Nonuniformly Hyperbolic Attractors Geometric and Probabilistic Aspects

This monograph offers a coherent, self-contained account of the theory of Sinai–Ruelle–Bowen measures and decay of correlations for nonuniformly hyperbolic dynamical systems.

A central topic in the statistical theory of dynamical systems, the book in particular provides a detailed exposition of the theory developed by L.-S. Young for systems admitting induced maps with certain analytic and geometric properties. After a brief introduction and preliminary results, Chapters 3, 4, 6 and 7 provide essentially the same pattern of results in increasingly interesting and complicated settings. Each chapter builds on the previous one, apart from Chapter 5 which presents a general abstract framework to bridge the more classical expanding and hyperbolic systems explored in Chapters 3 and 4 with the nonuniformly expanding and partially hyperbolic systems described in Chapters 6 and 7. Throughout the book, the theory is illustrated with applications.

A clear and detailed account of topicsof current research interest, this monograph will be of interest to researchers in dynamical systems and ergodic theory. In particular, beginning researchers and graduate students will appreciate the accessible, self-contained presentation.

1 Introduction.- 2 Preliminaries.- 3 Expanding Structures.- 4 Hyperbolic Structures.- 5 Inducing Schemes.- 6 Nonuniformly Expanding Attractors.- 7 Partially Hyperbolic Attractors.- Index.

José Ferreira Alves is a full Professor at the Department of Mathematics of the Faculty of Sciences of the University of Porto, Portugal. He obtained his PhD in Mathematics from the Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro (Brazil), in 1997. In 2000, he was a postdoc at the University of Maryland, USA, and during the academic year 2018/19 he was a Visiting Professor at Loughborough University, UK, with a grant from the Leverhulme Trust. His research interests lie in Dynamical Systems and Ergodic Theory, with an emphasis on the statistical properties of nonuniformly hyperbolic dynamics.