Finite measures of maximal entropy for an open set of partially hyperbolic diffeomorphisms

Mongez, Juan; Pacifico, Maria

doi:10.1090/tran/9230

Finite measures of maximal entropy for an open set of partially hyperbolic diffeomorphisms
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by Juan Carlos Mongez and Maria Jose Pacifico;

Trans. Amer. Math. Soc. 377 (2024), 8695-8720

DOI: https://doi.org/10.1090/tran/9230

Published electronically: August 9, 2024

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Abstract:

We consider partially hyperbolic diffeomorphisms $f$ with a one-dimensional central direction such that the unstable entropy is different from the stable entropy. Our main result proves that such maps have a finite number of ergodic measures of maximal entropy. Moreover, any $C^{1+}$ diffeomorphism near $f$ in the $C^1$ topology possesses at most the same number of ergodic measures of maximal entropy. These results extend the findings in Buzzi, Crovisier, and Sarig [Ann. of Math. (2) 195 (2022), pp. 421–508] to arbitrary dimensions and provides an open class of non-Axiom A systems of diffeomorphisms exhibiting a finite number of ergodic measures of maximal entropy. We believe our technique, essentially distinct from the one in Buzzi et al., is robust and may find applications in further contexts.

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Finite measures of maximal entropy for an open set of partially hyperbolic diffeomorphisms
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Abstract: