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Rigidity and Absolute Continuity of Foliations of Anosov Endomorphisms

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Abstract

We found a dichotomy involving rigidity and measure of maximal entropy of a \(C^{\infty }\)-special Anosov endomorphism of the 2-torus. Considering \(\widetilde{m} \) the measure of maximal entropy of a \(C^{\infty }\)-special Anosov endomorphism of the 2-torus, either \(\widetilde{m}\) satisfies the Pesin formula (in this case we get smooth conjugacy with the linearization) or there is a set Z,  such that \(\widetilde{m}(Z) = 1,\) but Z intersects every unstable leaf on a set of zero measure of the leaf. Also, we can characterize the absolute continuity of the intermediate foliation for a class of volume-preserving special Anosov endomorphisms of \(\mathbb {T}^3\).

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All relevant data supporting the findings of the manuscript “Rigidity and Absolute Continuity of Foliations of Anosov Endomorphisms” can be found in the references included within the manuscript. Moreover all papers and preprints quoted in the manuscript can be consulted from the following websites:

\(\bullet \)https://mathscinet.ams.org/mathscinet

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\(\bullet \)https://paperzz.com/doc/7088645/absolute-continuity-of-foliations-1.-introduction-in-what

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Funding

This research had financial support from NNSFC 12071202.

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  1. Instituto de Matemática e Computação (IMC), UNIFEI, Itajubá, MG, Brazil

    Fernando Micena

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  1. Fernando Micena
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F. Micena wrote and reviewed the manuscript.

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Correspondence to Fernando Micena.

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The author thanks Professors Raúl Ures and Jana Rodriguez-Hertz for their hospitality during the visit to the Department of Mathematics of SUSTech and all their support for developing this research. The author also thanks the mathematical SUSTech group, the hospitality of the department, and the wonderful work environment. This research had financial support from NNSFC 12071202.

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Micena, F. Rigidity and Absolute Continuity of Foliations of Anosov Endomorphisms. J Dyn Diff Equat (2024). https://doi.org/10.1007/s10884-024-10350-1

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