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<p>In this paper we prove that for topologically mixing metric Anosov flows their equilibrium states corresponding to Hölder potentials satisfy a strong rigidity property: they are determined only by their disintegrations on (strong) stable or unstable leaves. As a consequence we deduce: the corresponding horocyclic foliations of such systems are uniquely quasi-ergodic, provided that the corresponding Jacobian is Hölder, without any restriction on the dimension of the invariant distributions. This gives another proof of a result of Babillott-Ledrappier.</p>

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Rigidity of Equilibrium States and Unique Quasi-Ergodicity for Horocyclic Foliations

  • 期刊论文

  • 发表日期：

    2025年2月8日

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  • Pablo D. Carrasco,
  • Federico Rodriguez-Hertz

摘要

In this paper we prove that for topologically mixing metric Anosov flows their equilibrium states corresponding to Hölder potentials satisfy a strong rigidity property: they are determined only by their disintegrations on (strong) stable or unstable leaves. As a consequence we deduce: the corresponding horocyclic foliations of such systems are uniquely quasi-ergodic, provided that the corresponding Jacobian is Hölder, without any restriction on the dimension of the invariant distributions. This gives another proof of a result of Babillott-Ledrappier.