Periodic expansiveness of smooth surface diffeomorphisms and applications

We prove that periodic asymptotic expansiveness introduced in [13] implies the equidistribution of periodic points with respect to measures of maximal entropy. Then following Yomdin's approach [50] we show by using semi-algebraic tools that C1 interval maps and C1 surface diffeomorphisms satisfy this expansiveness property respectively for repelling and saddle hyperbolic points with Lyapunov exponents uniformly away from zero.