Large deviations for non-uniformly expanding maps

Abstract

We obtain large deviation results for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average decays to zero exponentially fast with the number of iterates involved. As easy by-products we deduce escape rates from subsets of the basins of physical measures for these types of maps. The rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states. The corrections added to the published version of this text appear in bold; see last section for a list of changes

36 pages, 1 figure. After many PhD students and colleagues having pointed several errors in the statements and proofs, this is a correction to published article answering those comments. List of main changes in a new last section

References in corpus (2)

• Stochastic stability at the boundary of expanding maps
• SRB measures for weakly expanding maps

Cited by in corpus (9)

• Large deviations for systems with non-uniform structure
• Robust exponential decay of correlations for singular-flows
• Large deviation principle for Benedicks-Carleson quadratic maps
• Weak Specification Properties and Large Deviations for Non-additive Potentials
• Upper large deviations bound for singular-hyperbolic attracting sets
• Statistical properties of Lorenz like flows, recent developments and perspectives
• Sharp large deviations for some hyperbolic systems
• Large deviation principles of one-dimensional maps for Hölder continuous potentials
• Pesin's Entropy Formula for $C^1$ non-uniformly expanding maps