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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)
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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.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 2 , April 2010 , pp. 469 - 487
Copyright
Copyright © Cambridge University Press 2009

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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