On $\mathrm{C}^1$-class local diffeomorphisms whose periodic points are nonuniformly expanding

TL;DR

This paper proves that a $C^1$ local diffeomorphism on a closed manifold is uniformly expanding on the closure of its periodic points if it exhibits nonuniform expansion on those points, using a sifting-shadowing technique.

Contribution

It introduces a novel method combining sifting-shadowing to establish uniform expansion from nonuniform expansion on periodic points.

Findings

Uniform expansion on the closure of periodic points

Extension of nonuniform expansion to uniform expansion

Application of sifting-shadowing technique

Abstract

Using a sifting-shadowing combination, we prove in this paper that an arbitrary $\mathrm{C}^1$-class local diffeomorphism $f$ of a closed manifold $M^n$ is uniformly expanding on the closure $\mathrm{Cl}_{M^n}(\mathrm{Per}(f))$ of its periodic point set $\mathrm{Per}(f)$, if it is nonuniformly expanding on $\mathrm{Per}(f)$.