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Finding periodic points of a map by use of a $$k$$-adic expansion. (English) Zbl 0965.37015
The main goal of this paper is to present an algebraic language which allows to put earlier known results in a unified framework and to simplify proofs. The authors give a new characterization of a sequence of Lefschetz numbers of iterates of a map. The basic observation is that the Lefschetz number $$L(f^m)$$ is the value at $$m$$ of a character of a virtual representation of $$\mathbb{Z}$$ given by the nonsingular part of the map induced by $$f$$ ($$f$$ is a map, $$f:X\to X$$, $$X$$ is paracompact) on the rational (complex) cohomology spaces of $$X$$. For a smooth transversal map they give a refined version of Matsuoka’s theorem on the parity of the number of periodic orbits of a transversal map. Moreover they show the existence of infinitely many prime periods provided the sequence of Lefschetz numbers of the iterates is unbounded.

##### MSC:
 37B30 Index theory for dynamical systems, Morse-Conley indices 37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. 55M20 Fixed points and coincidences in algebraic topology
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