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 - A, 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.  Google Scholar

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C. CurryJ. MayerJ. 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.  Google Scholar

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A. FreireA. Lopes and R. Mañé, An invariant measure for rational maps, Bol. Soc. Brasil. Mat., 14 (1983), 45-62.  doi: 10.1007/BF02584744.  Google Scholar

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J. Hawkins, Lebesgue ergodic rational maps in parameter space, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1423-1447.  doi: 10.1142/S021812740300731X.  Google Scholar

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J. Milnor, Dynamics in One Complex Variable (3rd ed.), Princeton Univ. Press, 2006.  Google Scholar

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S. Morosawa, On the residual Julia sets of rational functions, Ergodic Theory and Dynam. Sys., 17 (1997), 205-210.  doi: 10.1017/S0143385797069848.  Google Scholar

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S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge Univ. Press, 2000.  Google Scholar

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J. Qiao, Topological complexity of Julia sets, Sci. China Ser. A, 40 (1997), 1158-1165.  doi: 10.1007/BF02931834.  Google Scholar

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L. Tan and Y. Yin, Local connectivity of the Julia set for geometrically finite rational maps, Sci. China Ser. A, 39 (1996), 39-47.   Google Scholar

show all references

References:
[1]

A. Beardon, Iteration of Rational Functions, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4422-6.  Google Scholar

[2]

C. CurryJ. MayerJ. 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.  Google Scholar

[3]

C. CurryJ. 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.  Google Scholar

[4]

A. FreireA. Lopes and R. Mañé, An invariant measure for rational maps, Bol. Soc. Brasil. Mat., 14 (1983), 45-62.  doi: 10.1007/BF02584744.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

C. McMullen, Complex Dynamics and Renormalization, Princeton Univ. Press, 1994.  Google Scholar

[8]

J. Milnor, Dynamics in One Complex Variable (3rd ed.), Princeton Univ. Press, 2006.  Google Scholar

[9]

S. Morosawa, On the residual Julia sets of rational functions, Ergodic Theory and Dynam. Sys., 17 (1997), 205-210.  doi: 10.1017/S0143385797069848.  Google Scholar

[10]

S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge Univ. Press, 2000.  Google Scholar

[11]

J. Qiao, Topological complexity of Julia sets, Sci. China Ser. A, 40 (1997), 1158-1165.  doi: 10.1007/BF02931834.  Google Scholar

[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.   Google Scholar

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