Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-30T23:22:01.162Z Has data issue: false hasContentIssue false
  • English
  • Français

The Mañé–Conze–Guivarc’h lemma for intermittent maps of the circle

Published online by Cambridge University Press:  03 February 2009

IAN D. MORRIS
IAN D. MORRIS*
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the existence of solutions g to the functional inequality f≤g ∘T−g+β, where f is a prescribed continuous function, T is a weakly expanding transformation of the circle having an indifferent fixed point, and β is the maximum ergodic average of f. Using a method due to T. Bousch, we show that continuous solutions g always exist when the Hölder exponent of f is close to 1. In the converse direction, we construct explicit examples of continuous functions f with low Hölder exponent for which no continuous solution g exists. We give sharp estimates on the best possible Hölder regularity of a solution g given the Hölder regularity of f.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 5 , October 2009 , pp. 1603 - 1611
Copyright
Copyright © Cambridge University Press 2009

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]Barabanov, N. E.. On the Lyapunov exponent of discrete inclusions I. Avtomat. i Telemekh. 2 (1988), 40–46; English translation: Autom. Remote Control 49 (1988), 152–157.Google Scholar
[2]Bousch, T.. Le poisson n’a pas d’arêtes. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), 489–508.CrossRefGoogle Scholar
[3]Bousch, T.. La condition de Walters. Ann. Sci. École Norm. Sup. 34 (2001), 287–311.CrossRefGoogle Scholar
[4]Bousch, T.. Nouvelle preuve d’un théorème de Yuan et Hunt. Bull. Soc. Math. France 136(2) (2008), 227–242.CrossRefGoogle Scholar
[5]Bousch, T.. Le lemme de Mañé–Conze–Guivarc’h pour les systèmes amphidynamiques rectifiables. Preprint, 2007.Google Scholar
[6]Bousch, T. and Jenkinson, O.. Cohomology classes of dynamically non-negative C k functions. Invent. Math. 148 (2002), 207–217.CrossRefGoogle Scholar
[7]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.CrossRefGoogle Scholar
[8]Branton, S.. Sub-actions for Young towers. Preprint.Google Scholar
[9]Brémont, J.. Gibbs measures at temperature zero. Nonlinearity 16 (2003), 419–426.CrossRefGoogle Scholar
[10]Conze, J.-P. and Guivarc’h, Y.. Croissance des sommes ergodiques. Unpublished manuscript, circa 1993.Google Scholar
[11]Contreras, G., Lopes, A. and Thieullen, P.. Lyapunov minimizing measures for expanding maps of the circle. Ergod. Th. & Dynam. Sys. 21 (2001), 1379–1409.CrossRefGoogle Scholar
[12]Hu, H.. Decay of correlations for piecewise smooth maps with indifferent fixed points. Ergod. Th. & Dynam. Sys. 24 (2004), 495–524.CrossRefGoogle Scholar
[13]Hunt, B. and Ott, E.. Optimal periodic orbits of chaotic systems. Phys. Rev. Lett. 76 (1996), 2254–2257.CrossRefGoogle ScholarPubMed
[14]Jenkinson, O.. Geometric barycentres of invariant measures for circle maps. Ergod. Th. & Dynam. Sys. 21 (2001), 1429–1445.CrossRefGoogle Scholar
[15]Jenkinson, O.. Rotation, entropy and equilibrium states. Trans. Amer. Math. Soc. 353 (2001), 3713–3739.CrossRefGoogle Scholar
[16]Jenkinson, O., Mauldin, R. D. and Urbański, M.. Zero temperature limits of Gibbs-equilibrium states for countable alphabet subshifts of finite type. J. Stat. Phys. 119 (2005), 765–776.CrossRefGoogle Scholar
[17]Leizarowitz, A.. Infinite horizon autonomous systems with unbounded cost. Appl. Math. Optim. 13 (1985), 19–43.CrossRefGoogle Scholar
[18]Liverani, C., Saussol, B. and Vaienti, S.. A probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys. 19 (1999), 671–685.CrossRefGoogle Scholar
[19]Lopes, A. O. and Thieullen, P.. Sub-actions for Anosov diffeomorphisms. Geometric methods in dynamics II. Astérisque 287(xix) (2003), 135–146.Google Scholar
[20]Niţică, V. and Pollicott, M.. Transitivity of Euclidean extensions of Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 25 (2005), 257–269.CrossRefGoogle Scholar
[21]Peres, Y.. A combinatorial application of the maximal ergodic theorem. Bull. London Math. Soc. 20 (1988), 248–252.CrossRefGoogle Scholar
[22]Savchenko, S. V.. Homological inequalities for finite topological Markov chains. Funct. Anal. Appl. 33 (1999), 236–238.CrossRefGoogle Scholar
[23]Souza, R.. Sub-actions for weakly hyperbolic one-dimensional systems. Dyn. Syst. 18 (2003), 165–179.CrossRefGoogle Scholar
[24]Young, L.-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.CrossRefGoogle Scholar
[25]Yuan, G. and Hunt, B.. Optimal orbits of hyperbolic systems. Nonlinearity 12 (1999), 1207–1224.CrossRefGoogle Scholar