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Approximating the maximum ergodic average via periodic orbits

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Collier, D. and Morris, Ian D.. (2008) Approximating the maximum ergodic average via periodic orbits. Ergodic Theory and Dynamical Systems, Vol.28 (No.4). pp. 1081-1090. ISSN 0143-3857

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Official URL: http://dx.doi.org/10.1017/S014338570700082X

Abstract

Let sigma: Sigma(A) -> Sigma(A) be a subshift of finite type, let M-sigma be the set of all sigma-invariant Borel probability measures on Sigma(A), and let f : Sigma(A) -> R be a Holder continuous observable. There exists at least one or-invariant measure A which maximizes integral 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 integral f d mu be approximated by averages along periodic orbits of period less than p? We give an example of a Holder observable f for which this rate of approximation is slower than stretched-exponential in p.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Ergodic theory, Combinatorial dynamics, Mathematical optimization
Journal or Publication Title:	Ergodic Theory and Dynamical Systems
Publisher:	Cambridge University Press
ISSN:	0143-3857
Date:	August 2008
Volume:	Vol.28
Number:	No.4
Page Range:	pp. 1081-1090
Identification Number:	10.1017/S014338570700082X
Status:	Peer Reviewed
Access rights to Published version:	Open Access
References:	[1] Brémont, J.. Gibbs measures at temperature zero. Nonlinearity 16 (2003), 419–426. [2] Bousch, T.. Le poisson n’a pas d’arêtes. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), 489–508. [3] Bousch, T. and Jenkinson, O.. Cohomology classes of dynamically non-negative Ck functions. Invent. Math. 148 (2002), 207–217. [4] Bousch, T. and Mairesse, J.. Asymptotic height optimization for topical IFS, tetris heaps, and the finiteness conjecture. J. Amer. Math. Soc. 15 (2002), 77–111. [5] Bressaud, X. and Quas, A.. Rates of approximation of minimizing measures. Nonlinearity 20 (2007), 845–853. [6] Contreras, G., Lopes, A. and Thieullen, P.. Lyapunov minimizing measures for expanding maps of the circle. Ergod. Th. & Dynam. Sys. 5 (2001), 1379–1409. [7] Hunt, B. and Ott, E.. Optimal periodic orbits of chaotic systems occur at low period. Phys. Rev. E 54 (1996), 328–337. [8] Jenkinson, O.. Ergodic optimization. Disc. Cont. Dyn. Syst. 15 (2006), 197–224. [9] Jenkinson, O.. Rotation, entropy and equilibrium states. Trans. Amer. Math. Soc. 353 (2001), 3713–3739. [10] Parthasarathy, K. R.. On the category of ergodic measures. Illinois J. Math. 5 (1961), 648–656. [11] Yuan, G. and Hunt, B.. Optimal orbits of hyperbolic systems. Nonlinearity 12(4) (1999), 1207–1224.
URI:	http://wrap.warwick.ac.uk/id/eprint/864