Morris, Ian D.. (2009) The Mañé–Conze–Guivarc'h lemma for intermittent maps of the circle. Ergodic Theory and Dynamical Systems, Vol.29 (No.5). pp. 1603-1611. ISSN 0143-3857

Preview

PDF WRAP_Morris_Mane_Conze_Guivarc'h.pdf - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (160Kb)

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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Ergodic theory, Circle, Geometry, Plane, Lipschitz spaces, Curves, Plane
Journal or Publication Title:	Ergodic Theory and Dynamical Systems
Publisher:	Cambridge University Press
ISSN:	0143-3857
Date:	October 2009
Volume:	Vol.29
Number:	No.5
Page Range:	pp. 1603-1611
Identification Number:	10.1017/S0143385708000837
Status:	Peer Reviewed
Access rights to Published version:	Restricted or Subscription 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. [2] Bousch, T.. Le poisson n’a pas d’arêtes. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), 489–508. [3] Bousch, T.. La condition de Walters. Ann. Sci. École Norm. Sup. 34 (2001), 287–311. [4] Bousch, T.. Nouvelle preuve d’un théorème de Yuan et Hunt. Bull. Soc. Math. France 136(2) (2008), 227–242. [5] Bousch, T.. Le lemme de Mañé–Conze–Guivarc’h pour les systèmes amphidynamiques rectifiables. Preprint, 2007. [6] Bousch, T. and Jenkinson, O.. Cohomology classes of dynamically non-negative Ck functions. Invent. Math. 148 (2002), 207–217. [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. [8] Branton, S.. Sub-actions for Young towers. Preprint. [9] Brémont, J.. Gibbs measures at temperature zero. Nonlinearity 16 (2003), 419–426. [10] Conze, J.-P. and Guivarc’h, Y.. Croissance des sommes ergodiques. Unpublished manuscript, circa 1993. [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. [12] Hu, H.. Decay of correlations for piecewise smooth maps with indifferent fixed points. Ergod. Th. & Dynam. Sys. 24 (2004), 495–524. [13] Hunt, B. and Ott, E.. Optimal periodic orbits of chaotic systems. Phys. Rev. Lett. 76 (1996), 2254–2257. [14] Jenkinson, O.. Geometric barycentres of invariant measures for circle maps. Ergod. Th. & Dynam. Sys. 21 (2001), 1429–1445. [15] Jenkinson, O.. Rotation, entropy and equilibrium states. Trans. Amer. Math. Soc. 353 (2001), 3713–3739. [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. [17] Leizarowitz, A.. Infinite horizon autonomous systems with unbounded cost. Appl. Math. Optim. 13 (1985), 19–43. [18] Liverani, C., Saussol, B. and Vaienti, S.. A probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys. 19 (1999), 671–685. [19] Lopes, A. O. and Thieullen, P.. Sub-actions for Anosov diffeomorphisms. Geometric methods in dynamics II. Astérisque 287(xix) (2003), 135–146. [20] Niţică, V. and Pollicott, M.. Transitivity of Euclidean extensions of Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 25 (2005), 257–269. [21] Peres, Y.. A combinatorial application of the maximal ergodic theorem. Bull. London Math. Soc. 20 (1988), 248–252. [22] Savchenko, S. V.. Homological inequalities for finite topological Markov chains. Funct. Anal. Appl. 33 (1999), 236–238. [23] Souza, R.. Sub-actions for weakly hyperbolic one-dimensional systems. Dyn. Syst. 18 (2003), 165–179. [24] Young, L.-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188. [25] Yuan, G. and Hunt, B.. Optimal orbits of hyperbolic systems. Nonlinearity 12 (1999), 1207–1224.
URI:	http://wrap.warwick.ac.uk/id/eprint/695

Request changes to a record

Actions (login required)

View Item