# Ergodic theory

ergodicergodic theoremergodic transformationergodicityBirkhoff Ergodic Theoremergodic motionergodic systemsergodic theoremsErgodic Theory of Dynamical Systemsergodic-theoretical
Ergodic theory (Greek: έργον ergon "work", όδος hodos "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems.wikipedia
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### John von Neumann

von NeumannJ. von NeumannNeumann, John von
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.
He made major contributions to a number of fields, including mathematics (foundations of mathematics, functional analysis, ergodic theory, representation theory, operator algebras, geometry, topology, and numerical analysis), physics (quantum mechanics, hydrodynamics, and quantum statistical mechanics), economics (game theory), computing (Von Neumann architecture, linear programming, self-replicating machines, stochastic computing), and statistics.

### Poincaré recurrence theorem

Poincaré recurrencePoincaré recurrence timeRecurrence
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.
The theorem is commonly discussed in the context of ergodic theory, dynamical systems and statistical mechanics.

### Mixing (mathematics)

mixingstrong mixingtopological mixing
Stronger properties, such as mixing and equidistribution, have also been extensively studied.
The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems.

### Measure (mathematics)

measuremeasure theorymeasurable
Ergodic theory, like probability theory, is based on general notions of measure theory.
The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory.

### Eberhard Hopf

E. HopfHopf, Eberhard
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.
Eberhard Frederich Ferdinand Hopf (April 17, 1902 in Salzburg, Austria-Hungary – July 24, 1983 in Bloomington, Indiana) 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 geometry.

### Measure-preserving dynamical system

Kolmogorov–Sinai entropymeasure-preserving transformationentropy
An outstanding role in ergodic theory and its applications to stochastic processes is played by the various notions of entropy for dynamical systems.
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular.

### Lattice (discrete subgroup)

latticelatticesX-lattice
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).
Lattices are of interest in many areas of mathematics: geometric group theory (as particularly nice examples of discrete groups), in differential geometry (through the construction of locally homogeneous manifolds), in number theory (through arithmetic groups), in ergodic theory (through the study of homogeneous flows on the quotient spaces) and in combinatorics (through the construction of expanding Cayley graphs and other combinatorial objects).

### Markov chain

Markov processMarkov chainscontinuous-time Markov process
Markov chains form a common context for applications in probability theory.
A state i is said to be ergodic if it is aperiodic and positive recurrent.

### Haar measure

unimodularHaar integralHaar measures
Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory.

### Irrational rotation

irrationalIrrational anglerotation action
. An irrational rotation is a measure-preserving ergodic transformation, but it is not mixing.

### Ergodicity

ergodicergodic measurenon-ergodic
The concepts of ergodicity and the ergodic hypothesis are central to applications of ergodic theory.
The branch of mathematics that studies ergodic systems is known as ergodic theory.

### Liouville's theorem (Hamiltonian)

Liouville's theoremLiouville equationLiouville equations
There are related mathematical results in symplectic topology and ergodic theory.

A generalization of Birkhoff's theorem is Kingman's subadditive ergodic theorem.
In mathematics, Kingman's subadditive ergodic theorem is one of several ergodic theorems.

### Hilbert space

Hilbert spacesHilbertseparable Hilbert space
Let U be a unitary operator on a Hilbert space 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).
They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics).

### Ergodic hypothesis

ergodicproperties of statistical mechanicsnonergodic
The concepts of ergodicity and the ergodic hypothesis are central to applications of ergodic theory.
Ergodic theory is a branch of mathematics which deals with dynamical systems that satisfy a version of this hypothesis, phrased in the language of measure theory.

### Dynamical system

dynamical systemsdynamic systemdynamic systems
Ergodic theory (Greek: έργον ergon "work", όδος hodos "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems.

### Number theory

number theoristcombinatorial number theorytheory of numbers
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).
Its focus on issues of growth and distribution accounts in part for its developing links with ergodic theory, finite group theory, model theory, and other fields.

### Diophantine approximation

Diophantine approximationsLagrange's approximation theoremrational approximations
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).
In his plenary address at the International Mathematical Congress in Kyoto (1990), Grigory Margulis outlined a broad program rooted in ergodic theory that allows one to prove number-theoretic results using the dynamical and ergodic properties of actions of subgroups of semisimple Lie groups.

### Grigory Margulis

Grigori MargulisMargulisGregory Margulis
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.
Gregori Aleksandrovich Margulis (Григо́рий Алекса́ндрович Маргу́лис, first name often given as Gregory, Grigori or Grigory; born February 24, 1946) is a Russian-American mathematician known for his work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation.

### Marina Ratner

RatnerRatner, Marina
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.
Marina Evseevna Ratner (Мари́на Евсе́евна Ра́тнер; October 30, 1938 – July 7, 2017 ) was a professor of mathematics at the University of California, Berkeley who worked in ergodic theory.

### Ratner's theorems

Ratner's theorem
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 are a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990.

### Hillel Furstenberg

FurstenbergHarry FurstenbergFurstenberg, Hillel
Unique ergodicity of the flow was established by Hillel Furstenberg in 1972.
He is known for his application of probability theory and ergodic theory methods to other areas of mathematics, including number theory and Lie groups.

### Ornstein isomorphism theorem

Ornstein theoryBernoulli flow
In mathematics, the Ornstein isomorphism theorem is a deep result for ergodic theory.

### Yakov Sinai

Yakov G. SinaiYa. G. SinaiYa. Sinai
In 1967 D. V. Anosov and Ya. G. Sinai proved ergodicity of the geodesic flow on compact manifolds of variable negative sectional curvature.
In 2014, the Norwegian Academy of Science and Letters awarded him the Abel Prize, for his contributions to dynamical systems, ergodic theory, and mathematical physics.

### Mean sojourn time

mean waiting timesojourn
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: