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A natural space of functions for the Ruelle operator theorem

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Walters, Peter, 1943-. (2007) A natural space of functions for the Ruelle operator theorem. Ergodic Theory and Dynamical Systems, Vol.27 (No.4). pp. 1323-1348. ISSN 0143-3857

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Official URL: http://dx.doi.org/10.1017/S0143385707000028

Abstract

We study a new space, $R(X)$, of real-valued continuous functions on the space $X$ of sequences of zeros and ones. We show exactly when the Ruelle operator theorem holds for such functions. Any $g$-function in $R(X)$ has a unique $g$-measure and powers of the corresponding transfer operator converge. We also show Bow$(X,T)\neq W(X,T)$ and relate this to the existence of bounded measurable coboundaries, which are not continuous coboundaries, for the shift on the space of bi-sequences of zeros and ones.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Ruelle operators, Transfer operators, Operator theory
Journal or Publication Title:	Ergodic Theory and Dynamical Systems
Publisher:	Cambridge University Press
ISSN:	0143-3857
Date:	22 June 2007
Volume:	Vol.27
Number:	No.4
Page Range:	pp. 1323-1348
Identification Number:	10.1017/S0143385707000028
Status:	Peer Reviewed
Access rights to Published version:	Open Access
References:	T. Bousch. La condition de Walters. Ann. Sci. ´Ecole Norm. Sup. (4) 34 (2001), 287–311. R. Bowen. Some systems with unique equilibrium states. Math. Syst. Theory 8 (1974), 193–202. W. Feller. An Introduction to Probability Theory and Its Applications, 2nd edn. Vol. 1. Wiley, New York, 1962. M. E. Fisher. Physica 3 (1967), 255–283. F. Hofbauer. Examples for the nonuniqueness of the equilibrium state. Trans. Amer. Math. Soc. 228 (1977), 223–241. P. Hulse. PhD Thesis, University of Warwick, 1980. A. Johansson and A. Oberg. Square summability of variations of g-functions and uniqueness of g-measures. Math. Res. Lett. 10 (2003), 1–15. A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995. F. Ledrappier. Principe variationnel et syst`emes symboliques. Z. Wahr. Verw. Gebiete 30 (1974), 185–202. A. Quas. Rigidity of continuous coboundaries. Bull. London Math. Soc. 29 (1997), 595–600. P.Walters. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, Berlin, 1982. P.Walters. Ruelle’s operator theorem and g-measures. Trans. Amer. Math. Soc. 214 (1975), 375–387. P. Walters. Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978), 121–153. P. Walters. Convergence of the Ruelle operator for a function satisfying Bowen’s condition. Trans. Amer. Math. Soc. 353 (2001), 327–347. P. Walters. A necessary and sufficient condition for a two-sided continuous function to be cohomologous to a one-sided continuous function. Dyn. Sys. 18 (2003), 131–138. P. Walters. Regularity conditions and Bernoulli properties of equilibrium states and g-measures. J. London Math. Soc (2) 71 (2005), 379–396.
URI:	http://wrap.warwick.ac.uk/id/eprint/658