Invariant manifolds, entropy and billiards; smooth maps with singularities. With the collab. of F. Ledrappier and F. Przytycki
type: Item
instance of: scholarly article
title: Invariant manifolds, entropy and billiards; smooth maps with singularities. With the collab. of F. Ledrappier and F. Przytycki
zbMATH Open document ID: 0658.58001
author: Anatole Katok
author: Jean-Marie Strelcyn
published in: Lecture Notes in Mathematics
publication date: 1986-01-01T00:00:00Z
review text: In classical mechanics some dynamical systems have singularities. Some of these systems can be reduced to billiard systems, i.e. systems describing the motion of a point mass within a Riemannian manifold with boundary with reflection from the boundary. A theorem of Ya. B. Pesin assures the existence and absolute continuity of invariant families of stable and unstable manifolds for smooth dynamical systems showing behavior of hyperbolic type. The aim of this book is to generalize these results to a large class of dynamical systems with singularities, to correct errors and complete the proof of the mentioned theorem. As the authors resume, in this book they complete the lower right corner of the following diagram. \begin{tabular}{p{6 cm}cp{6 cm}} \\ The theory of Anosov systems and the related systems &\(\longrightarrow\) & The theory of billiards of Sinai and Bunimovich\\ \multicolumn{1}{c}{\(\downarrow\)} & & \multicolumn{1}{c}{\(\downarrow\)} \\ Pesin theory of diffeomorphisms of compact manifolds & \(\longrightarrow\) & Pesin theory of mappings with singularities \end{tabular}
Mathematics Subject Classification ID: 58-02
Mathematics Subject Classification ID: 37D99
Mathematics Subject Classification ID: 28D20
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 34F05
zbMATH DE Number: 4076091
zbMATH Keywords: Anosov systems
zbMATH Keywords: Pesin theory
zbMATH Keywords: classical mechanics
zbMATH Keywords: dynamical systems
zbMATH Keywords: singularities
zbMATH Keywords: billiard systems
zbMATH Keywords: invariant families
zbMATH Keywords: stable and unstable manifolds
zbMATH Keywords: hyperbolic type
reviewed by: I. Mihuţ
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4076091