Estimation of the number of periodic orbits
type: Item
instance of: scholarly article
title: Estimation of the number of periodic orbits
zbMATH Open document ID: 0855.55001
published in: Pacific Journal of Mathematics
publication date: 1997-02-03T00:00:00Z
review text: The author is concerned with estimating the asymptotic growth rate of the Nielsen numbers \(N(f^n)\) for a map \(f : X \to X\) of a compact connected polyhedron. Pick a point \(v \in X\) and a path \(w\) from \(v\) to \(f(v)\). Denote \(G : = \pi_1 (X,w)\) and let \(f_G : = w_* f_* : G \to G\). Two elements \(g,g' \in G\) are said to be \(f_G\)-conjugate if there is an \(h \in G\) with \(g' = f_G (h) gh^{-1}\). Let \(f(x) = x\) and pick a path \(c\) from \(v\) to \(x\). The \(G\)-coordinate of \(x\) is defined to be the \(f_G\)-conjugacy class of \(w (f \circ c) c^{-1}\). The \(G\)-coordinate is constant on each fixed point class \({\mathbf F}\), so one defines \(\text{cd}_G ({\mathbf F}, f)\) to be this common value. The generalized Lefschetz number (or Reidemeister trace) is then defined to be \(L_G (f) = \sum \text{ind} ({\mathbf F}, f) \text{cd}_G ({\mathbf F}, f)\) where the summation extends over all fixed point classes \({\mathbf F}\) of \(f\). Let \(T_f\) be the mapping torus of \(f\), let \(\varphi\) be the semiflow on \(T_f\) generated by \(f\) and \(\Gamma : = \pi_1 (T_f,v)\) and denote by \(\Gamma_c\) the set of conjugacy classes in \(\Gamma\). If \(f^n x = x\) the \(\Gamma\)-coordinate of \((x,n)\), \(\text{cd}_\Gamma (x,n)\), is defined to be the free homotopy class of the loop \([0,n] \to X\) defined by \(t \mapsto \varphi_tx\). Points \(x\) and \(x'\) with \(f^n x = x\) and \(f^n x' = x'\) are said to be in the same \(n\)-orbit class \({\mathbf O}^n\) if \(\text{cd}_\Gamma (x,n) = \text{cd}_\Gamma (x',n)\). This common value is again denoted by \(\text{cd}_\Gamma ({\mathbf O}^n)\). The generalized Lefschetz number of \(f^n\) is then defined to be \(L_\Gamma (f^n) = \sum \text{ind} ({\mathbf O}^n, f^n) \text{cd}_\Gamma ({\mathbf O}^n) \in \mathbb{Z} \Gamma_c\) the summation being over all \(n\)-orbit classes. Consider a commutative ring \(R\) with unity and a representation \(\rho : G \to \text{GL}_\ell (R)\). This extends to a representation \(\rho :…
Mathematics Subject Classification ID: 55M20
zbMATH DE Number: 910251
zbMATH Keywords: Nielsen numbers
zbMATH Keywords: generalized Lefschetz number
zbMATH Keywords: Reidemeister trace
zbMATH Keywords: Lefschetz zeta function
zbMATH Keywords: topological entropy
author: Boju Jiang
reviewed by: Christian Fenske
MaRDI profile type: Publication
full work available at URL: pjm.1996.172.151
OpenAlex ID: W1500418640
DOI: 10.2140/PJM.1996.172.151
description: scientific article; zbMATH DE number 910251