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