Birkhoff–Khinchin Theorem
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Ergodic theory is a branch of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
that studies
statistical Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
properties of deterministic
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
s; it is the study of
ergodicity In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies th ...
. In this context, "statistical properties" refers to properties which are expressed through the behavior of
time average In statistics, a moving average (rolling average or running average or moving mean or rolling mean) is a calculation to analyze data points by creating a series of averages of different selections of the full data set. Variations include: simple ...
s of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any
random In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. ...
perturbations,
noise Noise is sound, chiefly unwanted, unintentional, or harmful sound considered unpleasant, loud, or disruptive to mental or hearing faculties. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrat ...
, etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, is based on general notions of measure theory. Its initial development was motivated by problems of
statistical physics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
. 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 In mathematics and physics, the Poincaré recurrence theorem states that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (fo ...
, which claims that
almost all In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...
points in any subset of the
phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
eventually revisit the set. Systems for which the Poincaré recurrence theorem holds are
conservative system In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink o ...
s; thus all ergodic systems are conservative. 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 In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
and is related to the space average. Two of the most important theorems are those of Birkhoff (1931) and von Neumann which assert the existence of a time average along each trajectory. For the special class of ergodic systems, this 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 In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequences ...
, 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 In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es is played by the various notions of
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
for dynamical systems. The concepts of
ergodicity In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies th ...
and the
ergodic hypothesis In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., tha ...
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 (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, methods of ergodic theory have been used to study the
geodesic flow In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conne ...
on
Riemannian manifolds In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the n-sphere, hyperbolic space, and smooth surfaces in ...
, starting with the results of
Eberhard Hopf Eberhard Frederich Ferdinand Hopf (April 4, 1902 in Salzburg, Austria-Hungary – July 24, 1983 in Bloomington, Indiana, USA) was a German mathematician and astronomer, one of the founding fathers of ergodic theory and a pioneer of bifurcation the ...
for
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s of negative curvature.
Markov chain In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally ...
s form a common context for applications in
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
. Ergodic theory has fruitful connections with
harmonic analysis Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...
,
Lie theory In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact (mathematics), contact of spheres that have come to be called Lie theory. For instance, ...
(
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
, lattices in
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
s), and
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
(the theory of
diophantine approximations In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by ...
,
L-functions In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may gi ...
).


Ergodic transformations

Ergodic theory is often concerned with ergodic transformations. The intuition behind such transformations, which act on a given set, is that they do a thorough job "stirring" the elements of that set. E.g. if the set is a quantity of hot oatmeal in a bowl, and if a spoonful of syrup is dropped into the bowl, then iterations of the inverse of an ergodic transformation of the oatmeal will not allow the syrup to remain in a local subregion of the oatmeal, but will distribute the syrup evenly throughout. At the same time, these iterations will not compress or dilate any portion of the oatmeal: they preserve the measure that is density. The formal definition is as follows: Let be a
measure-preserving transformation In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special cas ...
on a
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
, with . Then is ergodic if for every in with (that is, is invariant), either or . The operator Δ here is the symmetric difference of sets, equivalent to the
exclusive-or Exclusive or, exclusive disjunction, exclusive alternation, logical non-equivalence, or logical inequality is a logical operator whose negation is the logical biconditional. With two inputs, XOR is true if and only if the inputs differ (one ...
operation with respect to set membership. The condition that the symmetric difference be measure zero is called being essentially invariant.


Examples

* An
irrational rotation In the mathematical theory of dynamical systems, an irrational rotation is a map : T_\theta : ,1\rightarrow ,1\quad T_\theta(x) \triangleq x + \theta \mod 1 , where is an irrational number. Under the identification of a circle with , or with t ...
of the
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
R/Z, ''T'': ''x'' → ''x'' + θ, where θ is
irrational Irrationality is cognition, thinking, talking, or acting without rationality. Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
, is ergodic. This transformation has even stronger properties of unique ergodicity, minimality, and
equidistribution In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequences ...
. 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'' (that is, the union of ''I'', ''T''(''I''), ..., ''T''''q''−1(''I''), which contains the image of ''I'' under any number of applications of ''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 Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
abelian group In mathematics, 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 the order in which they are written. That is, the group operation is commu ...
, ''μ'' the normalized
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfrà ...
, and ''T'' a group automorphism of ''G''. Let ''G''* be the
Pontryagin dual In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...
group, consisting of the continuous
characters Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to Theoph ...
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 In the mathematical field of representation theory, a trivial representation is a representation of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial representation of an associative or Lie algebra is ...
of ''G''. In particular, if ''G'' is the ''n''-dimensional
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
and the automorphism ''T'' is represented by a
unimodular matrix In mathematics, a unimodular matrix ''M'' is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix ''N'' that is its inverse (these are equi ...
''A'' then ''T'' is ergodic if and only if no
eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
of ''A'' is a
root of unity In mathematics, a root of unity is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory ...
. * A
Bernoulli shift In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical syst ...
is ergodic. More generally, ergodicity of the shift transformation associated with a sequence of i.i.d. random variables and some more general
stationary process In mathematics and statistics, a stationary process (also called a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose statistical properties, such as mean and variance, do not change over time. M ...
es follows from
Kolmogorov's zero–one law In probability theory, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, namely a ''tail event of independent σ-algebras'', will either almost surely happen or almost su ...
. * Ergodicity of a continuous dynamical system means that its trajectories "spread around" the
phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
. A system with a compact phase space which has a non-constant first integral cannot be ergodic. This applies, in particular, to
Hamiltonian system A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...
s with a first integral ''I'' functionally independent from the Hamilton function ''H'' and a compact level set ''X'' = 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 In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern differential ...
.


Ergodic theorems

Let ''T'': ''X'' → ''X'' be a
measure-preserving transformation In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special cas ...
on a
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
(''X'', Σ, ''μ'') and suppose ƒ is a ''μ''-integrable function, i.e. ƒ ∈ ''L''1(''μ''). Then we define the following ''averages'':
Time average: This is defined as the average (if it exists) over iterations of ''T'' starting from some initial point ''x'': : \hat f(x) = \lim_\; \frac \sum_^ f(T^k x).
Space average: If ''μ''(''X'') is finite and nonzero, we can consider the ''space'' or ''phase'' average of ƒ: : \bar f =\frac 1 \int f\,d\mu.\quad\text \mu(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 In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
. This is the celebrated ergodic theorem, in an abstract form due to
George David Birkhoff George David Birkhoff (March21, 1884November12, 1944) was one of the top American mathematicians of his generation. He made valuable contributions to the theory of differential equations, dynamical systems, the four-color problem, the three-body ...
. (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 In mathematics, the equidistribution theorem is the statement that the sequence :''a'', 2''a'', 3''a'', ... mod 1 is Equidistributed sequence, uniformly distributed on the circle \mathbb/\mathbb, when ''a'' is an irrational number. It is a spe ...
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 ƒ 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_\; \frac \sum_^ f(T^k x) = \frac 1 \int f\,d\mu for
almost all In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...
''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 ƒ(''x'') denote the
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
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. A generalization of Birkhoff's theorem is Kingman's subadditive ergodic theorem.


Probabilistic formulation: Birkhoff–Khinchin theorem

Birkhoff–Khinchin theorem. Let ƒ be measurable, ''E''(, ƒ, ) < ∞, and ''T'' be a measure-preserving map. Then
with probability 1 In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur has ...
: :\lim_\; \frac \sum_^ f(T^k x)=E(f \mid \mathcal)(x), where E(f, \mathcal) is the
conditional expectation In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on ...
given the σ-algebra \mathcal of invariant sets of ''T''. Corollary (Pointwise Ergodic Theorem): In particular, if ''T'' is also ergodic, then \mathcal is the trivial σ-algebra, and thus with probability 1: :\lim_\; \frac \sum_^ f(T^k x)=E(f).


Mean ergodic theorem

Von Neumann's mean ergodic theorem, holds in Hilbert spaces. Let ''U'' be a
unitary operator In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Non-trivial examples include rotations, reflections, and the Fourier operator. Unitary operators generalize unitar ...
on a
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
''H''; more generally, an isometric linear operator (that is, a not necessarily surjective linear operator satisfying ‖''Ux''‖ = ‖''x''‖ for all ''x'' in ''H'', or equivalently, satisfying ''U''*''U'' = I, but not necessarily ''UU''* = I). Let ''P'' be the
orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it we ...
onto  = ker(''I'' âˆ’ ''U''). Then, for any ''x'' in ''H'', we have: : \lim_ \sum_^ U^ x = P x, where the limit is with respect to the norm on ''H''. In other words, the sequence of averages :\frac \sum_^ U^n converges to ''P'' in the
strong operator topology In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ...
. Indeed, it is not difficult to see that in this case any x\in H admits an orthogonal decomposition into parts from \ker(I-U) and \overline respectively. The former part is invariant in all the partial sums as N grows, while for the latter part, from the
telescoping series In mathematics, a telescoping series is a series whose general term t_n is of the form t_n=a_-a_n, i.e. the difference of two consecutive terms of a sequence (a_n). As a consequence the partial sums of the series only consists of two terms of (a_n ...
one would have: :\lim_ \sum_^ U^n (I-U)=\lim_ (I-U^N)=0 This theorem specializes to the case in which the Hilbert space ''H'' consists of ''L''2 functions on a measure space and ''U'' is an operator of the form :Uf(x) = f(Tx) \, 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 Æ’ over sufficiently large time-scales is approximated by the orthogonal component of Æ’ which is time-invariant. In another form of the mean ergodic theorem, let ''Ut'' be a strongly continuous
one-parameter group In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism :\varphi : \mathbb \rightarrow G from the real line \mathbb (as an additive group) to some other topological group G. If \varphi is in ...
of unitary operators on ''H''. Then the operator :\frac\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. 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 ''U''), 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 ''U'' will converge to 0. Also, 0 is the only fixed point of ''U'', 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 (''X'', Σ, ''μ'') be as above a probability space with a measure preserving transformation ''T'', and let 1 ≤ ''p'' ≤ ∞. The conditional expectation with respect to the sub-σ-algebra Σ''T'' of the ''T''-invariant sets is a linear projector ''ET'' of norm 1 of the Banach space ''Lp''(''X'', Σ, ''μ'') onto its closed subspace ''Lp''(''X'', Σ''T'', ''μ''). The latter may also be characterized as the space of all ''T''-invariant ''Lp''-functions on ''X''. The ergodic means, as linear operators on ''Lp''(''X'', Σ, ''μ'') also have unit operator norm; and, as a simple consequence of the Birkhoff–Khinchin theorem, converge to the projector ''ET'' in the
strong operator topology In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ...
of ''Lp'' if 1 ≤ ''p'' ≤ ∞, and in the
weak operator topology In functional analysis, the weak operator topology, often abbreviated WOT,Ilijas Farah, Combinatorial Set Theory of C*-algebras' (2019), p. 80. is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional ...
if ''p'' = ∞. More is true if 1 < ''p'' ≤ ∞ then the Wiener–Yoshida–Kakutani ergodic dominated convergence theorem states that the ergodic means of ƒ ∈ ''Lp'' are dominated in ''Lp''; however, if ƒ ∈ ''L''1, the ergodic means may fail to be equidominated in ''Lp''. Finally, if ƒ is assumed to be in the Zygmund class, that is , ƒ, log+(, ƒ, ) is integrable, then the ergodic means are even dominated in ''L''1.


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 {{Multiple issues, {{original research, date=March 2012 {{confusing, date=March 2012 {{context, date=December 2024 {{unreferenced, date=November 2024 {{No footnotes, date=November 2024 The mean sojourn time (or sometimes mean waiting time) for an o ...
: : \frac = \frac 1\int \chi_A\, d\mu = \lim_\; \frac \sum_^ \chi_A(T^k x) for all ''x'' except for a set of measure
zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
, where χ''A'' is the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
of ''A''. The occurrence times of a measurable set ''A'' is defined as the set ''k''1, ''k''2, ''k''3, ..., of times ''k'' such that ''Tk''(''x'') is in ''A'', sorted in increasing order. The differences between consecutive occurrence times ''Ri'' = ''ki'' − ''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 ''k''0 = 0. : \frac \rightarrow \frac \quad\text (See
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur ha ...
.) That is, the smaller ''A'' is, the longer it takes to return to it.


Ergodic flows on manifolds

The ergodicity of the
geodesic flow In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conne ...
on
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s of variable negative
curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
and on compact manifolds of constant negative curvature of any dimension was proved by
Eberhard Hopf Eberhard Frederich Ferdinand Hopf (April 4, 1902 in Salzburg, Austria-Hungary – July 24, 1983 in Bloomington, Indiana, USA) was a German mathematician and astronomer, one of the founding fathers of ergodic theory and a pioneer of bifurcation the ...
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 flow 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 "contr ...
s 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 In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to âˆ’1. It is homogeneous, and satisfies the stronger property of being a symme ...
by the
action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...
of a
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an or ...
in the semisimple Lie group SO(n,1). Ergodicity of the geodesic flow on
Riemannian symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geomet ...
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 In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a po ...
. A simple criterion for the ergodicity of a homogeneous flow on a
homogeneous space In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...
of a
semisimple Lie group In mathematics, a simple Lie group is a connected space, connected nonabelian group, non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple ...
was given by
Calvin C. Moore Calvin C. Moore (November 2, 1936 – July 26, 2023) was an American mathematician who worked in the theory of operator algebras and topological groups. Moore graduated from Harvard University with a bachelor's degree in 1958 and with a Ph.D. i ...
in 1966. Many of the theorems and results from this area of study are typical of rigidity theory. In the 1930s
G. A. Hedlund Gustav Arnold Hedlund (May 7, 1904 – March 15, 1993), an American mathematician, was one of the founders of symbolic and topological dynamics. Biography Hedlund was born May 7, 1904, in Somerville, Massachusetts. He did his undergraduate stud ...
proved that the horocycle flow on a compact hyperbolic surface is minimal and ergodic. Unique ergodicity of the flow was established by
Hillel Furstenberg Hillel "Harry" Furstenberg (; born September 29, 1935) is a German-born American-Israeli mathematician and professor emeritus at the Hebrew University of Jerusalem. He is a member of the Israel Academy of Sciences and Humanities and U.S. Natio ...
in 1972.
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 ...
provide a major generalization of ergodicity for unipotent flows on the homogeneous spaces of the form Γ \ ''G'', where ''G'' is a
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
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 (; born August 1, 1970) is an Israeli mathematician, and a winner of the 2010 Fields Medal. Since 2004, he has been a professor at Princeton University. In 2009, he was appointed as a Professor at the Einstein Institute of Mat ...
, and he was awarded the
Fields medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...
in 2010 for this result.


See also

*
Chaos theory Chaos theory is an interdisciplinary area of Scientific method, scientific study and branch of mathematics. It focuses on underlying patterns and Deterministic system, deterministic Scientific law, laws of dynamical systems that are highly sens ...
*
Ergodic hypothesis In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., tha ...
*
Ergodic process In physics, statistics, econometrics and signal processing, a stochastic process is said to be in an ergodic regime if an observable's ensemble average equals the time average. In this regime, any collection of random samples from a process must ...
* Kruskal principle * Lindy effect *
Lyapunov time In mathematics, the Lyapunov time is the characteristic timescale on which a dynamical system is chaotic. It is named after the Russian mathematician Aleksandr Lyapunov. It is defined as the inverse of a system's largest Lyapunov exponent. Use T ...
– the time limit to the
predictability Predictability is the degree to which a correct prediction or forecast of a system's state can be made, either qualitatively or quantitatively. Predictability and causality Causal determinism has a strong relationship with predictability. Perfec ...
of the system * Maximal ergodic theorem * Ornstein isomorphism theorem *
Statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
*
Symbolic dynamics In mathematics, symbolic dynamics is the study of dynamical systems defined on a discrete space consisting of infinite sequences of abstract symbols. The evolution of the dynamical system is defined as a simple shift of the sequence. Because of t ...


References


Historical references

* . * . * . * . * . * . * . * .


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. . ''(See Chapter 6.)'' * * ''(A survey of topics in ergodic theory; with exercises.)'' * Karl Petersen. Ergodic Theory (Cambridge Studies in Advanced Mathematics). Cambridge: Cambridge University Press. 1990. * Françoise Pène,
Stochastic properties of dynamical systems
', Cours spécialisés de la SMF, Volume 30, 2022 * 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, . ''(An extensive survey of the ergodic properties of generalizations of the
equidistribution theorem In mathematics, the equidistribution theorem is the statement that the sequence :''a'', 2''a'', 3''a'', ... mod 1 is Equidistributed sequence, uniformly distributed on the circle \mathbb/\mathbb, when ''a'' is an irrational number. It is a spe ...
of
shift map In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function to its translation . In time series analysis, the shift operator is called the ''lag operator ...
s on the
unit interval In 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. It is often denoted ' (capital letter ). In addition to its role in real analysi ...
. Focuses on methods developed by Bourgain.)'' * A. N. Shiryaev, ''Probability'', 2nd ed., Springer 1996, Sec. V.3. . * ''(A detailed discussion about the priority of the discovery and publication of the ergodic theorems by Birkhoff and von Neumann, based on a letter of the latter to his friend Howard Percy Robertson.)'' * Andrzej Lasota, Michael C. Mackey, ''Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics''. Second Edition, Springer, 1994. * Manfred Einsiedler and Thomas Ward
Ergodic Theory with a view towards Number Theory
Springer, 2011. * Jane Hawkins, ''Ergodic Dynamics: From Basic Theory to Applications'', Springer, 2021.


External links


Ergodic Theory (16 June 2015)
Notes by Cosma Rohilla Shalizi
Ergodic theorem passes the test
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