Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-06-22T20:23:12.290Z Has data issue: false hasContentIssue false
  • English
  • Français

Ruelle operator theorem for non-expansive systems

Published online by Cambridge University Press:  23 June 2009

YUNPING JIANG  and
YUAN-LING YE
YUNPING JIANG
Affiliation:
Department of Mathematics, Queens College of CUNY, Flushing, NY 11367, USA Department of Mathematics, CUNY Graduate Center, 365 Fifth Avenue, New York, NY 10016, USA (email: yunping.jiang@qc.cuny.edu)
YUAN-LING YE
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, People’s Republic of China (email: ylye@scnu.edu.cn)
Get access Rights & Permissions [Opens in a new window]

Abstract

The Ruelle operator theorem has been studied extensively both in dynamical systems and iterated function systems. In this paper we study the Ruelle operator theorem for non-expansive systems. Our theorems give some sufficient conditions for the Ruelle operator theorem to be held for a non-expansive system.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 2 , April 2010 , pp. 469 - 487
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]Baladi, V., Jiang, Y. P. and Lanford III, O. E.. Transfer operators acting on Zygmund functions. Trans. Amer. Math. Soc. 348 (1996), 1599–1615.CrossRefGoogle Scholar
[2]Barnsley, M. F., Demko, S. G., Elton, J. H. and Geronimo, J. S.. Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities. Ann. Inst. Henri Poincaré 24 (1988), 367–394.Google Scholar
[3]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.CrossRefGoogle Scholar
[4]Dunford, N. and Schwartz, J. T.. Linear Operators. Part I. Wiley-Interscience, New York, 1958.Google Scholar
[5]Fan, A. H.. A proof of the Ruelle operator theorem. Rev. Math. Phys. 7 (1995), 1241–1247.CrossRefGoogle Scholar
[6]Fan, A. H. and Jiang, Y. P.. On Ruelle–Perron–Frobenius operators I. Ruelle Theorem. Commun. Math. Phys. 223 (2001), 125–141.CrossRefGoogle Scholar
[7]Fan, A. H. and Jiang, Y. P.. On Ruelle-Perron-Frobenius operators II. Convergence speeds. Commun. Math. Phys. 223 (2001), 143–159.CrossRefGoogle Scholar
[8]Fan, A. H. and Lau, K. S.. Iterated function system and Ruelle operator. J. Math. Anal. Appl. 231 (1999), 319–344.CrossRefGoogle Scholar
[9]Hata, M.. On the structure of self-similar sets. Japan J. Appl. Math. 2 (1985), 381–414.CrossRefGoogle Scholar
[10]Hennion, H.. Sur un théorèm spectral et son application aux noyaux lipchitziens. Proc. Amer. Math. Soc. 118 (1993), 627–634.Google Scholar
[11]Hutchinson, J. E.. Fractal and self-similarity. Indian Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
[12]Jiang, Y.. A proof of existence and simplicity of a maximal eigenvalue for Ruelle–Perron–Frobenius operators. Lett. Math. Phys. 48(3) (1999), 211–219.CrossRefGoogle Scholar
[13]Jiang, Y. and Maume-Deschamps, V.. RPF operators for non-Hölder potentials on an arbitrary metric space, Unpublished Note.Google Scholar
[14]Lasota, A. and Yorke, J. A.. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973), 481–488.CrossRefGoogle Scholar
[15]Lau, K. S. and Ye, Y. L.. Ruelle operator with nonexpansive IFS. Studia Math. 148 (2001), 143–169.CrossRefGoogle Scholar
[16]Liverani, C., Saussol, B. and Vaienti, S.. A probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys. 19 (1999), 671–685.CrossRefGoogle Scholar
[17]Mauldin, R. D. and Urbański, M.. Parabolic iterated function systems. Ergod. Th. & Dynam. Sys. 20 (2000), 1423–1448.CrossRefGoogle Scholar
[18]Nussbaum, R. D.. The radius of the essential spectrum. Duke Math. J. 37 (1970), 473–478.CrossRefGoogle Scholar
[19]Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187/188 (1990), 1–268.Google Scholar
[20]Pomeau, Y. and Manneville, P.. Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 74(2) (1980), 189–197.CrossRefGoogle Scholar
[21]Prellberg, T. and Slawny, J.. Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions. J. Statist. Phys. 66 (1991), 503–514.CrossRefGoogle Scholar
[22]Ruelle, D.. Statical mechanics of a one-dimensional lattice gas. Commun. Math. Phys. 9 (1968), 267–278.CrossRefGoogle Scholar
[23]Rugh, H. H.. Intermittency and regularized Fredholm determinants. Invent. Math. 135(1) (1999), 1–24.CrossRefGoogle Scholar
[24]Urbański, M.. Parabolic Cantor sets. Fund. Math. 151 (1996), 241–277.Google Scholar
[25]Walters, P.. Ruelle’s operator theorem and g-measure. Trans. Amer. Math. Soc. 214 (1975), 375–387.Google Scholar
[26]Walters, P.. Convergence of the Ruelle operator for a function satisfying Bowen’s condition. Trans. Amer. Math. Soc. 353 (2001), 327–347.CrossRefGoogle Scholar
[27]Ye, Y. L.. Decay of correlations for weakly expansive dynamical systems. Nonlinearity 17 (2004), 1377–1391.CrossRefGoogle Scholar
[28]Ye, Y. L.. Non-hyperbolic dynamical systems on [0, 1]. Preprint, 2008.Google Scholar
[29]Young, L. S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.CrossRefGoogle Scholar
[30]Yuri, M.. Invariant measures for certain multi-dimensional maps. Nonlinearity 7 (1994), 1093–1124.CrossRefGoogle Scholar