Periodic expansiveness of smooth surface diffeomorphisms and applications

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