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Approximating the maximum ergodic average via periodic orbits

Published online by Cambridge University Press:  01 August 2008

D. COLLIER
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK (email: [email protected])
I. D. MORRIS
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: [email protected])

Abstract

Let σA→ΣA be a subshift of finite type, let be the set of all σ-invariant Borel probability measures on ΣA, and let be a Hölder continuous observable. There exists at least one σ-invariant measure μ which maximizes . The following question was asked by B. R. Hunt, E. Ott and G. Yuan: how quickly can the maximum of the integrals 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.

Type
Research Article
Copyright
Copyright © 2008 Cambridge University Press

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References

[1]Brémont, J.. Gibbs measures at temperature zero. Nonlinearity 16 (2003), 419426.CrossRefGoogle Scholar
[2]Bousch, T.. Le poisson n’a pas d’arêtes. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), 489508.CrossRefGoogle Scholar
[3]Bousch, T. and Jenkinson, O.. Cohomology classes of dynamically non-negative C k functions. Invent. Math. 148 (2002), 207217.CrossRefGoogle Scholar
[4]Bousch, T. and Mairesse, J.. Asymptotic height optimization for topical IFS, tetris heaps, and the finiteness conjecture. J. Amer. Math. Soc. 15 (2002), 77111.CrossRefGoogle Scholar
[5]Bressaud, X. and Quas, A.. Rates of approximation of minimizing measures. Nonlinearity 20 (2007), 845853.CrossRefGoogle Scholar
[6]Contreras, G., Lopes, A. and Thieullen, P.. Lyapunov minimizing measures for expanding maps of the circle. Ergod. Th. & Dynam. Sys. 5 (2001), 13791409.Google Scholar
[7]Hunt, B. and Ott, E.. Optimal periodic orbits of chaotic systems occur at low period. Phys. Rev. E 54 (1996), 328337.Google Scholar
[8]Jenkinson, O.. Ergodic optimization. Disc. Cont. Dyn. Syst. 15 (2006), 197224.CrossRefGoogle Scholar
[9]Jenkinson, O.. Rotation, entropy and equilibrium states. Trans. Amer. Math. Soc. 353 (2001), 37133739.CrossRefGoogle Scholar
[10]Parthasarathy, K. R.. On the category of ergodic measures. Illinois J. Math. 5 (1961), 648656.Google Scholar
[11]Yuan, G. and Hunt, B.. Optimal orbits of hyperbolic systems. Nonlinearity 12(4) (1999), 12071224.Google Scholar