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A relation between entropy and transitivity of Anosov diffeomorphisms

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Abstract

It is known that transitive Anosov diffeomorphisms have a unique measure of maximal entropy (MME). Here we discuss the converse question. Transitivity of Anosov diffeomorphisms can be reached under suitable hypotheses on Lyapunov exponents on the set of periodic points and the structure of the MME. In another way, assuming together the uniqueness of MME and that every point is regular, in Oseledec’s Theorem sense, also we can get the transitivity of Anosov diffeomorphisms in this setting.

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Authors and Affiliations

  1. FPM- Instituto de Matemática e Computação, IMC-UNIFEI, Itajubá, MG, Brazil

    Fernando Pereira Micena

  2. JRH- SUSTech International Center for Mathematics, Department of Mathematics, Southern University of Science and Technology of China, No 1088, Xueyuan Rd., Xili, Nanshan District, Shenzhen, Guangdong, 518055, China

    Jana Rodriguez-Hertz

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  1. Fernando Pereira Micena
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  2. Jana Rodriguez-Hertz
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Correspondence to Fernando Pereira Micena.

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FPM was partially supported by NNSFC 12071202 and JRH was supported by NSFC 11871262, NSFC 11871394, NSFC 12161141002, NSFC 12250710130.

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Micena, F.P., Rodriguez-Hertz, J. A relation between entropy and transitivity of Anosov diffeomorphisms. J Dyn Control Syst 29, 1241–1249 (2023). https://doi.org/10.1007/s10883-022-09620-2

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  • DOI: https://doi.org/10.1007/s10883-022-09620-2

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