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The study of hyperbolic systems is a core theme of modern dynamics. On surfaces the theory of the fine scale structure of hyperbolic invariant sets and their measures can be described in a very complete and elegant way, and is the subject of this book, largely self-contained, rigorously and clearly written. It covers the most important aspects of the subject and is based on several scientific works of the leading research workers in this field. This book fills a gap in the literature of dynamics. We highly recommend it for any Ph.D student interested in this area. The authors are well-known experts in smooth dynamical systems and ergodic theory.
List of contents
HR structures.- Solenoid functions.- Self-renormalizable structures.- Rigidity.- Gibbs measures.- Measure scaling functions.- Measure solenoid functions.- Cocycle-gap pairs.- Hausdorff realizations.- Extended Livsic-Sinai eigenvalue formula.- Arc exchange systems and renormalization.- Golden tilings (in collaboration with J.P. Almeida and A. Portela).- Pseudo-Anosov diffeomorphisms in pseudo-surfaces.
About the author
Alberto Pinto Awards& Honours:
2008 Recognition of the scientific merit of the works done in Psychology Sciences by the Psychology School of University of Minho
2007 Espinho Town hall Golden Medal by scientific merit
1984Prize Augusto Martins, University of Porto.
1981 One of the six winners of the Mini-Olympics of Mathematics in Portugal.
David Rand Awards& Honours
1986 Whitehead Prize of the London Mathematical Society.
1988 Wolfson Research Award.
1988 Founding editor of Nonlinearity
2004 Fellow, Institute of Mathematics and Its Applications (by invitation)
2005 Britten Lecturer. McMaster University
2006 EPSRC Senior Research Fellowship
Summary
The study of hyperbolic systems is one of the core themes of modern dynamical systems. This book plays an important role in filling a gap in the present literature on hyperbolic dynamics and is highly recommended for all PhD students interested in this field.
Additional text
From the reviews:
“The book is a mostly self contained text on the theory of the fine scale structures of hyperbolic diffeomorphisms on surfaces. It is aimed at researchers and Ph. D students interested in the topic. Most of the text is based on the research work of the authors but it also contains related topics and background material. It is clearly written and very well structured.” (Isabel Lugão Rios, Mathematical Reviews, Issue 2010 e)
“The main theme of the book Fine Structures of Hyperbolic Diffeomorphisms, by Pinto, Rand and Ferreira, is the rigidity and flexibility of two-dimensional diffeomorphisms on hyperbolic basic sets and properties of invariant measures that are related to the geometry of these invariant sets. … The book under review is based on a series of articles by the authors and is aimed at experts in the field. The theorems are clearly stated and complete proofs are provided.” (W. De Melo, Bulletin of the American Mathematical Society, Vol. 48 (1), January, 2011)
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From the reviews:
"The book is a mostly self contained text on the theory of the fine scale structures of hyperbolic diffeomorphisms on surfaces. It is aimed at researchers and Ph. D students interested in the topic. Most of the text is based on the research work of the authors but it also contains related topics and background material. It is clearly written and very well structured." (Isabel Lugão Rios, Mathematical Reviews, Issue 2010 e)
"The main theme of the book Fine Structures of Hyperbolic Diffeomorphisms, by Pinto, Rand and Ferreira, is the rigidity and flexibility of two-dimensional diffeomorphisms on hyperbolic basic sets and properties of invariant measures that are related to the geometry of these invariant sets. ... The book under review is based on a series of articles by the authors and is aimed at experts in the field. The theorems are clearly stated and complete proofs are provided." (W. De Melo, Bulletin of the American Mathematical Society, Vol. 48 (1), January, 2011)
