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.