Dimension and product structure of hyperbolic measures

We prove that every hyperbolic measure invariant under a C1+alpha diffeomor phism of a smooth Riemannian manifold possesses asymptotically "almost" loc al product structure, i.e., its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials. This has not been known even for measures supported on locally maximal hyp erbolic sets. Using this property of hyperbolic measures we prove the long-standing Eckma nn-Ruelle conjecture in dimension theory of smooth dynamical systems: the p ointwise dimension of every hyperbolic measure invariant under a C1+alpha d iffeomorphism exists almost everywhere. This implies the crucial fact that virtually all the characteristics of dimension type of the measure (includi ng the Hausdorff dimension, box dimension, and information dimension) coinc ide. This provides the rigorous mathematical justification of the concept o f fractal dimension for hyperbolic measures.