A forward Ergodic Closing Lemma and the Entropy Conjecture for nonsingular endomorphisms away from tangencies

Abstract

We prove a Ergodic Closing Lemma for nonsingular \begin{document}$ C^1 $\end{document} endomorphisms, claiming that the set of eventually strongly closable points is a total probability set. The forward means that the closing perturbation is involved along a finite part of the orbit of a point in a total probability set, which is the same perturbation as in Mane's Ergodic Closing Lemma for \begin{document}$ C^1 $\end{document} diffeomorphisms. As an application, Shub's Entropy Conjecture for nonsingular \begin{document}$ C^1 $\end{document} endomorphisms away from homoclinic tangencies is proved, extending the result for \begin{document}$ C^1 $\end{document} diffeomorphisms by Liao, Viana and Yang.