Distribution of periodic orbits of symbolic and axiom A flows
type: Item
instance of: scholarly article
title: Distribution of periodic orbits of symbolic and axiom A flows
zbMATH Open document ID: 0637.58013
DOI: 10.1016/0196-8858(87)90012-1
published in: Advances in Applied Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let A be an \(n\times n\) 0-1 matrix. For given matrix A, \(\Sigma_{\hat A}\) denotes the set of all doubly infinite sequences and \(C(\Sigma_ A)\) the space of continuous complex-valued functions on \(\Sigma_ A\). For \(f\in C(\Sigma_{\hat A})\) let \((\Sigma^ f_{\hat A},\sigma^ f)\) denote the symbolic flow, defined on the suspension space \(\Sigma^ f_ A\) by usual manner. Let G be a bounded Borel measurable real-valued function on \(\Sigma^ f_ A\). Periodic orbits of \((\Sigma^ f_ A,\sigma^ f)\) are denoted by \(\tau\), \(\tau\) (G) denotes the integral of G, with respect to time, over one period for \(\tau\). Thus \(\tau\) (1) is the period of \(\tau\), and \(\tau\) (G)/\(\tau\) (1) is the mean value of G over \(\tau\). Define measure \(Q^ f\) on \({\mathbb{R}}^+\) by \(Q^ f(I)=\#\{\tau: \tau (1)\in I\}\). The main results of this paper concern the asymptotic behavior of this measure, when the flow is weakly mixing. Also further analogous measures are constructed and investigated.
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 37D15
Mathematics Subject Classification ID: 28D99
zbMATH DE Number: 4038843
zbMATH Keywords: subshift of finite type
zbMATH Keywords: invariant measure
zbMATH Keywords: symbolic flow
zbMATH Keywords: suspension space
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: weakly mixing
author: Steven P. Lalley
reviewed by: A. Klíč
MaRDI profile type: Publication
full work available at URL: 0196-8858(87)90012-1
OpenAlex ID: W2080231903
Wikidata QID: Q114683983
cites work: Q3272101
cites work: Periodic Orbits for Hyperbolic Flows
cites work: The Equidistribution of Closed Geodesics
cites work: Symbolic Dynamics for Hyperbolic Flows
cites work: Equilibrium states and the ergodic theory of Anosov diffeomorphisms
cites work: Q3232397
cites work: Q5531120
cites work: Q3874765
cites work: Q4729089
cites work: Applications of ergodic theory to the investigation of manifolds of negative curvature
cites work: Bowen's equidistribution theory and the Dirichlet density theorem
cites work: An analogue of the prime number theorem for closed orbits of Axiom A flows
cites work: A complex Ruelle-Perron-Frobenius theorem and two counterexamples
cites work: Q4185090
cites work: Asymptotic behavior of periodic orbits of the horocycle flow and eisenstein series
cites work: Q3237356
cites work: Q5564831
description: scientific article; zbMATH DE number 4038843