  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 $C^\infty$ 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.
