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On the unstable directions and Lyapunov exponents of Anosov endomorphisms

Volume 235 / 2016

Fernando Micena, Ali Tahzibi Fundamenta Mathematicae 235 (2016), 37-48 MSC: Primary 37-XX; Secondary 37D20. DOI: 10.4064/fm92-10-2015 Published online: 23 March 2016

Abstract

Unlike in the invertible setting, Anosov endomorphisms may have infinitely many unstable directions. Here we prove, under the transitivity assumption, that an Anosov endomorphism of a closed manifold $M$ is either special (that is, every $x \in M$ has only one unstable direction), or for a typical point in $M$ there are infinitely many unstable directions. Another result is the semi-rigidity of the unstable Lyapunov exponent of a $C^{1+\alpha }$ codimension one Anosov endomorphism that is $C^1$-close to a linear endomorphism of $\mathbb {T}^n$ for $(n \geq 2).$

Authors

  • Fernando MicenaDepartamento de Matemática
    IM-UFAL
    Maceió, AL, Brazil
    e-mail
  • Ali TahzibiDepartamento de Matemática
    ICMC-USP
    São Carlos, SP, Brazil
    e-mail

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