The C1 closing lemma for generic C1 endomorphisms

Given a compact m-dimensional manifold M and 1⩽r⩽∞, consider the space Cr(M) of self mappings of M. We prove here that for every map f in a residual subset of C1(M), the C1 closing lemma holds. In particular, it follows that the set of periodic points is dense in the nonwandering set of a generic C1 map. The proof is based on a geometric result asserting that for generic Cr maps the future orbit of every point in M visits the critical set at most m times.