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On the pointwise dimension
of hyperbolic measures: a proof of the Eckmann-Ruelle conjecture
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## On the pointwise dimension

of hyperbolic measures:

a proof of the Eckmann-Ruelle conjecture

### Luis Barreira, Yakov Pesin, and J\"org Schmeling

**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.

*Copyright American Mathematical Society 1996*

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#### Article Info

- ERA Amer. Math. Soc.
**02** (1996), pp. 69-72
- Publisher Identifier: S 1079-6762(96)00007-3
- 1991
*Mathematics Subject Classification*. Primary 58F11,
28D05
- Received by the editors May 13, 1996
- Communicated by Svetlana Katok
- Comments (When Available)

**Luis Barreira**

Department of Mathematics,
The Pennsylvania State University,
University Park, PA 16802, U.S.A.

*E-mail address:* `luis@math.psu.edu`

**Yakov Pesin**

Department of Mathematics,
The Pennsylvania State University,
University Park, PA 16802, U.S.A.

*E-mail address:* `pesin@math.psu.edu`

**J\"org Schmeling**

Weierstrass Institute of Applied Analysis and Stochastics,
Mohrenstrasse 39, D-10117 Berlin, Germany

*E-mail address:* `schmeling@wias-berlin.de`

This paper was written while L. B. was on leave from Instituto
Superior T\'ecnico, 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 Leopoldina-Forderpreis.
The work of Ya. P. was partially supported by the National Science
Foundation grant #DMS9403723.

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