Luis Barreira

A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems

A non-additive version of the thermodynamic formalism is developed. This allows us to obtain lower and upper bounds for the dimension of a broad class of Cantor-like sets. These are constructed with a decreasing sequence of closed sets that may satisfy no asymptotic behavior. Moreover, they can be coded by arbitrary symbolic dynamics, and the geometry of the construction may depend on all the symbolic past. Applications include estimates of dimension for hyperbolic sets of maps that need not be differentiable.

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

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 C1+α diffeomorphism of a smooth Riemannian manifold. Let M be a smooth Riemannian manifold without boundary, and f : M → M a C diffeomorphism of M for some α > 0. Also let μ be an f -invariant Borel probability measure on M with a compact support. Given a set Z ⊂M , we denote respectively by dimH Z, dimBZ, and dimBZ the Hausdorff 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 Hausdorff dimension of μ and lower and upper box dimensions of μ. We denote them by dimHμ, dimBμ, and dimBμ, respectively. We have dimHμ = inf{dimH Z | μ(Z) = 1}, dimBμ = lim δ→0 inf{dimBZ | μ(Z) ≥ 1− δ}, dimBμ = lim δ→0 inf{dimBZ | μ(Z) ≥ 1− δ}. It follows from the definitions that dimHμ ≤ dimBμ ≤ dimBμ. In [Y], Young found a criterion that guarantees the coincidence of the Hausdorff dimension and lower and upper box dimensions of measures. We define the lower Received by the editors May 13, 1996. 1991 Mathematics Subject Classification. Primary 58F11, 28D05.