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 Free Access

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. Rufus Bowen, Some systems with unique equilibrium states, Math. Systems Theory 8 (1974/75), no. 3, 193–202. MR 0399413
  • 4. Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
  • 5. Rufus Bowen and Brian Marcus, Unique ergodicity for horocycle foliations, Israel J. Math. 26 (1977), no. 1, 43–67. MR 0451307
  • 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 𝜎-finite invariant measures, Ann. of Math. (2) 54 (1951), 595–608. MR 0045193
  • 8. Nicolai T. A. Haydn, Canonical product structure of equilibrium states, Random Comput. Dynam. 2 (1994), no. 1, 79–96. MR 1265227
  • 9. N. T. A. Haydn and D. Ruelle, Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification, Comm. Math. Phys. 148 (1992), no. 1, 155–167. MR 1178139
  • 10. C. T. Ionescu Tulcea and G. Marinescu, Théorie ergodique pour des classes d’opérations non complètement continues, Ann. of Math. (2) 52 (1950), 140–147 (French). MR 0037469
  • 11. F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985), no. 3, 509–539. MR 819556, 10.2307/1971328
  • 12. F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Math. (2) 122 (1985), no. 3, 540–574. MR 819557, 10.2307/1971329
  • 13. V. A. Rohlin, On the fundamental ideas of measure theory, Mat. Sbornik N.S. 25(67) (1949), 107–150 (Russian). MR 0030584
    V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation 1952 (1952), no. 71, 55. MR 0047744
  • 14. Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
  • 15. David Ruelle, Thermodynamic formalism for maps satisfying positive expansiveness and specification, Nonlinearity 5 (1992), no. 6, 1223–1236. MR 1192516

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
Email: Renaud.Le-Plaideur@univ-ubs.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02479-4
Received by editor(s): June 30, 1997
Published electronically: November 17, 1999
Article copyright: © Copyright 2000 American Mathematical Society