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Approximating the maximum ergodic average via periodic orbits

Published online by Cambridge University Press:  01 August 2008

D. COLLIER  and
I. D. MORRIS
D. COLLIER
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK (email: dcollier@maths.manchester.ac.uk)
I. D. MORRIS
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: Ian.Morris@warwick.ac.uk)
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Abstract

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Let σ:ΣA→ΣA be a subshift of finite type, let $\mathcal {M}_\sigma $ be the set of all σ-invariant Borel probability measures on ΣA, and let $f\colon \Sigma _A \to \mathbb {R}$ be a Hölder continuous observable. There exists at least one σ-invariant measure μ which maximizes $\int f \,d\mu $. The following question was asked by B. R. Hunt, E. Ott and G. Yuan: how quickly can the maximum of the integrals $\int f \,d\mu $ be approximated by averages along periodic orbits of period less than p? We give an example of a Hölder observable f for which this rate of approximation is slower than stretched-exponential in p.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 4 , August 2008 , pp. 1081 - 1090
Copyright
Copyright © 2008 Cambridge University Press

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