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

Abstract

We consider partially hyperbolic diffeomorphisms f 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 + C^{1+} diffeomorphism near f f in the C 1 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.