Ergodic Theory of Differentiable Dynamical Systems

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


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.


Lyapunov Exponent Invariant Measure Ergodic Theory Unstable Manifold Borel Probability Measure 
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© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

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