Ergodic theory

From Wikipedia, the free encyclopedia

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics.

A central aspect of ergodic theory is the behavior of a dynamical system when it is allowed to run long. This is expressed through ergodic theorems which assert that, under certain conditions, the time average of a function along the trajectories exists almost everywhere and is related to the space average. Two most important examples are the ergodic theorems of Birkhoff and von Neumann. For the special class of ergodic systems, the time average is the same for almost all initial points: statistically speaking, the system that evolves for a long time "forgets" its initial state. Stronger properties, such as mixing and equidistribution have also been extensively studied. The problem of metric classification of systems is another important part of the abstract ergodic theory. An outstanding role in ergodic theory and its applications to stochastic processes is played by the various notions of entropy for dynamical systems.

Applications of ergodic theory to other parts of mathematics usually involve establishing ergodicity properties for systems of special kind. In geometry, methods of ergodic theory have been used to study the geodesic flow on Riemannian manifolds, starting with the results of Eberhard Hopf for Riemann surfaces of negative curvature. Markov chains form a common context for applications in probability theory. Ergodic theory has fruitful connections with harmonic analysis, Lie theory (representation theory, lattices in algebraic groups), and number theory (the theory of diophantine approximations, L-functions).

[edit] Ergodic transformations

Let T: X → X be a measure-preserving transformation on a measure space (X, Σ, μ), usually assumed to have finite measure. An element A of Σ is T-invariant mod 0 if T⁻¹(A) differs from A by a set of measure zero:

$\mu(T^{-1}(A)\bigtriangleup A)=0,$

where $\bigtriangleup$ denotes the symmetric difference. If this is true then A is Tⁿ-invariant mod 0 for all n.

A measure-preserving transformation T as above is ergodic if for every T-invariant element mod 0 measurable set A, either A or its complement X\A has measure zero. In older literature, ergodic transformations were called metrically transitive.

These definitions have natural analogues for the case of measurable flows and, more generally, measure-preserving semigroup actions. Let {T^t} be a measurable flow on (X, Σ, μ). An element A of Σ is invariant mod 0 under {T^t} if

$\mu(T^{t}(A))\bigtriangleup A=0$

for each t ∈ R. Measurable sets invariant mod 0 under a flow or a semigroup action form the invariant subalgebra of Σ, and the corresponding measure-preserving dynamical system is ergodic if the invariant subalgebra is the trivial σ-algebra consisting of the sets of measure 0 and their complements in X. If the measure is normalized, μ(X)=1, so that (X, Σ, μ) is a probability space, then all invariant mod 0 sets must have measure 0 or 1.

Conceptually, ergodicity of a dynamical system is a certain irreducibility property, akin to the notions of irreducible representation in algebra and prime number in arithmetic. A general measure-preserving transformation or flow on a Lebesgue space admits a canonical decomposition into its ergodic components, each of which is ergodic.

[edit] Examples

An irrational rotation of the circle R/Z, T: x → x+θ, where θ is irrational, is ergodic. This transformation has even stronger properties of unique ergodicity, minimality, and equidistribution. By contrast, if θ = p/q is rational (in lowest terms) then T is periodic, with period q, and thus cannot be ergodic: for any interval I of length a, 0 < a < 1/q, its orbit under T is a T-invariant mod 0 set that is a union of q intervals of length a, hence it has measure qa strictly between 0 and 1.

Let G be a compact abelian group, μ the normalized Haar measure, and T a group automorphism of G. Let G^* be the Pontryagin dual group, consisting of the continuous characters of G, and T^* be the corresponding adjoint automorphism of G^*. The automorphism T is ergodic if and only if the equality (T^*)ⁿ(χ)=χ is possible only when n = 0 or χ is the trivial character of G. In particular, if G is the n-dimensional torus and the automorphism T is represented by an integral matrix A then T is ergodic if and only if no eigenvalue of A is a root of unity.

A Bernoulli shift is ergodic. More generally, ergodicity of the shift transformation associated with a sequence of i.i.d. random variables and some more general stationary processes follows from Kolmogorov's zero-one law.

Ergodicity of a continuous dynamical system means that its trajectories "spread around" the phase space. A system with a compact phase space which has a non-constant first integral cannot be ergodic. This applies, in particular, to Hamiltonian systems with a first integral I functionally independent from the Hamilton function H and a compact level set X = {(p,q): H(p,q)=E} of constant energy. Liouville's theorem implies the existence of a finite invariant measure on X, but the dynamics of the system is constrained to the level sets of I on X, hence the system possesses invariant sets of positive but less than full measure. A property of continuous dynamical systems that is the opposite of ergodicity is complete integrability.

[edit] Ergodic theorems

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 $μ$ -integrable function f, 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 $μ(X)$ is finite and nonzero, we can consider the "space average" or "phase average" of f, defined as

$\bar f =\frac 1{\mu(X)} \int f\,d\mu$ . (For a probability space,

μ(X) = 1

)

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 limit in the definition of the time average of f exists for almost every x and that the (almost everywhere defined) limit function $\hat f$ is integrable:

$\hat f \in L^1(\mu)$

Furthermore, $\hat f$ is T-invariant, that is to say

$\hat f \circ T= \hat f$

holds almost everywhere, and if $μ(X)$ is finite, then the normalization is the same:

$\int \hat f\, d\mu = \int f\, d\mu.$

In particular, if T is ergodic, then $\hat f$ must be a constant (almost everywhere), and so one has that

$\bar f = \hat f$

almost everywhere. Joining the first to the last claim and assuming that $μ(X)$ 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 $(X,Σ,μ)$ 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.

[edit] Probabilistic formulation: Birkhoff-Khinchin theorem

Birkhoff-Khinchin theorem. Let $f$ be measurable, $E(|f|)<+\infty$ , and $T$ be a measure-preserving operator. Then

$\lim_{n\rightarrow\infty}\; \frac{1}{n} \sum_{k=0}^{n-1} f\left(T^k x\right)=E(f|\mathcal{C}),$

where $E(f|\mathcal{C})$ is the conditional expectation given the $σ$ -algebra $\mathcal{C}$ of invariant sets of $T$ .

Corollary (Pointwise ergodic theorem) In particular, if $T$ is also ergodic, then $\mathcal{C}$ is the trivial $σ$ -algebra, and thus

$\lim_{n\rightarrow\infty}\; \frac{1}{n} \sum_{k=0}^{n-1} f\left(T^k x\right)=E(f)$ a.s.

[edit] Mean Ergodic Theorem

Another form of the ergodic theorem, von Neumann's mean ergodic theorem, holds in Hilbert spaces.^[1]

Let $U$ be a unitary operator on a Hilbert space $H$ . Let $P$ be the orthogonal projection onto $\{\psi \in H| U\psi=\psi\} = \operatorname{Ker}(\operatorname{id}-U)$ .

Then, for any $x \in H$ , we have:

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

where the limit is with respect to the norm on H. In other words, the sequence of averages

$\frac{1}{N} \sum_{n=0}^{N-1}U^n$

converges to P in the strong operator topology.

This theorem specializes to the case in which the Hilbert space H consists of L² functions on a measure space and U is an operator of the form

U f (x) = f (T x)

where T is a measure-preserving automorphism of X, thought of in applications as representing a time-step of a discrete dynamical system.^[2] The ergodic theorem then asserts that the average behavior of a function f over sufficiently large time-scales is approximated by the orthogonal component of f which is time-invariant.

In another form of the mean ergodic theorem, let U_t be a strongly continuous one-parameter group of unitary operators on H. Then the operator

$\frac{1}{T}\int_0^T U_t\,dt$

converges in the strong operator topology as T → ∞. In fact, this result also extends to the case of strongly continuous one-parameter semigroup of contractive operators on a reflexive space.

[edit] Sojourn time

Let $(X,Σ,μ)$ be a measure space such that $μ(X)$ 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 $χ A$ 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 T^k(x) is in A, sorted in increasing order. The differences between consecutive occurrence times R_i = k_i − k_i−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.

[edit] Ergodic flows on manifolds

The ergodicity of the geodesic flow on compact Riemann surfaces of variable negative curvature and on compact manifolds of constant negative curvature of any dimension was proved by Eberhard Hopf in 1939, although special cases had been studied earlier: see for example, Hadamard's billiards (1898) and Artin billiard (1924). The relation between geodesic flows on Riemann surfaces and one-parameter subgroups on SL(2,R) was described in 1952 by S. V. Fomin and I. M. Gelfand. The article on Anosov flows provides an example of ergodic flows on SL(2,R) and on Riemann surfaces of negative curvature. Much of the development described there generalizes to hyperbolic manifolds, since they can be viewed as quotients of the hyperbolic space by the action of a lattice in the semisimple Lie group SO(n,1). Ergodicity of the geodesic flow on Riemannian symmetric spaces was demonstrated by F. I. Mautner in 1957. In 1967 D. V. Anosov and Ya. G. Sinai proved ergodicity of the geodesic flow on compact manifolds of variable negative sectional curvature. 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.

In the 1930s G. A. Hedlund proved that the horocycle flow on a compact hyperbolic surface is minimal and ergodic. Unique ergodicity of the flow was established by Hillel Furstenberg in 1972. Ratner's theorems provide a major generalization of ergodicity for unipotent flows on the homogeneous spaces of the form Γ\G, where G is a Lie group and Γ is a lattice in G.

[edit] See also

[edit] References

^ I: Functional Analysis : Volume 1 by Michael Reed, Barry Simon,Academic Press; REV edition (1980)
^ (Walters 1982)

[edit] Historical references

Birkhoff, George David (1931), "Proof of the ergodic theorem", Proc Natl Acad Sci USA 17: 656–660, doi:10.1073/pnas.17.12.656, http://www.pnas.org/cgi/reprint/17/12/656 .
Birkhoff, George David (1942), "What is the ergodic theorem?", American Mathematical Monthly 49 (4): 222–226, doi:10.2307/2303229, http://www.jstor.org/stable/2303229 .
von Neumann, John (1932), "Proof of the Quasi-ergodic Hypothesis", Proc Natl Acad Sci USA 18: 70–82, doi:10.1073/pnas.18.1.70 .
von Neumann, John (1932), "Physical Applications of the Ergodic Hypothesis", Proc Natl Acad Sci USA 18: 263–266, doi:10.1073/pnas.18.3.263, http://www.jstor.org/stable/86260 .
Hopf, Eberhard (1939), "Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung", Leipzig Ber. Verhandl. Sächs. Akad. Wiss. 91: 261–304 .
Fomin, Sergei V.; Gelfand, I. M. (1952), "Geodesic flows on manifolds of constant negative curvature", Uspehi Mat. Nauk 7 (1): 118–137 .
Mautner, F. I. (1957), "Geodesic flows on symmetric Riemann spaces", Ann. Of Math. 65: 416–431, doi:10.2307/1970054 .
Moore, C. C. (1966), "Ergodicity of flows on homogeneous spaces", Amer. J. Math. 88: 154–178, doi:10.2307/2373052 .

[edit] Modern references

D.V. Anosov (2001), "Ergodic theory", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
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.)
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.)
A.N. Shiryaev, Probability, 2nd ed., Springer 1996, Sec. V.3. ISBN 0-387-94549-0.
http://www.cscs.umich.edu/~crshalizi/notebooks/ergodic-theory.html

[0] I: Functional Analysis : Volume 1 by Michael Reed, Barry Simon,Academic Press; REV edition (1980)

[1] (Walters 1982)

[1]

[2]

Ergodic theory

From Wikipedia, the free encyclopedia

Contents

[edit] Ergodic transformations

[edit] Examples

[edit] Ergodic theorems

[edit] Probabilistic formulation: Birkhoff-Khinchin theorem

[edit] Mean Ergodic Theorem

[edit] Sojourn time

[edit] Ergodic flows on manifolds

[edit] See also

[edit] References

[edit] Historical references

[edit] Modern references

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