On positive Lyapunov exponents along $E^{cu}$ and non-uniformly expanding for partially hyperbolic systems (2112.11149v3)
Abstract: In this paper we consider $C{1}$ diffeomorphisms on compact Riemannian manifolds of any dimension that admit a dominated splitting $E{cs} \oplus E{cu}.$ We prove that if the Lyapunov exponents along $E{cu}$ are positive for Lebesgue almost every point, then a map $f$ is non-uniformly expanding along $E{cu}$ under the assumption that the cocycle $Df_{|E{cu}(f)}{-1}$ has a dominated splitting with index 1 on the support of an ergodic Lyapunov maximizing observable measure. As a result, there exists a physical SRB measure for a $C{1+\alpha}$ diffeomorphism map $f$ that admits a dominated splitting $E{s} \oplus E{cu}$ under assumptions that $f$ has non-zero Lyapunov exponents for Lebesgue almost every point and that the cocycle $Df_{|E{cu}(f)}{-1}$ has a dominated splitting with index 1 on the support of an ergodic Lyapunov maximizing observable measure.