Periodic expansiveness of smooth surface diffeomorphisms and applications

Abstract

We prove that periodic asymptotic expansiveness implies the equidistribution of periodic points to measures of maximal entropy. Then following Yomdin’s approach we show by using semi-algebraic tools that $C^1$ interval maps and $C^1$ surface diffeomorphisms satisfy this expansiveness property respectively for repelling and saddle hyperbolic points with Lyapunov exponents uniformly away from zero.