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.
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