Shadowing and hyperbolicity

Description

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. Front Matter ....Pages i-xiv Main Definitions and Basic Results (Sergei Yu. Pilyugin, Kazuhiro Sakai)....Pages 1-35 Lipschitz and Hölder Shadowing and Structural Stability (Sergei Yu. Pilyugin, Kazuhiro Sakai)....Pages 37-124 C1 Interiors of Sets of Systems with Various Shadowing Properties (Sergei Yu. Pilyugin, Kazuhiro Sakai)....Pages 125-179 Chain Transitive Sets and Shadowing (Sergei Yu. Pilyugin, Kazuhiro Sakai)....Pages 181-208 Back Matter ....Pages 209-218