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Dimension, entropy and Lyapunov exponents

  • Lai-Sang Young (a1)
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.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] A. S. Besicovitch . Sets of fractional dimension II: On the sum of digits of real numbers represented in the dyadic system. Math. Ann. 110 (1934), 321329.

[3] R. Bowen . Some systems with unique equilibrium states. Math. Systems Theory. 8 (1974), 193202.

[10] A. Katok . Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. I.H.E.S. 51 (1980), 137174.

[12] F. Ledrappier . Some relations between dimension and Lyapunov exponents. Commun. Math. Phys. 81 (1981), 229238.

[17] Ja. Pesin . Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surveys. 32 (1977), 55114.

[19] D. Ruelle . An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Mat. 9 (1978), 8387.

[20] D. Ruelle . Ergodic theory of differentiable dynamical systems. Publ. Math. I.H.E.S. 50 (1979), 2758.

[21] S. Smale . Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747817.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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