On the pointwise dimension of hyperbolic measures: a proof of the Eckmann-Ruelle conjecture
Authors:
Luis Barreira, Yakov Pesin and Jörg Schmeling
Journal:
Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 69-72
MSC (1991):
Primary 58F11, 28D05
DOI:
https://doi.org/10.1090/S1079-6762-96-00007-8
MathSciNet review:
1405971
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Abstract: We prove the long-standing Eckmann–Ruelle conjecture in dimension theory of smooth dynamical systems. We show that the pointwise dimension exists almost everywhere with respect to a compactly supported Borel probability measure with non-zero Lyapunov exponents, invariant under a $C^{1+\alpha }$ diffeomorphism of a smooth Riemannian manifold.
- J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys. 57 (1985), no. 3, 617–656. MR 800052, DOI https://doi.org/10.1103/RevModPhys.57.617
- Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
- François Ledrappier, Dimension of invariant measures, Proceedings of the conference on ergodic theory and related topics, II (Georgenthal, 1986) Teubner-Texte Math., vol. 94, Teubner, Leipzig, 1987, pp. 116–124. MR 931135
- F. Ledrappier and M. Misiurewicz, Dimension of invariant measures for maps with exponent zero, Ergodic Theory Dynam. Systems 5 (1985), no. 4, 595–610. MR 829860, DOI https://doi.org/10.1017/S0143385700003187
- 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, DOI https://doi.org/10.2307/1971328
- Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann–Ruelle conjecture, Comm. Math. Phys. (to appear).
- Ya. Pesin and C. Yue, The Hausdorff dimension of measures with non-zero Lyapunov exponents and local product structure, PSU preprint.
- Lai Sang Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 109–124. MR 684248, DOI https://doi.org/10.1017/s0143385700009615
- J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys. 57 (3) (1985), 617–656.
- K. Falconer, Fractal geometry. Mathematical foundations and applications, John Wiley & Sons, 1990.
- F. Ledrappier, Dimension of invariant measures, Proceedings of the conference on ergodic theory and related topics, II (Georgenthal, 1986), Teubner-Texte Math., vol. 94, Leipzig, 1987, pp. 116–124.
- F. Ledrappier and M. Misiurewicz, Dimension of invariant measures for maps with exponent zero, Ergod. Theory and Dyn. Syst. 5 (1985), 595–610.
- 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 (3) (1985), 509–539; The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Math. (2) 122 (3) (1985), 540–574.
- Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann–Ruelle conjecture, Comm. Math. Phys. (to appear).
- Ya. Pesin and C. Yue, The Hausdorff dimension of measures with non-zero Lyapunov exponents and local product structure, PSU preprint.
- L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems 2 (1982), 109–124.
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Additional Information
Luis Barreira
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, U.S.A.
MR Author ID:
601208
Email:
luis@math.psu.edu
Yakov Pesin
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, U.S.A.
MR Author ID:
138355
Email:
pesin@math.psu.edu
Jörg Schmeling
Affiliation:
Weierstrass Institute of Applied Analysis and Stochastics, Mohrenstrasse 39, D–10117 Berlin, Germany
Email:
schmeling@wias-berlin.de
Keywords:
Eckmann–Ruelle conjecture,
hyperbolic measure,
pointwise dimension
Received by editor(s):
May 13, 1996
Additional Notes:
This paper was written while L. B. was on leave from Instituto Superior Técnico, Department of Mathematics, at Lisbon, Portugal, and J. S. was visiting Penn State. L. B. was supported by Program PRAXIS XXI, Fellowship BD 5236/95, JNICT, Portugal. J. S. was supported by the Leopoldina-Forderpreis. The work of Ya. P. was partially supported by the National Science Foundation grant #DMS9403723.
Communicated by:
Svetlana Katok
Article copyright:
© Copyright 1996
American Mathematical Society
