Dimension, entropy and Lyapunov exponents

1 March 1982

journal article
research article
Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems

Vol. 2 (1) , 109-124
https://doi.org/10.1017/s0143385700009615

Abstract

We consider diffeomorphisms of surfaces leaving invariant an ergodic Borel probability measure μ. Define HD (μ) to be the infimum of Hausdorff dimension of sets having full μ-measure. We prove a formula relating HD (μ) to the entropy and Lyapunov exponents of the map. Other classical notions of fractional dimension such as capacity and Rényi dimension are discussed. They are shown to be equal to Hausdorff dimension in the present context.

Keywords

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