Symbolic extensions for nonuniformly entropy expanding maps

Abstract

Abstract. We call nonuniformly entropy expanding map any C 1 map defined on a compact manifold whose ergodic measures with positive entropy have only nonnegative Lyapunov exponents. We prove that a 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[14].

$\mathcal{C}^{2}$ surface diffeomorphisms have symbolic extensions

“…This conjecture was proved by T.Downarowicz and A.Maass in dimension one [15]. In [10] we prove the conjecture in any dimension up to a factor d and an additional term corresponding to the sum of the negative Lyapunov exponents of ν (then to apply the Estimate Theorem we are reduced to the class of nonuniformly entropy expanding maps because of this remaining term). Theorem 4 corresponds to the case of C 2 surface diffeomorphisms up to a factor 2.…”

“…This fact is still true for surface local diffeomorphisms. Indeed the author proved in [10] the following "local invertible" version of Ruelle-Margulis inequality :…”

“…Lemma 9 [10] Let M be a compact surface and let T : M → M be a local diffeomorphism. There exists ǫ > 0 such that for any ergodic measure ν, we have :…”

“…The results of T.Downarowicz and A.Maass has been extended by the author in any dimension to nonuniformly entropy expanding maps (i.e. C 1 maps whose ergodic measures with positive entropy have nonnegative Lyapunov exponent) of class C r with 1 < r < +∞ [10]. The existence of symbolic extensions for general C r maps with 1 < r < +∞ is still an open question.…”

Asymptotich-expansiveness rate ofC∞maps

“…In fact one has again in this case h loc (f, ε) ≤ v * 1 (f, 2ε) for any ε small enough. Indeed it was proved for local diffeomorphisms in [12] (Theorem 5) that there exists ε > 0 such that any invariant measure µ with h loc (µ, ε) > 0 has at least one negative Lyapunov exponent.…”