Titlebook: Nonuniformly Hyperbolic Attractors; Geometric and Probab José F. Alves Book 2020 The Editor(s) (if applicable) and The Author(s), under exc

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Springer Monographs in Mathematicshttp://image.papertrans.cn/n/image/667922.jpg

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https://doi.org/10.1007/978-3-030-62814-7Nonuniformly expanding maps; Inducing schemes; Young towers; Partially hyperbolic diffeomorphisms; Gibbs

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Introduction,In this chapter, we present an introduction to physical measures and decay of correlations, addressing particularly the special case of SRB measures for systems with some positive Lyapunov exponent.

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Preliminaries,In this chapter, we present several concepts and results to be used throughout the text. In particular, we introduce some key notions related to partitions in a measure space, such as generating partition or measurable partition, in the sense of Rohlin. The notion of Jacobian and results on the basin of a measure are also addressed here.

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Expanding Structures,In this chapter, we introduce Gibbs-Markov maps and describe Young’s construction of a tower extension associated with an induced Gibbs-Markov map. We derive, in particular, the existence of ergodic absolutely continuous invariant probability measures, and conclusions on the decay of correlations with respect to such measures.

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Inducing Schemes,In this chapter, we develop a general framework for the construction of inducing schemes suitable for application to nonuniformly expanding maps and partially hyperbolic diffeomorphisms.

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Nonuniformly Expanding Attractors,In this chapter, we construct Gibbs-Markov induced maps for nonuniformly expanding maps, deduce the existence of SRB measures for such maps and provide rates for the decay of correlations with respect to these measures in terms of the tail of expansion.

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