An improvement of the $C^{1}$ closing lemma for endomorphisms

Abstract

L. Wen proved the $C^{1}$ closing lemma for endomorphisms with finitely many singularities. The arguments and the tools of Wen are available for endomorphisms which have infinitely many singularities but at most finitely many ones in the nonwandering sets. By refining the argument of Wen we prove the $C^{1}$ closing lemma for endomorphisms with finitely many singularities in the nonwandering sets. By using this lemma we can slightly improve characterization of $C^{1}$ absolutely $\Omega$-stable endomorphisms. That is, for an endomorphism $f$ with finitely many singularities in the nonwandering set, $f$ is $C^{1}$ absolutely $\Omega$-stable if and only if $f$ has a neighborhood $\mathscr{U}$ such that every $g$ in $\mathscr{U}$ satisfies weak Axiom A.