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Dimension, entropy and Lyapunov exponents

Published online by Cambridge University Press:  13 August 2009

Lai-Sang Young
Department of Mathematics, Michigan State University, East Lansing, MI 48824
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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.

Research Article
Copyright © Cambridge University Press 1982



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