Ergodic theory
Encyclopedia
Ergodic theory is a branch of mathematics
that studies dynamical system
s with an invariant measure
and related problems. Its initial development was motivated by problems of statistical physics
.
A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem
, which claims that almost all points in any subset of the phase space
eventually revisit the set. More precise information is provided by various 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 of the most important examples are 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 process
es is played by the various notions of entropy for dynamical systems.
The concepts of ergodicity
and the ergodic hypothesis
are central to applications of ergodic theory. The underlying idea is that for certain systems the time average of their properties is equal to the average over the entire space. 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 surface
s of negative curvature. Markov chain
s form a common context for applications in probability theory
. Ergodic theory has fruitful connections with harmonic analysis
, Lie theory
(representation theory
, lattices
in algebraic group
s), and number theory
(the theory of diophantine approximations, L-functions).
Let T: X → X be a measure-preserving transformation on a measure space (X, Σ, μ), with
. A measure-preserving transformation T as above is ergodic if for every
with 
, then either 
or 
.
be a measure-preserving transformation on a measure space 
. One may then consider the "time average" of a 
-integrable function f, i.e. 
. The "time average" is defined as the average (if it exists) over iterations of T starting from some initial point x.
If
is finite and nonzero, we can consider the "space average" or "phase average" of f, defined as
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
is integrable:
Furthermore,
is T-invariant, that is to say
holds almost everywhere, and if
is finite, then the normalization is the same:
In particular, if T is ergodic, then
must be a constant (almost everywhere), and so one has that
almost everywhere. Joining the first to the last claim and assuming that
is finite and nonzero, one has that
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
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.
be measurable, 
, and 
be a measure-preserving map. Then
where
is the conditional expectation
given the
-algebra 
of invariant sets of 
.
Corollary (Pointwise ergodic theorem)
In particular, if
is also ergodic, then 
is the trivial 
-algebra, and thus
Let
be a unitary operator
on a Hilbert space
; more generally, an isometric linear operator (that is, a not necessarily surjective linear operator satisfying 
for all 
, or equivalently, satisfying 
but not necessarily 
). Let 
be the orthogonal projection onto 
.
Then, for any
, we have:
where the limit is with respect to the norm on H. In other words, the sequence of averages
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
where T is a measure-preserving endomorphism of X, thought of in applications as representing a time-step of a discrete dynamical system. 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 Ut be a strongly continuous one-parameter group
of unitary operators on H. Then the operator
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.
Remark: Some intuition for the mean ergodic theorem can be developed by considering the case where complex numbers of unit length are regarded as unitary transformations on the complex plane (by left multiplication). If we pick a single complex number of unit length (which we think of as
), it is intuitive that its powers will fill up the circle. Since the circle is symmetric around 0, it makes sense that the averages of the powers of 
will converge to 0. Also, 0 is the only fixed point of 
, and so the projection onto the space of fixed points must be the zero operator (which agrees with the limit just described).
be as above a probability space with a measure preserving transformation T, and let 
. The conditional expectation with respect to the sub-σ-algebra 
of the T-invariant sets is a linear projector 
of norm 1 of the Banach space 
onto its closed subspace 
The latter may also be characterized as the space of all T-invariant 
-functions on X. The ergodic means, as linear operators on 
also have unit operator norm; and, as a simple consequence of the Birkhoff–Khinchin theorem, converge to the projector 
in the strong operator topology of 
if 
and in the weak operator topology if 
. More is true if 
then the Wiener–Yoshida–Kakutani ergodic dominated convergence theorem states that the ergodic means of 
are dominated in 
; however, if 
, the ergodic means may fail to be equidominated in 
. Finally, if f is assumed to be in the Zygmund class, that is 
is integrable, then the ergodic means are even dominated in 
.
be a measure space such that 
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
:
for all x except for a set of measure
zero, where
is the indicator function of 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.
(See almost surely
.) That is, the smaller A is, the longer it takes to return to it.
Riemann surface
s 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 space
s 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 Calvin 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.
In the last 20 years, there have been many works trying to find a measure-classification theorem similar to Ratner's theorems but for diagonalizable actions, motivated by conjectures of Furstenberg and Margulis.
An important partial result (solving those conjectures with an extra assumption of positive entropy) was proved by Elon Lindenstrauss
, and he was awarded the Fields medal
in 2010 for this result.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
that studies dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s with an invariant measure
Invariant measure
In mathematics, an invariant measure is a measure that is preserved by some function. Ergodic theory is the study of invariant measures in dynamical systems...
and related problems. Its initial development was motivated by problems of statistical physics
Statistical physics
Statistical physics is the branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic...
.
A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem
Poincaré recurrence theorem
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. The Poincaré recurrence time is the length of time elapsed until the recurrence. The result applies to physical systems in which...
, which claims that almost all points in any subset of the phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
eventually revisit the set. More precise information is provided by various ergodic theorems which assert that, under certain conditions, the time average of a function along the trajectories exists almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
and is related to the space average. Two of the most important examples are ergodic theorems of Birkhoff
George David Birkhoff
-External links:* − from National Academies Press, by Oswald Veblen....
and von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
. 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
Mixing (mathematics)
In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, etc....
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 process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
es is played by the various notions of entropy for dynamical systems.
The concepts of ergodicity
Ergodicity
In mathematics, the term ergodic is used to describe a dynamical system which, broadly speaking, has the same behavior averaged over time as averaged over space. In physics the term is used to imply that a system satisfies the ergodic hypothesis of thermodynamics.-Etymology:The word ergodic is...
and the ergodic hypothesis
Ergodic hypothesis
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 of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a...
are central to applications of ergodic theory. The underlying idea is that for certain systems the time average of their properties is equal to the average over the entire space. Applications of ergodic theory to other parts of mathematics usually involve establishing ergodicity properties for systems of special kind. In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, methods of ergodic theory have been used to study the geodesic flow on Riemannian manifolds, starting with the results of Eberhard Hopf
Eberhard Hopf
Eberhard Frederich Ferdinand Hopf was a mathematician and astronomer, one of the founding fathers of ergodic theory and a pioneer of bifurcation theory who also made significant contributions to the subjects of partial differential equations and integral equations, fluid dynamics, and differential...
for Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s of negative curvature. Markov chain
Markov chain
A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
s form a common context for applications in probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
. Ergodic theory has fruitful connections with harmonic analysis
Harmonic analysis
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 the notions of Fourier series and Fourier transforms...
, Lie theory
Lie theory
Lie theory is an area of mathematics, developed initially by Sophus Lie.Early expressions of Lie theory are found in books composed by Lie with Friedrich Engel and Georg Scheffers from 1888 to 1896....
(representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, lattices
Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure...
in algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s), and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
(the theory of diophantine approximations, L-functions).
Ergodic transformations
Ergodic theory is often concerned with ergodic transformations.Let T: X → X be a measure-preserving transformation on a measure space (X, Σ, μ), with
. A measure-preserving transformation T as above is ergodic if for every with , then either or .Examples
- An irrational rotationIrrational rotationIn mathematical theory of dynamical systems, an irrational rotation is a mapwhere θ is an irrational number. Under the identification of a circle with R/Z, or with the interval [0, 1] with the boundary points glued together, this map becomes a rotation of a circle by a proportion θ of a full...
of the circle R/Z, T: x → x+θ, where θ is irrationalIrrational numberIn mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
, 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 compactCompact groupIn mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
abelian groupAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
, μ the normalized Haar measureHaar measureIn mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
, and T a group automorphism of G. Let G* be the Pontryagin dual group, consisting of the continuous charactersCharacter (mathematics)In mathematics, a character is a special kind of function from a group to a field . There are at least two distinct, but overlapping meanings...
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 integralIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
matrixMatrix (mathematics)In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
A then T is ergodic if and only if no eigenvalue of A is a root of unityRoot of unityIn mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
.
- 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 processStationary processIn the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...
es follows from Kolmogorov's zero-one lawKolmogorov's zero-one lawIn probability theory, Kolmogorov's zero-one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, called a tail event, will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one.Tail...
.
- Ergodicity of a continuous dynamical system means that its trajectories "spread around" the phase spacePhase spaceIn mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in 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 systemHamiltonian systemIn physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....
s 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 theoremLiouville's theorem (Hamiltonian)In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics...
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 integrabilityIntegrable systemIn mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems.In the general theory of differential systems, there is Frobenius integrability, which refers to overdetermined systems. In the classical theory of Hamiltonian dynamical...
.
Ergodic theorems
Let be a measure-preserving transformation on a measure space . One may then consider the "time average" of a -integrable function f, i.e. . The "time average" is defined as the average (if it exists) over iterations of T starting from some initial point x.If
is finite and nonzero, we can consider the "space average" or "phase average" of f, defined asIn 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
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
. This is the celebrated ergodic theorem, in an abstract form due to George David Birkhoff
George David Birkhoff
-External links:* − from National Academies Press, by Oswald Veblen....
. (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
Equidistribution theorem
In mathematics, the equidistribution theorem is the statement that the sequenceis uniformly distributed on the unit interval, when a is an irrational number...
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
is integrable:Furthermore,
is T-invariant, that is to sayholds almost everywhere, and if
is finite, then the normalization is the same:In particular, if T is ergodic, then
must be a constant (almost everywhere), and so one has thatalmost everywhere. Joining the first to the last claim and assuming that
is finite and nonzero, one has thatfor almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
x, i.e., for all x except for a set of measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
zero.
For an ergodic transformation, the time average equals the space average almost surely.
As an example, assume that the measure space
models the particles of a gas as above, and let f(x) denotes the velocityVelocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
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.
Probabilistic formulation: Birkhoff–Khinchin theorem
Birkhoff–Khinchin theorem. Let be measurable, , and be a measure-preserving map. Thenwhere
is the conditional expectationConditional expectation
In probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution....
given the
-algebra of invariant sets of .Corollary (Pointwise ergodic theorem)
In particular, if
is also ergodic, then is the trivial -algebra, and thusMean ergodic theorem
Von Neumann's mean ergodic theorem, holds in Hilbert spaces.Let
be a unitary operatorUnitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
on a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
; more generally, an isometric linear operator (that is, a not necessarily surjective linear operator satisfying for all , or equivalently, satisfying but not necessarily ). Let be the orthogonal projection onto .Then, for any
, we have:where the limit is with respect to the norm on H. In other words, the sequence of averages
converges to P in the strong operator topology
Strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the weakest locally convex topology on the set of bounded operators on a Hilbert space such that the evaluation map sending an operator T to the real number \|Tx\| is continuous for each vector...
.
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
where T is a measure-preserving endomorphism of X, thought of in applications as representing a time-step of a discrete dynamical system. 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 Ut be a strongly continuous one-parameter group
One-parameter group
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphismfrom the real line R to some other topological group G...
of unitary operators on H. Then the operator
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.
Remark: Some intuition for the mean ergodic theorem can be developed by considering the case where complex numbers of unit length are regarded as unitary transformations on the complex plane (by left multiplication). If we pick a single complex number of unit length (which we think of as
), it is intuitive that its powers will fill up the circle. Since the circle is symmetric around 0, it makes sense that the averages of the powers of will converge to 0. Also, 0 is the only fixed point of , and so the projection onto the space of fixed points must be the zero operator (which agrees with the limit just described).Convergence of the ergodic means in the Lp norms
Let be as above a probability space with a measure preserving transformation T, and let . The conditional expectation with respect to the sub-σ-algebra of the T-invariant sets is a linear projector of norm 1 of the Banach space onto its closed subspace The latter may also be characterized as the space of all T-invariant -functions on X. The ergodic means, as linear operators on also have unit operator norm; and, as a simple consequence of the Birkhoff–Khinchin theorem, converge to the projector in the strong operator topology of if and in the weak operator topology if . More is true if then the Wiener–Yoshida–Kakutani ergodic dominated convergence theorem states that the ergodic means of are dominated in ; however, if , the ergodic means may fail to be equidominated in . Finally, if f is assumed to be in the Zygmund class, that is is integrable, then the ergodic means are even dominated in .Sojourn time
Let be a measure space such that 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 timeMean sojourn time
The mean sojourn time for an object in 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.- Calculation :...
:
for all x except for a set of measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
zero, where
is the indicator function of 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.
(See almost surely
Almost surely
In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
.) 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 compactCompact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s of variable negative curvature
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...
and on compact manifolds of constant negative curvature
Hyperbolic manifold
In mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold...
of any dimension was proved by Eberhard Hopf
Eberhard Hopf
Eberhard Frederich Ferdinand Hopf was a mathematician and astronomer, one of the founding fathers of ergodic theory and a pioneer of bifurcation theory who also made significant contributions to the subjects of partial differential equations and integral equations, fluid dynamics, and differential...
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
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
by the action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of a lattice
Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure...
in the semisimple Lie group SO(n,1). Ergodicity of the geodesic flow on Riemannian symmetric space
Riemannian symmetric space
In differential geometry, representation theory and harmonic analysis, a symmetric space is a smooth manifold whose group of symmetries contains an inversion symmetry about every point. There are two ways to formulate the inversion symmetry, via Riemannian geometry or via Lie theory...
s 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
Sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...
. A simple criterion for the ergodicity of a homogeneous flow on a homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
of a semisimple Lie group was given by Calvin 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
Ratner's theorems
In mathematics, Ratner's theorems are a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990. The theorems grew out of Ratner's earlier work on horocycle flows. The study of the dynamics of unipotent flows played a decisive...
provide a major generalization of ergodicity for unipotent flows on the homogeneous spaces of the form Γ\G, where G is a Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
and Γ is a lattice in G.
In the last 20 years, there have been many works trying to find a measure-classification theorem similar to Ratner's theorems but for diagonalizable actions, motivated by conjectures of Furstenberg and Margulis.
An important partial result (solving those conjectures with an extra assumption of positive entropy) was proved by Elon Lindenstrauss
Elon Lindenstrauss
Elon Lindenstrauss is an Israeli mathematician, and a winner of the 2010 Fields Medal.Since 2004, he has been a professor at Princeton University...
, and he was awarded the Fields medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
in 2010 for this result.
See also
- Chaos theoryChaos theoryChaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
- Ergodic hypothesisErgodic hypothesisIn 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 of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a...
- Ergodic processErgodic processIn signal processing, a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long sample of the process.- Specific definitions :...
- Maximal ergodic theorem
- Statistical mechanicsStatistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
- Symbolic dynamicsSymbolic dynamicsIn mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics given by the shift operator...
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 theoremEquidistribution theoremIn mathematics, the equidistribution theorem is the statement that the sequenceis uniformly distributed on the unit interval, when a is an irrational number...
of shift maps on the unit intervalUnit intervalIn mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
. Focuses on methods developed by Bourgain.) - A.N. ShiryaevAlbert ShiryaevAl'bert Nikolayevich Shiryaev is a Soviet and Russian mathematician. He is known for his work in probability theory, statistics and financial mathematics....
, Probability, 2nd ed., Springer 1996, Sec. V.3. ISBN 0-387-94549-0.
External links
- Ergodic Theory (29 October 2007) Notes by Cosma Rohilla Shalizi
- Ergodic theorem passes the test From Physics World