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

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 ${\mathrm{C}\mathrm{l}}_{M^n}(\mathrm{P}\mathrm{e}\mathrm{r}(f))$ of its periodic point set $\mathrm{P}\mathrm{e}\mathrm{r}(f)$, if it is nonuniformly expanding on $\mathrm{P}\mathrm{e}\mathrm{r}(f)$.