The space of dynamical systems with the \(C^ 0\)-topology (Q1320429)

This book is an introduction to the theory of \(C^ 0\)-small perturbations of dynamical systems which was developed intensively over the last 20 years. It consists of 5 chapters and 3 appendices. Chapter 0 contains definitions and preliminary results. In particular, there are introduced: the space \(Z (M)\) of continuous discrete dynamical systems on a smooth closed manifold \(M\) -- the main object in the book, and the set of diffeomorphisms which satisfy the STC (the strong transversality condition) that plays a crucial role in the book. Chapter 1 is devoted to generic properties of systems in \(Z(M)\). Necessary and sufficient conditions for topological stability are given in chapter 2. Combining results of Z. Nitecki and M. Shub it is shown that the topologically stable systems are dense in CLD\((M)\) (the closure of the set of diffeomorphisms on \(M\) in \(Z(M)\)). In the special case \(M=S^ 1\) a system \(\varphi \in Z(S^ 1)\) is topologically stable when it is topologically conjugate to a Morse-Smale diffeomorphism. Chapter 3 is devoted to stability of attractors under \(C^ 0\)-small perturbations of the system. In chapter 4 limit sets are studied. It is a pleasure to read this book since the main ideas are described clearly and the more technical proofs are left out of the text: in the appendices \(A,B\) and \(C\) or in the references.