Invariant manifolds, entropy and billiards; smooth maps with singularities. With the collab. of F. Ledrappier and F. Przytycki
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}
scientific article; zbMATH DE number 4076091