Overview
- Provides a survey of current research and new approaches in the theory of shadowing of approximate trajectories of dynamical systems
- Contains novelty approach for proving hyperbolicity by using the sifting method of Liao, which is both powerful and self-contained
- A main feature is the direct, straightforward approach to the results, well written, in an easy-to-read, comfortable pace
- Can be used right after a course of introduction to dynamical systems or even hyperbolic dynamics
- Includes supplementary material: sn.pub/extras
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2193)
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Table of contents (4 chapters)
Keywords
About this book
Focusing on the theory of shadowing of approximate trajectories (pseudotrajectories) of dynamical systems, this book surveys recent progress in establishing relations between shadowing and such basic notions from the classical theory of structural stability as hyperbolicity and transversality.
Special attention is given to the study of "quantitative" shadowing properties, such as Lipschitz shadowing (it is shown that this property is equivalent to structural stability both for diffeomorphisms and smooth flows), and to the passage to robust shadowing (which is also equivalent to structural stability in the case of diffeomorphisms, while the situation becomes more complicated in the case of flows).
Relations between the shadowing property of diffeomorphisms on their chain transitive sets and the hyperbolicity of such sets are also described.
The book will allow young researchers in the field of dynamical systems to gain a better understanding of new ideas in the global qualitative theory. It will also be of interest to specialists in dynamical systems and their applications.
Reviews
“This book gives an up-to-date account of results on the relations between shadowing and such basic notions from the classical theory of structural stability as hyperbolicity and transversality. … The style of presentation is very clear and, in my opinion, the book is quite suitable for researchers in the field of dynamical systems to understand the global qualitative theory from different points of view.” (Yujun Zhu, Mathematical Reviews, July, 2018)
Authors and Affiliations
Bibliographic Information
Book Title: Shadowing and Hyperbolicity
Authors: Sergei Yu Pilyugin, Kazuhiro Sakai
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-319-65184-2
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG 2017
Softcover ISBN: 978-3-319-65183-5Published: 02 September 2017
eBook ISBN: 978-3-319-65184-2Published: 31 August 2017
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: XIV, 218
Number of Illustrations: 5 b/w illustrations