Physical measures for partially hyperbolic diffeomorphisms with mixed hyperbolicity * *Y Cao is partially supported by National Key R&D Program of China(2022YFA1005802) and NSFC 12371194. Z Mi is partially supported by National Key R&D Program of China(2022YFA1007800) and NSFC 12271260. J Yang was partially supported by CNPq, FAPERJ, PRONEX and NNSFC Grants 12271538, 12071202 and 12161141002.

Abstract

We study the partially hyperbolic diffeomorphisms whose centre direction admits the u-definite property in the sense that all the central Lyapunov exponents of each ergodic Gibbs u-state are either all positive or all negative. We prove that for this kind of partially hyperbolic diffeomorphisms, there are finitely many physical measures, whose basins cover a full Lebesgue measure subset of the ambient space. We also provide examples of diffeomorphisms whose centres are u-definite, where the supports of different ergodic Gibbs u-states have nontrivial intersections.