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Smooth Ergodic Theory

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Smooth ergodic theoryis the study of the statistical and geometric properties of measures invariant undera smooth transformation or flow. The study of smooth ergodic theory is as old as the study of abstract ergodic theory, having its origins inBolzmann's Ergodic Hypothesis in the late 19th Century. As a response to Boltzmann's hypothesis, which was formulated in the context of HamiltonianMechanics, Birkhoff and von Neumann defined ergodicity in the 1930s and proved their foundational ergodic theorems. The study of ergodic properties ofsmooth systems saw an advance in the work of Hadamard and E. Hopf in the 1930s their study of geodesic flows for negatively curved surfaces. Beginning inthe 1950s, Kolmogorov, Arnold and Moser developed a perturbative theory producing obstructions to ergodicity in Hamiltonian systems, known asKolmogorov–Arnold–Moser (KAM) Theory. Beginning in the 1960s with the work of Anosov and Sinai on hyperbolic systems, the study of...

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Abbreviations

Conservative, dissipative:

Conservative dynamical systems (on a compact phase space) are those that preserve a finite measure equivalent to volume. Hamiltonian dynamical systems are important examples of conservative systems. Systems that are not conservative are called dissipative. Finding physically meaningful invariant measures for dissipative maps is a central object of study in smooth ergodic theory.

Distortion estimate:

A key technique in smooth ergodic theory, a distortion estimate for a smooth map f gives a bound on the variation of the jacobian of f n in a given region, for n arbitrarily large. The jacobian of a smooth map at a point x is the absolute value of the determinant of derivative at x, measured in a fixed Riemannian metric. The jacobian measures the distortion of volume under f in that metric.

Hopf argument:

A technique developed by Eberhard Hopf for proving that a conservative diffeomorphism or flow is ergodic. The argument relies on the Ergodic Theorem for invertible transformations, the density of continuous functions among integrable functions, and the existence of stable and unstable foliations for the system. The argument has been used, with various modifications, to establish ergodicity for hyperbolic, partially hyperbolic and nonuniformly hyperbolic systems.

Hyperbolic:

A compact invariant set \( { \Lambda\subset M } \) for a diffeomorphism \( { f\colon M\to M } \) is hyperbolic if, at every point in ?, the tangent space splits into two subspaces, one that is uniformly contracted by the derivative of f, and another that is uniformly expanded. Expanding maps and Anosov diffeomorphisms are examples of globally hyperbolic maps. Hyperbolic diffeomorphisms and flows are the archetypical smooth systems displaying chaotic behavior, and their dynamical properties are well-understood. Nonuniform hyperbolicity and partial hyperbolicity are two generalizations of hyperbolicity that encompass a broader class of systems and display many of the chaotic features of hyperbolic systems.

Sinai–Ruelle–Bowen (SRB) measure:

The concept of SRB measure is a rigorous formulation of what it means for an invariant measure to be “physically meaningful”. An SRB measure attracts a large set of orbits into its support, and its statistical features are reflected in the behavior of these attracted orbits.

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Authors and Affiliations

  1. Northwestern University, Evanston, USA

    Amie Wilkinson

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  1. Amie Wilkinson
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Wilkinson, A. (2009). Smooth Ergodic Theory. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_484

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