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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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