Ergodic theory

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In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. An older term for this property was metrically transitive. Ergodic theory, the study of ergodic transformations, grew out of an attempt to prove the ergodic hypothesis of statistical physics. Much of the early work in what is now called chaos theory was pursued almost entirely by mathematicians, and published under the title of "ergodic theory", as the term "chaos theory" was not introduced until the middle of the 20th century.

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[edit] Ergodic theorem

Let T:X\to X be a measure-preserving transformation on a measure space (X,Σ,μ). One may then consider the "time average" of a well-behaved function f (more precisely, f must be L1-integrable with respect to measure μ, i.e. f\in L^1(\mu)). The "time average" is defined as the average (if it exists) over iterations of T starting from some initial point x.

\hat f(x) = \lim_{n\rightarrow\infty}\; \frac{1}{n} \sum_{k=0}^{n-1} f\left(T^k x\right)

Consider also the "space average" or "phase average" of f, defined as

\bar f = \int f\,d\mu

where μ is the measure of the probability space.

In general the time average and space average may be different. But if the transformation is ergodic, and the measure is invariant, then the time average is equal to the space average almost everywhere. This is the celebrated ergodic theorem, in an abstract form due to George David Birkhoff. (Actually, Birkhoff's paper considers not the abstract general case but only the case of dynamical systems arising from differential equations on a smooth manifold.) The equidistribution theorem is a special case of the ergodic theorem, dealing specifically with the distribution of probabilities on the unit interval.

More precisely, the pointwise or strong ergodic theorem states that there exists a

f^*\in L^1(\mu)

such that

f^*(x)=\hat{f}(x)

for almost all x\in X. Furthermore, f * is T-invariant, so that

f^* \circ T=f^*

almost everywhere. The normalization must be the same,

\int f^*\, d\mu = \int f\, d\mu.

This, combined with the T-invariance of f * implies that f * is constant almost everywhere, and so one has that

\bar f = f^*

almost everywhere. Joining the first to the last claim, one then has that

\lim_{n\rightarrow\infty}\; \frac{1}{n} \sum_{k=0}^{n-1} f\left(T^k x\right) = \int f\,d\mu

for almost every x. For an ergodic transformation, the time average equals the space average almost surely.

[edit] Sojourn time

The time spent in a measurable set A is called the sojourn time. An immediate consequence of the ergodic theorem is that the measure of A is equal to the mean sojourn time.

\mu(A) = \int \chi_A\, d\mu = \lim_{n\rightarrow\infty}\; \frac{1}{n} \sum_{k=0}^{n-1} \chi_A\left(T^k x\right)

where χA is the indicator function on A.

Let the occurrence times of a measurable set A be defined as the set k1, k2, k3, ..., of times k such that Tk(x) is in A, sorted in increasing order. The differences between consecutive occurrence times Ri = ki − ki−1 are called the recurrence times of A. Another consequence of the ergodic theorem is that the average recurrence time of A is inversely proportional to the measure of A, assuming that the initial point x is in A, so that k0 = 0.

\frac{R_1 + \cdots + R_n}{n} \rightarrow \frac{1}{\mu(A)} \quad\mbox{(almost surely)}

(See almost surely.) That is, the smaller A is, the longer it takes to return to it.

[edit] Ergodic flows on manifolds

The ergodicity of the geodesic flow on manifolds of constant negative curvature was discovered by Eberhard Hopf in 1939, although special cases were studied earlier; see for example, Hadamard's billiards (1898) and Artin's billiards (1924). The relation between geodesic flows and one-parameter subgroups on SL(2,R) was given by S. V. Fomin and I. M. Gelfand in 1952. Ergodicity of geodesic flow in symmetric spaces was given by F. I. Mautner in 1957. A simple criterion for the ergodicity of a homogeneous flow on a homogeneous space of a semisimple Lie group was given by C. C. Moore in 1966. Many of the theorems and results from this area of study are typical of rigidity theory.

The article on Anosov flows provides an example of ergodic flows on SL(2,R) and more generally on Riemann surfaces of negative curvature. Much of the development given there generalizes to hyperbolic manifolds of constant negative curvature, as these can be viewed as the quotient of a simply connected hyperbolic space modulo a lattice in SO(n,1).

[edit] See also

[edit] References

[edit] Historical references

  • G. D. Birkhoff, Proof of the ergodic theorem, (1931), Proc Natl Acad Sci U S A, 17 pp 656-660.
  • J. von Neumann, Proof of the Quasi-ergodic Hypothesis, (1932), Proc Natl Acad Sci U S A, 18 pp 70-82.
  • J. von Neumann, Physical Applications of the Ergodic Hypothesis, (1932), Proc Natl Acad Sci U S A, 18 pp 263-266.
  • E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, (1939) Leipzig Ber. Verhandl. Sächs. Akad. Wiss. 91, p.261-304.
  • S. V. Fomin and I. M. Gelfand, Geodesic flows on manifolds of constant negative curvature, (1952) Uspehi Mat. Nauk 7 no. 1. p. 118-137.
  • F. I. Mautner, Geodesic flows on symmetric Riemann spaces, (1957) Ann. of Math. 65 p. 416-431.
  • C. C. Moore, Ergodicity of flows on homogeneous spaces, (1966) Amer. J. Math. 88, p.154-178.

[edit] Modern references

  • D.V. Anosov, "Ergodic theory" SpringerLink Encyclopaedia of Mathematics (2001)
  • This article incorporates material from ergodic theorem on PlanetMath, which is licensed under the GFDL.
  • Vladimir Igorevich Arnol'd and André Avez, Ergodic Problems of Classical Mechanics. New York: W.A. Benjamin. 1968.
  • Leo Breiman, Probability. Original edition published by Addison-Wesley, 1968; reprinted by Society for Industrial and Applied Mathematics, 1992. ISBN 0-89871-296-3. (See Chapter 6.)
  • Peter Walters, An introduction to ergodic theory, Springer, New York, 1982, ISBN 0-387-95152-0.
  • Tim Bedford, Michael Keane and Caroline Series, eds. (1991). Ergodic theory, symbolic dynamics and hyperbolic spaces. Oxford University Press. ISBN 0-19-853390-X.  (A survey of topics in ergodic theory; with exercises.)
  • Joseph M. Rosenblatt and Máté Weirdl, Pointwise ergodic theorems via harmonic analysis, (1993) appearing in Ergodic Theory and its Connections with Harmonic Analysis, Proceedings of the 1993 Alexandria Conference, (1995) Karl E. Petersen and Ibrahim A. Salama, eds., Cambridge University Press, Cambridge, ISBN 0-521-45999-0. (An extensive survey of the ergodic properties of generalizations of the equidistribution theorem of shift maps on the unit interval. Focuses on methods developed by Bourgain.)

[edit] External links

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