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Convergence of the Ruelle operator for a function satisfying Bowen's condition

Author: Peter Walters
Journal: Trans. Amer. Math. Soc. 353 (2001), 327-347
MSC (2000): Primary 37D35; Secondary 28D20, 37A30, 37B10
Published electronically: September 13, 2000
MathSciNet review: 1783787
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We consider a positively expansive local homeomorphism $T\colon X\to X$ satisfying a weak specification property and study the Ruelle operator $\mathcal{L}_\varphi$ of a real-valued continuous function $\varphi$satisfying a property we call Bowen's condition. We study convergence properties of the iterates $\mathcal{L}_\varphi^n$ and relate them to the theory of equilibrium states.

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  • [A-H] N. Aoku and K. Hiraide, Topological Theory of Dynamical Systems, North-Holland, 1994. MR 95m:58095
  • [B1] R. Bowen, Some systems with unique equilibrium states, Math. Systems Theory 8 (1974), 193-202. MR 53:3257
  • [B2] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., vol. 470, Springer, Berlin, 1975. MR 56:1364
  • [B-K1] M. Bramson and S. Kalikow, Nonuniqueness in $g$-functions, Israel J. Math. 84 (1993), 153-160. MR 94h:28011
  • [B-K2] M. Brin and A. Katok, On local entropy, in Geometric Dynamics, Lecture Notes in Math., Vol. 1007, Springer, Berlin, 1983. MR 85c:58063
  • [F] Ai Hua Fan, A proof of the Ruelle operator theorem, Rev. Math. Phys. 7 (1995), 1241-1247. MR 97e:28034
  • [H] F. Hofbauer, Examples for the nonuniqueness of the equilibrium state, Trans. Amer. Math. Soc. 228 (1977), 223-241. MR 55:8312
  • [L] F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques, Z. Wahr. und Verw. Gebiete 30 (1974), 185-202. MR 53:8384
  • [P] K. Parthasarathy, Introduction to Probability and Measure, Macmillan, London, 1977. MR 58:31322a
  • [Q] A. Quas, Rigidity of continuous coboundaries, Bull. London Math. Soc. 29 (1997), 595-600. MR 99c:28054
  • [Re] W. Reddy, Expanding maps on compact metric spaces, Topology Appl. 13 (1982), 327-334. MR 83d:54070
  • [Ru] D. Ruelle, Thermodynamic formalism for maps satisfying positive expansiveness and specification, Nonlinearity 5 (1992), 1223-1236. MR 94a:58115
  • [W1] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978), 121-153. MR 57:6371
  • [W2] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Math., vol. 79, Springer, Berlin, 1982. MR 84e:28017

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Additional Information

Peter Walters
Affiliation: University of Warwick, Mathematics Institute, Coventry CV4 7AL, England

Keywords: Transfer operator, equilibrium state, entropy
Received by editor(s): August 9, 1999
Published electronically: September 13, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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