Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Local product structure for Equilibrium States

Author: Renaud Leplaideur
Journal: Trans. Amer. Math. Soc. 352 (2000), 1889-1912
MSC (2000): Primary 37D20, 37D35
Published electronically: November 17, 1999
MathSciNet review: 1661262
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The usual way to study the local structure of Equilibrium State of an Axiom-A diffeomorphism or flow is to use the symbolic dynamic and to push results on the manifold. A new geometrical method is given. It consists in proving that Equilibrium States for Hölder-continuous functions are related to other Equilibrium States of some special sub-systems satisfying a sort of expansiveness. Using different kinds of extensions the local product structure of Gibbs-measure is proven.

References [Enhancements On Off] (What's this?)

  • 1. M. Babillot and F. Ledrappier, Geodesic paths and horocycle flow on abelian covers, Centre de Mathématiques de l'École Polytechnique.
  • 2. L. Barreira, Y. Pesin, and J. Schmeling, Dimension of hyperbolic measures - A proof of the Eckmann-Ruelle conjecture, Weierstraß-Institut für Angewandte Analysis und Stochastik, 1996.
  • 3. R. Bowen, Some Systems with Unique Equilibrium States, Mathematical Systems Theory 8 (1974), no. 3, 193-202. MR 53:3257
  • 4. -, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, 1975. MR 56:1364
  • 5. R. Bowen and B. Marcus, Unique Ergodicity for Horocycle Foliations, Israel Journal of Mathematics 26 (1977), no. 1, 43-67. MR 56:9594
  • 6. A. Broise, F. Dal'bo, and M. Peigné, Méthode de opérateurs de transferts : Transformations dilatantes de l'intervalle et dénombrement de géodésiques fermées.
  • 7. Yael Naim Dowker, Finite and $\sigma$-finite invariant measures, Annals of Mathematics 54 (1951), no. 3, 595-608. MR 13:543a
  • 8. Nicolai T.A. Haydn, Canonical product structure of equilibrium states, Random and Computational Dynamics 2 (1994), no. 1, 79-96. MR 95c:58133
  • 9. N.T.A. Haydn and D. Ruelle, Equivalence of Gibbs and Equilibrium States for Homeomorphisms Satisfying Expansiveness and Specification, Commun. Math. Phys. 148 (1992), 155-167. MR 93h:58084
  • 10. C.T. Ionescu Tulcea and G. Marinescu, Théorie ergodique pour des classes d'opérations non complètement continues, Annals of Mathematics 52 (1950), no. 1, 140-147. MR 12:266g
  • 11. F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms Part I: Characterization of measures satisfying Pesin's entropy formula, Annals of Mathematics 122 (1985), 509-539. MR 87i:58101a
  • 12. -, The metric entropy of diffeomorphisms Part II: Relations between entropy,exponents and dimension, Annals of Mathematics 122 (1985), 540-574. MR 87i:58101b
  • 13. V.A. Rohlin, On the fundamental ideas of measure theory, A.M.S.Translation 10 (1962), 1-52. MR 11:18f; MR 13:924e
  • 14. Walter Rudin, Real and Complex Analysis, Third edition, McGraw-Hill Book Company, 1987.MR 88k:00002
  • 15. D. Ruelle, Thermodynamic formalism for maps satisfying positive expansiveness and specification, Nonlinearity 5 (1992), 1223-1236. MR 94a:58115

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37D20, 37D35

Retrieve articles in all journals with MSC (2000): 37D20, 37D35

Additional Information

Renaud Leplaideur
Affiliation: Laboratoire de Mathématique et Applications des Mathématiques, Université de Bretagne-Sud, 1, rue de la Loi, 56000 Vannes, France

Received by editor(s): June 30, 1997
Published electronically: November 17, 1999
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society