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Ergodic Theory of Differentiable Dynamical Systems

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Real and Complex Dynamical Systems

Part of the book series: NATO ASI Series ((ASIC,volume 464))

Abstract

These notes are about the dynamics of systems with hyperbolic properties. The setting for the first half consists of a pair (f, µ), where f is a diffeomorphism of a Riemannian manifold and µ is an f-invariant Borel probability measure. After a brief review of abstract ergodic theory, Lyapunov exponents are introduced, and families of stable and unstable manifolds are constructed. Some relations between metric entropy, Lyapunov exponents and Hausdorff dimension are discussed. In the second half we address the following question: given a differentiable mapping, what are its natural invariant measures? We examine the relationship between the expanding properties of a map and its invariant measures in the Lebesgue measure class. These ideas are then applied to the construction of Sinai-Ruelle-Bowen measures for Axiom A attractors. The nonuniform case is discussed briefly, but its details are beyond the scope of these notes.

The author is partially supported by the National Science Foundation.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, CA, 90024, USA

    Lai-Sang Young

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  1. Lai-Sang Young
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  1. Mathematical Institute, The Technical University of Denmark, Building 303, DK-2800, Lyngby, Denmark

    Bodil Branner  & Poul Hjorth  & 

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Young, LS. (1995). Ergodic Theory of Differentiable Dynamical Systems. In: Branner, B., Hjorth, P. (eds) Real and Complex Dynamical Systems. NATO ASI Series, vol 464. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8439-5_12

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  • DOI: https://doi.org/10.1007/978-94-015-8439-5_12

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4565-2

  • Online ISBN: 978-94-015-8439-5

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