In this talk we show that the positive Lyapunov exponents with a uniform 1-gap property imply non-uniformly expanding for partially hyperbolic systems, which provides an affirmative answer to a question posed by Alves, Bonatti, and Viana (Invent. Math. 140(2): 351-398, 2000). As a result, we show that there exists a physical SRB measure for a $C^{1+\alpha}$ diffeomorphism map $f$ that admits a dominated splitting under assumptions that $f$ has non-zero Lyapunov exponents for Lebesgue almost every point and the Lyapunov spectrum has a uniform 1-gap property.

Meeting ID: 852 4277 3200 Passcode: 103121