Abstract
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
MathematicsHausdorff dimensionLyapunov exponentDimension functionInfimum and supremumEffective dimensionMinkowski–Bouligand dimensionPacking dimensionErgodic theoryInvariant measurePure mathematicsHausdorff measureEntropy (arrow of time)Outer measureMathematical analysisInvariant (physics)Fractal dimensionFractalMathematical physicsChaotic
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Publication Info
- Year
- 1982
- Type
- article
- Volume
- 2
- Issue
- 1
- Pages
- 109-124
- Citations
- 633
- Access
- Closed
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Cite This
Lai-Sang Young
(1982).
Dimension, entropy and Lyapunov exponents.
Ergodic Theory and Dynamical Systems
, 2
(1)
, 109-124.
https://doi.org/10.1017/s0143385700009615
Identifiers
- DOI
- 10.1017/s0143385700009615