Ergodic theory

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 last quarter of the 20th century.

Less formally, ergodic theory is generally about what happens in dynamical systems when they are allowed to run for long periods of time. Markov chains are a particularly simple and common context for applications.

Ergodic transformation
Let $$T:X\to X$$ be a measure-preserving transformation on a measure space $$(X,\Sigma,\mu)$$. An element A of $$\Sigma$$ is T-invariant if A differs from $$T^{-1}(A)$$ by a set of measure zero, i.e. if
 * $$\mu(A\bigtriangleup T^{-1}(A))=0$$

where $$A\bigtriangleup B$$ denotes the set-theoretic symmetric difference of A and B.

The transformation T is said to be ergodic if for every T-invariant element A of $$\Sigma$$, either A or X\A has measure zero.

Ergodic transformations capture a very common phenomenon in statistical physics. For instance, if one thinks of the measure space as a model for the particles of some gas contained in a bounded recipient, with X being a finite set of positions that the particles fill at any time and $$\mu$$ the counting measure on X, and if T(x) is the position of the particle x after one unit of time, then the assertion that T is ergodic means that any part of the gas which is not empty nor the whole recipient is mixed with its complement during one unit of time. This is of course a reasonable assumption from a physical point of view.

Ergodic theorem (Individual or Birkhoff)
Let $$T:X\to X$$ be a measure-preserving transformation on a measure space $$(X,\Sigma,\mu)$$. One may then consider the "time average" of a well-behaved function f (more precisely, f must be L1-integrable with respect to measure $$\mu$$, 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) $$

If $$\mu(X)$$ is finite and nonzero, we can consider the "space average" or "phase average" of f, defined as


 * $$ \bar f =\frac 1{\mu(X)} \int f\,d\mu $$.

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 the time average of f converges almost everywhere and 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, and if $$\mu(X)$$ is finite, then the normalization is the same:


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

In general, if T is ergodic anf if $$f^*$$ is T-invariant, $$f^*$$ is constant almost everywhere, and so one has that


 * $$\bar f = f^*$$

almost everywhere. Joining the first to the last claim and assuming that $$\mu(X)$$ is finite and nonzero, one has that


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

for almost all x, i.e., for all x except for a set of measure zero.

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

As an example, assume that the measure space $$(X,\Sigma,\mu)$$ models the particles of a gas as above, and let f(x) denotes the velocity of the particle at position x. Then the pointwise ergodic theorems says that the average velocity of all particles at some given time is equal to the average velocity of one particle over time.

Mean Ergodic Theorem (Von Neumann's Ergodic Theorem)
From Functional Analysis Point of View

Let $$U$$ be a unitary operator on a Hilbert space $$H$$. Let $$P$$ be the orthogonal projection onto

{$$\psi | \psi \in H, U\psi=\psi$$}.

Then for any $$f \in H$$,

$$ \lim_{N \to \infty} {1 \over N} \sum_{n=0}^{N-1} U^{n} f = P f$$

Sojourn time
Let $$(X,\Sigma,\mu)$$ be a measure space such that $$\mu(X)$$ is finite and nonzero. The time spent in a measurable set A is called the sojourn time. An immediate consequence of the ergodic theorem is that, in an ergodic system, the relative measure of A is equal to the mean sojourn time:


 * $$ \frac{\mu(A)}{\mu(X)} = \frac 1{\mu(X)}\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 $$\chi_A$$ is the indicator function of A, for all x except for a set of measure zero.

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 &minus; ki&minus;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{\mu(X)}{\mu(A)}

\quad\mbox{(almost surely)}$$

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

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).

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.

Modern references

 * 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.
 * (A survey of topics in ergodic theory; with exercises.)
 * Karl Petersen. Ergodic Theory (Cambridge Studies in Advanced Mathematics). Cambridge: Cambridge University Press. 1990.
 * 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.)
 * Karl Petersen. Ergodic Theory (Cambridge Studies in Advanced Mathematics). Cambridge: Cambridge University Press. 1990.
 * 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.)