Backward Volume Contraction for Endomorphisms with Eventual Volume Expansion

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 for Lebesgue almost every point. 1. Statement of results Let M be a compact Riemannian manifold and let Leb be a volume form on M that we call Lebesgue measure. We take f : M → M any smooth map. Let 0 < a1 ≤ a2 ≤ a3 ≤ . . . be a sequence converging to infinity. We define h(x) = min{n > 0: | detDf(x)| ≥ an}, (1) if this minimum exists, and h(x) = ∞, otherwise. For n ≥ 1, we take Γn = {x ∈ M : h(x) ≥ n}. (2) Theorem 1.1. Assume that h ∈ L(Leb), for some p > 3, and take γ < (p− 3)/(p− 1). Choose any sequence 0 < b1 ≤ b2 ≤ b3 ≤ . . . such that bkbn ≥ bk+n for every k, n ∈ N, and assume that there is n0 ∈ N such that bn ≤ min {an,Leb(Γn) −γ} for every n ≥ n0. Then, for Leb almost every x ∈ M , there exists Cx > 0 such that | detDf (y)| > Cxbn for every y ∈ f−n(x). We say that f : M → M is eventually volume expanding if there exists λ > 0 such that for Lebesgue almost every x ∈ M sup n≥1 1 n log | detDf(x)| > λ. (3) Let h and Γn be defined as in (1) and (2), associated to the sequence an = e . Date: February 1, 2008. Work carried out at the Federal University of Bahia. Partially supported by FCT through CMUP and UFBA. 1 2 JOSÉ F. ALVES, VILTON PINHEIRO, AND ARMANDO CASTRO Corollary 1.2. If f is eventually volume expanding, then for Lebesgue almost every point x ∈ M there are Cx > 0 and σn → ∞ such that | detDf(y)| > Cxσn for every y ∈ f −n(x). Moreover, given α > 0 there is β > 0 such that (1) if Leb(Γn) ≤ O(e −αn), then we may take σn ≥ e ; (2) if Leb(Γn) ≤ O(e −αn ) for some τ > 0, then we may take σn ≥ e βn ; (3) if Leb(Γn) ≤ O(n −α) and α > 2, then we may take σn ≥ n . Specific rates will be obtained in Section 4 for some eventually volume expanding endomorphisms. In particular, non-uniformly expanding maps such as quadratic maps and Viana maps will be considered. 2. Concatenated collections Let (Un)n be a collection of measurable subsets of M whose union covers a full Lebesgue measure subset of M . We say that (Un)n is a concatenated collection if: x ∈ Un and f (x) ∈ Um ⇒ x ∈ Un+m. Given x ∈ ⋃ n≥1 Un, we define u(x) as the minimum n ∈ N for which x ∈ Un. Note that by definition we have x ∈ Uu(x). We define the chain generated by x ∈ ⋃ n≥1 Un as C(x) = {x, f(x), . . . , f u(x)−1(x)}. Lemma 2.1. Let (Un)n be a concatenated collection. If