Markov structures for non-uniformly expanding maps on compact manifolds in arbitrary dimension

Abstract

We consider non-uniformly expanding maps on compact Riemannian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an induced Markov tower structure for which the decay of the tail of the return time function can be controlled in terms of the time generic points needed to achieve some uniform expanding behavior. As a consequence we obtain some rates for the decay of correlations of those maps and conditions for the validity of the Central Limit Theorem. 1. Dynamical and geometrical assumptions Let M be a compact Riemannian manifold of dimension d ≥ 1 with a normalized Riemannian volume | · |, which we call Lebesgue measure. Let f : M → M be a C local diffeomorphism for all x ∈ M C, where C is some critical set, which may include points at which the derivative Dfx is degenerate, as well as points of discontinuity and points at which the derivative is infinite. We assume the following natural non-degeneracy condition on C, which generalizes the notion of non-flat critical points for smooth one-dimensional maps. Definition 1. The critical set C ⊂ M is non-degenerate if |C| = 0 and there is a constant β > 0 such that for every x ∈M C we have dist(x, C) . ‖Dfxv‖/‖v‖ . dist(x, C)−β for all v ∈ TxM , and the functions log detDf and log ‖Df−1‖ are locally Lipschitz with Lipschitz constant . dist(x, C)−β . We now state our two dynamical assumptions: the first is on the growth of the derivative and the second is on the approach rate of orbit to the critical set. Notice that for a linear map A, the condition ‖A‖ > 1 only provides information about the existence of some expanded direction, whereas the condition ‖A−1‖ 0) implies that every direction is expanded. Received by the editors November 5, 2002. 2000 Mathematics Subject Classification. Primary 37D20, 37D50, 37C40. Work carried out at the Federal University of Bahia, University of Porto and Imperial College, London. Partially supported by CMUP, PRODYN, SAPIENS and UFBA. c ©2003 American Mathematical Society