A dichotomy for measures of maximal entropy near time-one maps of transitive Anosov flows

Abstract

We show that time-one maps of transitive Anosov flows of compact manifolds\nare accumulated by diffeomorphisms robustly satisfying the following dichotomy:\neither all of the measures of maximal entropy are non-hyperbolic, or there are\nexactly two ergodic measures of maximal entropy, one with a positive central\nexponent and the other with a negative central exponent.\n We establish this dichotomy for certain partially hyperbolic diffeomorphisms\nisotopic to the identity whenever both of their strong foliations are minimal.\nOur proof builds on the approach developed by Margulis for Anosov flows where\nhe constructs suitable families of measures on the dynamical foliations.\n

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