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.

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