The $C^1$ closing lemma for generic $C^1$ endomorphisms

Abstract

Given a compact $m$-dimensional manifold $M$ and $1⩽r⩽\infty$, consider the space $C^r(M)$ of self mappings of $M$. We prove here that for every map $f$ in a residual subset of $C^1(M)$, the $C^1$ closing lemma holds. In particular, it follows that the set of periodic points is dense in the nonwandering set of a generic $C^1$ map. The proof is based on a geometric result asserting that for generic $C^r$ maps the future orbit of every point in $M$ visits the critical set at most $m$ times.