Univ. Press, 1994. [8] J. Milnor, Dynamics in One Complex Variable (3rd ed.), Princeton Univ. Press, 2006. [9] S. Morosawa, On the residual Julia sets of rational functions, Ergodic Theory and Dynam. Sys., 17 (1997), 205-210.  doi: 10.1017/S0143385797069848. [10] S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge Univ. Press, 2000. [11] J. Qiao, Topological complexity of Julia sets, Sci. China Ser. A, 40 (1997), 1158-1165.  doi: 10.1007/BF02931834. [12] L. Tan and Y. Yin, Local connectivity of the Julia set for geometrically finite rational maps, Sci. China Ser. A, 39 (1996), 39-47.
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