Backward volume contraction for endomorphisms with eventual volume expansion

Abstract

We consider smooth maps on compact Riemannian manifolds. We prove that under some mild condition of eventual volume expansion Lebesgue almost everywhere we have uniform backward volume contraction on every pre-orbit of Lebesgue almost every point. To cite this article: J.F. Alves et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).  2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Contraction en arrière pour des endomorphisms en expansion. Nous considérons des transformations différentiables sur des varietés Riemannienes compactes. Nous montrons que dans une certaine condition modérée d'expansion de volume nous pouvons déduire que pour Lebesgue presque chaque point nous avons contraction uniforme de volume en arrière de chaque pré-orbite. Pour citer cet article :

Continuity of SRB measure and entropy for Benedicks–Carleson quadratic maps

“…Theorem A alone, also allows us to apply corollary 1.2 from [ACP04] to obtain backward volume contraction.…”

“…We also refer to the recent work [ACP04] from which we conclude, by the non-uniformly expanding character of these maps, that for almost every x ∈ I and any y on a pre-orbit of x, one has an exponential growth of the derivative of y.…”

Adapted random perturbations for non-uniformly expanding maps

“…The choices of U n and the sequence b n ensure that n≥n 0 n−1 j=0 λ f j (q −1 (n)) < ∞; see [5,Section 3]. Moreover, in the one-dimensional setting, this series coincides with the one in the statement of the lemma.…”

“…Hence there exists β > 0 and N ∈ N such that b n = n β satisfies b n ≤ min{a n , λ(W n ) −ǫ } for all n ≥ N and some 0 < ǫ < p−3 p−1 . In addition, we clearly have b n b k ≥ b k+n for all big enough k, n ∈ N. In this setting, U n = {x ∈ M : | det D f n (x)| ≥ b n } is such that • ∪ n≥1 U n has full Lebesgue measure, since T n = {x ∈ M : n is a (σ, δ)-hyperbolic time} satisfies h −1 (n) ⊂ T n ⊂ U n ; and • if x ∈ U n and f n (x) ∈ U m , then x ∈ U n+m (i.e., (U n ) n≥1 is a concatenated collection as defined in [5]). In addition, lettingq(x) = min{n ≥ 1 : x ∈ U n }, we have againq(x) ≤ h(x) in general.…”

“…is a concatenated collection as defined in [5]). In addition, letting q(x) = min{n ≥ 1 : x ∈ U n }, we have again q(x) ≤ h(x) in general.…”