On the Pointwise Dimensionof Hyperbolic Measures : a Proof of the Eckmann { Ruelle

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+ diieomorphism of a smooth Riemannian manifold. Let M be a smooth Riemannian manifold without boundary, and f : M ! M a C 1+ diieomorphism of M for some > 0. Also let be an f-invariant Borel probability measure on M with a compact support. Given a set Z M, we denote respectively by dim H Z, dim B Z, and dim B Z the Hausdorr dimension of Z and the lower and upper box dimensions of Z (see for example F]). We will be mostly interested in subsets of positive measure invariant under f. To characterize their structure we use the notions of Hausdorr dimension of and lower and upper box dimensions of. We denote them by dim H , dim B , and dim B , respectively. We have dim H = inffdim H Z j (Z) = 1g; dim B = lim !0 It follows from the deenitions that dim H dim B dim B : In Y], Young found a criterion that guarantees the coincidence of the Hausdorr dimension and lower and upper box dimensions of measures. We deene the lower