# Periodic expansiveness of smooth surface diffeomorphisms and applications

## TL;DR

This paper demonstrates that periodic asymptotic expansiveness leads to the equidistribution of periodic points for smooth surface diffeomorphisms and interval maps, using semi-algebraic tools and hyperbolic dynamics.

## Contribution

It establishes a link between periodic asymptotic expansiveness and equidistribution of periodic points, extending Yomdin's approach to smooth surface diffeomorphisms and interval maps.

## Key findings

- Periodic asymptotic expansiveness implies equidistribution of periodic points.
- Smooth surface diffeomorphisms exhibit this property at saddle hyperbolic points.
- C^ interval maps satisfy this property at repelling points.

## Abstract

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.