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Abstract

A nonuniformly entropy expanding map is any $\mathcal{C}^1$ map defined on a compact manifold whose ergodic measures with positive entropy have only nonnegative Lyapunov exponents. We prove that a $\mathcal{C}^r$ nonuniformly entropy expanding map $T$ with $r>1$ has a symbolic extension and we give an explicit upper bound of the symbolic extension entropy in terms of the positive Lyapunov exponents by following the approach of T. Downarowicz and A. Maass [Invent. Math. 176 (2009)].