Ergodic theory - Citizendia

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position In Mathematics, an invariant measure is a measure that is preserved by some function. Its initial development was motivated by problems of statistical physics. Statistical physics is one of the fundamental theories of Physics, and uses methods of Statistics in solving physical problems

A central aspect of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long period of time. 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. 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 Two most important examples are the ergodic theorems of Birkhoff and von Neumann. George David Birkhoff ( 21 March 1884, Overisel Michigan - 12 November 1944, Cambridge Massachusetts) was an American 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. In Mathematics, mixing is an abstract concept originating from Physics: the attempt to describe the irreversible Thermodynamic process of mixing 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. A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory. In Mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of Dynamical systems, and Ergodic theory in particular

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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M Eberhard Frederich Ferdinand Hopf ( April 4, 1902 Salzburg Austria – July 24, 1983 Bloomington Indiana was a Mathematician In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional Markov chains form a common context for applications in probability theory. In Mathematics, a Markov chain, named after Andrey Markov, is a Stochastic process with the Markov property. Probability theory is the branch of Mathematics concerned with analysis of random phenomena 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). Harmonic analysis is the branch of Mathematics that studies the representation of functions or signals as the superposition of basic Waves It investigates and generalizes Lie theory is an area of Mathematics, developed initially by Sophus Lie. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Lie theory and related areas of Mathematics, a lattice in a Locally compact Topological group is a Discrete subgroup with the property In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes In Number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of Real numbers by Rational The theory of L -functions has become a very substantial and still largely Conjectural, part of contemporary Number theory.

1 Ergodic transformation
2 Ergodic theorem (Individual or Birkhoff)
3 Mean Ergodic Theorem (Von Neumann's Ergodic Theorem)
4 Sojourn time
5 Ergodic flows on manifolds
6 See also
7 References
8 Historical references
9 Modern references

Ergodic transformation

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. In Mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of Dynamical systems, and Ergodic theory in particular A probability space, in Probability theory, is the conventional Mathematical model of Randomness. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with An older term for this property was metrically transitive.

Let $T:X\to X$ be a measure-preserving transformation on a measure space $(X,Σ,μ)$ . In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with An element A of $Σ$ 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. In Mathematics, the symmetric difference of two sets is the set of elements which are in one of the sets but not in both

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

Ergodic transformations capture a very common phenomenon in statistical physics. Statistical physics is one of the fundamental theories of Physics, and uses methods of Statistics in solving physical problems 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 $μ$ 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 page is about the physical properties of gas as a state of matter In Mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a Subset is taken to be the number 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 $(X,Σ,μ)$ . In Mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of Dynamical systems, and Ergodic theory in particular In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with 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. Mathematicians (and those in related sciences very frequently speak of whether a mathematical object &mdash a Number, a function, a set, a space In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding 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

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

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. 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 This is the celebrated ergodic theorem, in an abstract form due to George David Birkhoff. George David Birkhoff ( 21 March 1884, Overisel Michigan - 12 November 1944, Cambridge Massachusetts) was an American (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. In Mathematics, the equidistribution theorem is the statement that the sequence a, 2 a, 3 a,.

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 $μ(X)$ is finite, then the normalization is the same:

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

In general, if T is ergodic and 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 $μ(X)$ 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. See also Generic property In Mathematics, the phrase almost all has a number of specialised uses e. , for all x except for a set of measure zero. In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to

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. In Physics, velocity is defined as the rate of change of Position. 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)

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 $H$ . In Functional analysis, a branch of Mathematics, a unitary operator is a Bounded linear operator U H → This article assumes some familiarity with Analytic geometry and the concept of a limit. Let $P$ be the orthogonal projection onto $\{\psi \in H| U\psi=\psi\} = \operatorname{Ker}(\operatorname{id}-U)$ .

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

$\lim_{N \to \infty} {1 \over N} \sum_{n=0}^{N-1} U^{n} f = P f$ , where the limit is in the L² sense.

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. In Mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to

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. In Probability theory, one says that an event happens almost surely (a ) 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 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). In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional In Differential geometry, the Gaussian curvature or Gauss curvature of a point on a Surface is the product of the Principal curvatures In Mathematics, a hyperbolic n -manifold is a complete Riemannian n-manifold of constant Sectional curvature -1 Eberhard Frederich Ferdinand Hopf ( April 4, 1902 Salzburg Austria – July 24, 1983 Bloomington Indiana was a Mathematician In Physics and Mathematics, the Hadamard dynamical system or Hadamard's billiards is a chaotic Dynamical system, a type of Dynamical In Mathematics and Physics, the Artin billiard is a type of a dynamical billiard first studied by Emil Artin in 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. In Mathematics, the Special linear group SL2( R) is the group of all real 2 × 2 matrices with Determinant one Sergei Vasilovich Fomin ( 9 December 1917 &ndash 17 August 1975) was a Russian Mathematician whoamong his other accomplishmentswas Israïl Moiseevich Gelfand (Израиль Моисеевич Гельфанд ישראל געלפֿאַנד (born on) is a Mathematician who has contributed substantially The article on Anosov flows provides an example of ergodic flows on SL(2,R) and on Riemann surfaces of negative curvature. In Mathematics, more particularly in the fields of Dynamical systems and Geometric topology, an Anosov map on a Manifold M is a certain In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional 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). In Mathematics, a hyperbolic n -manifold is a complete Riemannian n-manifold of constant Sectional curvature -1 In Mathematics, hyperbolic n -space, denoted H n, is the maximally symmetric Simply connected, n -dimensional In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Lie theory and related areas of Mathematics, a lattice in a Locally compact Topological group is a Discrete subgroup with the property In Mathematics, the indefinite orthogonal group, O( p, q) is the Lie group of all Linear transformations of a n = p 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. Dmitri Victorovich Anosov (b November 30, 1936 in Moscow) (Аносов Дмитрий Викторович is a Russian mathematician known for his Yakov Grigorevich Sinai (Яков Григорьевич Синай born September 21 1935) is one of the most influential Mathematicians of the twentieth In Riemannian geometry, the sectional curvature is one of the ways to describe the Curvature of Riemannian manifolds. 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. In Mathematics, particularly in the theories of Lie groups Algebraic groups and Topological groups a homogeneous space for a group In Mathematics, a Lie algebra is semisimple if it is a Direct sum of Simple Lie algebras i Many of the theorems and results from this area of study are typical of rigidity theory. In Mathematics, suppose C is a collection of mathematical objects (for instance sets or functions

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. Hillel (Harry Furstenberg (הלל (הארי פורסטנברג is an Israeli Mathematician, a member of the Israel Academy of Sciences and Humanities and U 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. In Mathematics, Ratner's theorems is a group of major theorems in Ergodic theory concerning unipotent flows on Homogeneous spaces proved by Marina Ratner In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Lie theory and related areas of Mathematics, a lattice in a Locally compact Topological group is a Discrete subgroup with the property

References

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

Historical references

G. D. Birkhoff, Proof of the ergodic theorem, (1931), Proc Natl Acad Sci U S A, 17 pp 656-660. In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that Dynamical systems theory is an area of Applied mathematics used to describe the behavior of complex Dynamical systems usually by employing Differential In Physics and Thermodynamics, the ergodic hypothesis says that over long periods of time the time spent by a particle in some region of the Phase space For functional analysis as used in psychology see the Functional analysis (psychology article The maximal ergodic theorem is a Theorem in Ergodic theory, a discipline within Mathematics. The mean sojourn time for a system is a mathematical term for the amount of time an object is expected to spend in a system before leaving the system for good In Mathematics, the Poincaré recurrence theorem states that certain systems will after a sufficiently long time return to a state very close to the initial state Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In Mathematics, a Markov chain, named after Andrey Markov, is a Stochastic process with the Markov property. George David Birkhoff ( 21 March 1884, Overisel Michigan - 12 November 1944, Cambridge Massachusetts) was an American
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. Eberhard Frederich Ferdinand Hopf ( April 4, 1902 Salzburg Austria – July 24, 1983 Bloomington Indiana was a Mathematician 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. Sergei Vasilovich Fomin ( 9 December 1917 &ndash 17 August 1975) was a Russian Mathematician whoamong his other accomplishmentswas Israïl Moiseevich Gelfand (Израиль Моисеевич Гельфанд ישראל געלפֿאַנד (born on) is a Mathematician who has contributed substantially 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

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. The Encyclopaedia of Mathematics is a large reference work in Mathematics. PlanetMath is a free, collaborative online Mathematics Encyclopedia.
Vladimir Igorevich Arnol'd and André Avez, Ergodic Problems of Classical Mechanics. Vladimir Igorevich Arnol'd or Arnold (Влади́мир И́горевич Арно́льд born June 12, 1937 in Odessa, Ukrainian SSR 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. In Mathematics, the equidistribution theorem is the statement that the sequence a, 2 a, 3 a,. 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, that is the set of all Real numbers x such that zero is less than or equal to x Focuses on methods developed by Bourgain. )

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