Periodic expansiveness of smooth surface diffeomorphisms and applications

Abstract

We prove that periodic asymptotic expansiveness introduced in [13] implies the equidistribution of periodic points along measures of maximal entropy. Then following Yomdin's approach [50] we show by using semi-algebraic tools that $C^{\infty }$ interval maps and $C^{\infty }$ surface diffeomorphisms satisfy this expansiveness property respectively for repelling and saddle hyperbolic points with Lyapunov exponents uniformly away from zero.