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April  2020, 40(4): 2285-2313. doi: 10.3934/dcds.2020114

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

 Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo, Japan

Received  June 2019 Revised  November 2019 Published  January 2020

We prove a forward Ergodic Closing Lemma for nonsingular $C^1$ 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 forward orbit of a point in a total probability set, which is the same perturbation as in Mañé's Ergodic Closing Lemma for $C^1$ diffeomorphisms. As an application, Shub's Entropy Conjecture for nonsingular $C^1$ endomorphisms away from homoclinic tangencies is proved, extending the result for $C^1$ diffeomorphisms by Liao, Viana and Yang.

Citation: Shuhei Hayashi. A forward Ergodic Closing Lemma and the Entropy Conjecture for nonsingular endomorphisms away from tangencies. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2285-2313. doi: 10.3934/dcds.2020114
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