Overview

This is the first systematic and self-contained textbook on homotopy methods in the study of periodic points of a map. A modern exposition of the classical topological fixed-point theory with a complete set of all the necessary notions as well as new proofs of the Lefschetz-Hopf and Wecken theorems are included.

Periodic points are studied through the use of Lefschetz numbers of iterations of a map and Nielsen-Jiang periodic numbers related to the Nielsen numbers of iterations of this map. Wecken theorem for periodic points is then discussed in the second half of the book and several results on the homotopy minimal periods are given as applications, e.g. a homotopy version of the Aarkovsky theorem, a dynamics of equivariant maps, and a relation to the topological entropy. Students and researchers in fixed point theory, dynamical systems, and algebraic topology will find this text invaluable.

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9789048105236

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<p>From the reviews of the first edition: <p> This book contains an up-to-date exposition of the topological fixed and periodic point theories associated with the names of Lefschetz and Hopf and of Nielsen. The phrase homotopy methods in its title refers to the fact that the foundations of these theories lie in algebraic topology and thus depend on tools that are homotopy invariant.<br>The feature that most sets the book apart from its predecessors is the presentation, occupying about one half of its more than 300-page length, of the theory of periodic points that is based on algebraic topology. A chapter on the sequence of integers that arise as the Lefschetz numbers of the iterates of a map informs the reader about what is known concerning such sequences and how this knowledge leads to information regarding its periodic points. The most distinctive chapters are concerned with the Nielsen theory of periodic points and with homotopy minimal periods. The first of these chapters contai

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