Recurrence and Topology

The main topic of this book is recurrence of dynamical systems on compact metric spaces (the authors consider continuous flows). The book starts with an introduction to flows, their special points and corresponding invariant sets (from fixed points and sets via periodic points and recurrent points to nonwandering and chain recurrent points). The next chapter deals with irreducible sets of flows (minimal sets, transitive sets, chain-transitive sets, attracting sets and repelling sets). The last chapter describes certain test functions (the potential function, the Hamiltonian function, the Morse function and several kinds of Lyapunov functions). The chapter ends with the fundamental theorem of dynamical systems: every continuous flow on a compact metric space has a complete Lyapunov function.

Every chapter contains exercises. Three short appendices contain a discussion of discrete dynamical systems, recurrence of circle rotation, and completeness and compactness of Hausdorff hyperspaces. In ‘Afterwords’, the main ideas of the book and relations to hyperbolic systems are briefly described, and suggestions for further reading are given. The specific feature of the book is that concepts (and many results) are demonstrated on flows of explicit systems of differential equations.

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