Fractal Dimensions for Poincare Recurrences

In this book, the authors introduce and study a new characteristic of complexity of motions in dynamical systems that has recently appeared. The characteristic measures an average return time and can be expressed in the framework of the dimension theory in dynamical systems.To show it, the authors summarized the current state of the modern dynamical systems theory and the theory of fractal dimensions. They use ideas and notions from symbolic and smooth dynamics, like topological entropy and topological pressure, Lyapunov exponents, etc. Therefore the book can serve as a text for special topic courses for graduate students. Being written on a high level of mathematical rigorousness, the book can be useful for people from applied sciences as well. An euristic chapter and examples of applications of the developed theory show how to use dimensions for Poincare recurrences in specific problems of nonlinear dynamics. In particular, an impressive result is related to distributions of Poincare recurrences in a non purely chaotic Hamiltonian systems when the asymptotic law has polynomial tails. The exponent in the law is expressed by means of the dimension for Poincare recurremces. This dimension can also serve very effectivelly to characterize synchronization regimes in coupled chaotic systems. These two applications provide a complete justification of the introduced chractaeristics. The authors have written this book in a clear and detailed way, the exposition is proper and scrupulous.Afraimovich, Valentin is the author of 'Fractal Dimensions for Poincare Recurrences', published 2006 under ISBN 9780444521897 and ISBN 0444521895.