Finding periodic points of a map by use of a \(k\)-adic expansion
type: Item
instance of: scholarly article
title: Finding periodic points of a map by use of a \(k\)-adic expansion
zbMATH Open document ID: 0965.37015
author: Wacław Marzantowicz
author: Piotr Maciej Przygodzki
published in: Discrete and Continuous Dynamical Systems
publication date: 2000-08-16T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37B30
Mathematics Subject Classification ID: 37C30
Mathematics Subject Classification ID: 55M20
zbMATH DE Number: 1492596
zbMATH Keywords: Lefschetz number
zbMATH Keywords: iteration of map
zbMATH Keywords: periodic points
zbMATH Keywords: \(k\)-adic expansion
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: dcds.1999.5.495
OpenAlex ID: W2005037441
DOI: 10.3934/DCDS.1999.5.495
description: scientific article; zbMATH DE number 1492596