Periods and Lefschetz zeta functions
type: Item
instance of: scholarly article
title: Periods and Lefschetz zeta functions
zbMATH Open document ID: 0821.58035
author: V. Pereyra
published in: Pacific Journal of Mathematics
publication date: 1995-09-19T00:00:00Z
review text: The object of this paper is to prove the following result: If \(f : M \to M\) is a transversal map on a compact manifold whose Lefschetz zeta function \(Z_ f(t)\) has all its poles and zeros equal to roots of unity, and if \(Z_ f(t)\) has an irreducible factor of the form \((1 \pm t^ n)^{\pm 1}\), then (a) \(n\) is a periodic point of \(f\) if \(n\) is odd, and (b) either \(n \over 2\) or \(n\) is a periodic point if \(n\) is even. (A periodic point of \(f\) is a member of the set \(\text{Per}(f) = \{m \in \mathbb{Z}\mid f\) has a periodic orbit of minimal period \(m\)\}.) The assumption that all poles and zeros of the Lefschetz zeta function are roots of unity is satisfied by three basic classes of transversal maps. One is the set of maps whose set of periods \(\text{Per}(f)\) is finite. Another is the self-maps of compact, connected surfaces with \(\text{Per}(f)\) finite or zero topological entropy. The third is the self-maps of the \(n\)-dimensional torus with \(\text{Per}(f)\) finite or zero topological entropy.
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 54C70
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 695063
zbMATH Keywords: cyclotomic polynomials
zbMATH Keywords: periods of transversal maps
zbMATH Keywords: Lefschetz zeta function
zbMATH Keywords: topological entropy
reviewed by: William J. Satzer Jr.
MaRDI profile type: Publication
full work available at URL: pjm.1994.165.51
OpenAlex ID: W2042020321
DOI: 10.2140/PJM.1994.165.51
description: scientific article; zbMATH DE number 695063