Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-27T06:15:30.416Z Has data issue: false hasContentIssue false
  • English
  • Français

Ergodic optimization of super-continuous functions on shift spaces

Published online by Cambridge University Press:  14 October 2011

ANTHONY QUAS  and
JASON SIEFKEN
ANTHONY QUAS
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria BC, Canada V8W 3R4 (email: aquas@uvic.ca, siefkenj@uvic.ca)
JASON SIEFKEN
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria BC, Canada V8W 3R4 (email: aquas@uvic.ca, siefkenj@uvic.ca)
Get access Rights & Permissions [Opens in a new window]

Abstract

Ergodic optimization is the process of finding invariant probability measures that maximize the integral of a given function. It has been conjectured that ‘most’ functions are optimized by measures supported on a periodic orbit, and it has been proved in several separable spaces that an open and dense subset of functions is optimized by measures supported on a periodic orbit. All known positive results have been for separable spaces. We give in this paper the first positive result for a non-separable space, the space of super-continuous functions on the full shift, where the set of functions optimized by periodic orbit measures contains an open dense subset.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 6 , December 2012 , pp. 2071 - 2082
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bousch, T.. Le poisson n’a pas d’arêtes. Ann. Inst. Henri Poincaré Probab. Stat. 36 (2000), 197–224.CrossRefGoogle Scholar
[2]Bousch, T.. La condition de Walters. Ann. Sci. Éc. Norm . Supér. (4) 34 (2001), 287–311.CrossRefGoogle Scholar
[3]Contreras, G., Lopes, A. O. and Thieullen, P.. Lyapunov minimizing measures for expanding maps of the circle. Ergod. Th. & Dynam. Sys. 21 (2001), 1379–1409.CrossRefGoogle Scholar
[4]Hunt, B. R. and Ott, E.. Optimal periodic orbits of chaotic systems. Phys. Rev. Lett. 76(13) (1996), 2254–2257.CrossRefGoogle ScholarPubMed
[5]Hunt, B. R. and Ott, E.. Optimal periodic orbits of chaotic systems occur at low period. Phys. Rev. E 54(1) (1996), 328–337.CrossRefGoogle ScholarPubMed
[6]Jenkinson, O.. Ergodic optimization. Discrete Contin. Dyn. Syst. (2006), 197–224.CrossRefGoogle Scholar
[7]Walters, P.. Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1985), 127–153.Google Scholar
[8]Yang, T.-H., Hunt, B. R. and Ott, E.. Optimal periodic orbits of continuous time chaotic systems. Phys. Rev. E 62(2) (2000), 1950–1959.CrossRefGoogle ScholarPubMed
[9]Yuan, G. and Hunt, B. R.. Optimal orbits of hyperbolic systems. Nonlinearity 12 (1999), 1207–1224.CrossRefGoogle Scholar