On positive Lyapunov exponents along $E^{cu}$ and non-uniformly expanding for partially hyperbolic systems

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.