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

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 C¹ 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.