Finiteness of physical measures for diffeomorphisms with multi 1-D centers

Yongluo Cao; Zeya Mi

paper

Finiteness of physical measures for diffeomorphisms with multi 1-D centers

arXiv:2306.06575

Abstract

Let $f$ be a $C^{2}$ diffeomorphism on compact Riemannian manifold $M$ with partially hyperbolic splitting $T M = E^{u} \oplus E_{1}^{c} \oplus \dots \oplus E_{k}^{c} \oplus E^{s},$ where $E^{u}$ is uniformly expanding, $E^{s}$ is uniformly contracting, and $dim E_{i}^{c} = 1, 1 \leq i \leq k, k \geq 1$ . We prove the finiteness of ergodic physical(SRB) measures of $f$ under the hyperbolicity of Gibbs $u$ -states.