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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of ergodic theory. ... In mathematics, a probability space or probability measure is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i. ... Statistical physics, one of the fundamental theories of physics, uses methods of statistics in solving physical problems. ... A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ...

Ergodic theorem

Let 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. ). The "time average" is defined as the average (if it exists) over iterations of T starting from some initial point x. In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of ergodic theory. ... In mathematics, a measure is a function that assigns a number, e. ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

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

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 mean is equal to the space mean almost everywhere. This is the celebrated ergodic theorem, in an abstract form due to George David Birkhoff. The equidistribution theorem is a special case of the ergodic theorem, dealing specifically with the distribution of probabilities on the unit interval. In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... George David Birkhoff George David Birkhoff (21 March 1884, Overisel, Michigan - 12 November 1944, Cambridge, Massachusetts) was an American mathematician, best known for what is now called the ergodic theorem. ... In mathematics, the equidistribution theorem is the statement that the sequence a, 2a, 3a, ... mod 1 is uniformly distributed on the unit interval, when a is an irrational number. ...

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

such that

for almost all . Furthermore, f ^* is T-invariant, so that In mathematics, the phrase almost all has a number of specialised uses. ...

almost everywhere. The normalization must be the same,

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

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

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

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.

where χ_A is the indicator function on A. In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ...

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.

(See almost surely.) That is, the smaller A is, the longer it takes to return to it. In mathematics, specifically, in probability theory, the phrase almost surely is a concise, precise way to state except on a set or event of probability measure zero. ...

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. This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ... In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. ... Eberhard Frederich Ferdinand Hopf (1902–1983) was an Austrian mathematician who made significant contributions in topology and ergodic theory. ... 1939 (MCMXXXIX) was a common year starting on Sunday (link will take you to calendar). ... In physics and mathematics, the Hadamard dynamical system or Hadamards billiards is a chaotic dynamical system, a type of dynamical billiards. ... In mathematics and physics, the Artin billiards are a type of dynamical billiards first studied by Emil Artin in 1924. ... Sergei Vasilovich Fomin (9 December 1917 &ndash 17 August 1975) was a Russian mathematician who, among his other accomplishments was a co-author with Kolmogorov in Introductory real analysis, a book that is widely read in Russian and English. ... Israel Moiseevich Gelfand (Израиль Моисеевич Гельфанд) (born 1913 in Okny, Kherson in Ukraine then part of the Russian Empire) is a prolific mathematician in the field of functional analysis, which he interprets in a broad sense as the mathematics of quantum mechanics. ... 1952 (MCMLII) was a Leap year starting on Tuesday (link will take you to calendar). ... In mathematics, the term symmetric space has several different meanings. ... Year 1957 (MCMLVII) was a common year starting on Tuesday of the Gregorian calendar. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ... In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... C. C. Moore may refer to: Charles Chilton Moore, (1837–1906), American atheist Charles Calvin Moore, (1866–1958), Governor of Idaho, 1923-1927 This is a disambiguation page: a list of articles associated with the same title. ... 1966 (MCMLXVI) was a common year starting on Saturday (the link is to a full 1966 calendar). ... In mathematics, suppose C is a collection of mathematical objects (for instance sets or functions). ...

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). In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold M is a certain type of mapping, from M to itself, with rather clearly marked local directions of expansion and contraction. Anosov diffeomorphisms were introduced by D. V. Anosov, who proved... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ... In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. ... See lattice for other meanings of this term, both within and without mathematics. ... In mathematics, the generalized orthogonal group, O(p,q) is the Lie group of all linear transformations of a n = p + q dimensional real vector space which leave invariant a nondegenerate, symmetric bilinear form of signature (p, q). ...

See also

• Mean sojourn time
• Poincaré recurrence theorem
• Ergodic literature

In theories like that of queuing (e. ... The Poincaré recurrence theorem states that a system having a finite amount of energy and confined to a finite spatial volume will, after a sufficiently long time, return to an arbitrarily small neighborhood of its initial state. ... Ergodic literature is literature that requires special effort to comprehend or read, perhaps due to a non linear structure. ...

References

Historical references

• G. D. Birkhoff, Proof of the ergodic theorem, (1931), Proceedings of the National Academy of Sciences USA, 17 pp 656-660.
• 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.

George David Birkhoff George David Birkhoff (21 March 1884, Overisel, Michigan - 12 November 1944, Cambridge, Massachusetts) was an American mathematician, best known for what is now called the ergodic theorem. ... Eberhard Frederich Ferdinand Hopf (1902–1983) was an Austrian mathematician who made significant contributions in topology and ergodic theory. ... Sergei Vasilovich Fomin (9 December 1917 – 17 August 1975) was a Russian mathematician who, among his other accomplishments was a co-author with Kolmogorov of Introductory real analysis, a book that is widely read in Russian and English. ... Israel Moiseevich Gelfand (Russian: ) (born in 1913) is a prolific mathematician in the field of functional analysis, which he interprets in a broad sense as the mathematics of quantum mechanics. ...

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.)

PlanetMath is a free, collaborative, online mathematics encyclopedia. ... Vladimir Igorevich Arnold (Влади́мир И́горевич Арно́льд, born June 12, 1937 in Odessa, USSR) is one of the worlds most prolific mathematicians. ... In mathematics, the equidistribution theorem is the statement that the sequence a, 2a, 3a, ... mod 1 is uniformly distributed on the unit interval, when a is an irrational number. ... In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...

External links

• What Is Ergodicity? An intuitive description of the concept.