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113 relations: Abelian group, Albert Shiryaev, Algebraic group, Almost all, Almost everywhere, Almost surely, Ancient Greek, Anosov diffeomorphism, Artin billiard, Bernoulli scheme, C0-semigroup, Calvin C. Moore, Chaos theory, Character (mathematics), Circle group, Compact group, Compact space, Conditional expectation, Diophantine approximation, Dmitri Anosov, Dynamical system, Eberhard Hopf, Eigenvalues and eigenvectors, Elon Lindenstrauss, Equidistributed sequence, Equidistribution theorem, Ergodic hypothesis, Ergodic process, Ergodicity, Fields Medal, Friederich Ignaz Mautner, Gaussian curvature, Geodesic, Geometry, George David Birkhoff, Grigory Margulis, Group action, Group isomorphism, Gustav A. Hedlund, Haar measure, Hadamard's dynamical system, Hamiltonian system, Harmonic analysis, Hilbert space, Hillel Furstenberg, Homogeneous space, Hyperbolic manifold, Hyperbolic space, Indefinite orthogonal group, Independent and identically distributed random variables, ..., Indicator function, Integrable system, Invariant measure, Irrational number, Irrational rotation, Israel Gelfand, John von Neumann, Kingman's subadditive ergodic theorem, Kolmogorov's zero–one law, L-function, Lattice (discrete subgroup), Lebesgue measure, Lie group, Lie theory, Lindy effect, Liouville's theorem (Hamiltonian), Lyapunov time, Marina Ratner, Markov chain, Mathematics, Maximal ergodic theorem, Mean sojourn time, Measure space, Measure-preserving dynamical system, Mixing (mathematics), Number theory, One-parameter group, Ornstein isomorphism theorem, Phase space, Poincaré recurrence theorem, Pontryagin duality, Predictability, Probability theory, Projection (linear algebra), Ratner's theorems, Representation theory, Riemann surface, Riemannian manifold, Rigidity (mathematics), Root of unity, Sectional curvature, Semisimple Lie algebra, Sergei Fomin, Shift operator, SL2(R), Springer Science+Business Media, Stationary process, Statistical mechanics, Statistical physics, Stochastic process, Strong operator topology, Symbolic dynamics, Symmetric space, Telescoping series, Torus, Trivial representation, Unimodular matrix, Unit interval, Unitary operator, Velocity, Vladimir Arnold, Weak operator topology, Yakov Sinai. Expand index (63 more) »
Abelian group
In 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 the order in which they are written.
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Albert Shiryaev
Albert Nikolayevich Shiryaev (Альбе́рт Никола́евич Ширя́ев; born October 12, 1934) is a Soviet and Russian mathematician.
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Algebraic group
In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.
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Almost all
In mathematics, the term "almost all" means "all but a negligible amount".
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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.
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Almost surely
In probability theory, one says that an event happens almost surely (sometimes abbreviated as a.s.) if it happens with probability one.
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Ancient Greek
The Ancient Greek language includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD.
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Anosov diffeomorphism
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold M is a certain type of mapping, from M to itself, with rather clearly marked local directions of "expansion" and "contraction".
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Artin billiard
In mathematics and physics, the Artin billiard is a type of a dynamical billiard first studied by Emil Artin in 1924.
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Bernoulli scheme
In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes.
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C0-semigroup
In mathematics, a C0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function.
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Calvin C. Moore
Calvin C. Moore (born November 2, 1936 in New York City) is an American mathematician who works in the theory of operator algebras and topological groups.
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Chaos theory
Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions.
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Character (mathematics)
In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers).
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Circle group
In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers.
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Compact group
In mathematics, a compact (topological) group is a topological group whose topology is compact.
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Compact space
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
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Conditional expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of "conditions" is known to occur.
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Diophantine approximation
In number theory, the field of Diophantine approximation deals with the approximation of real numbers by rational numbers.
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Dmitri Anosov
Dmitri Victorovich Anosov (Дми́трий Ви́кторович Ано́сов; November 30, 1936 – August 5, 2014) was a Soviet and Russian mathematician, known for his contributions to dynamical systems theory.
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Dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space.
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Eberhard Hopf
Eberhard Frederich Ferdinand Hopf (April 17, 1902, Salzburg, Austria-Hungary – July 24, 1983, 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.
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Eigenvalues and eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.
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Elon Lindenstrauss
Elon Lindenstrauss (אילון לינדנשטראוס, born August 1, 1970) is an Israeli mathematician, and a winner of the 2010 Fields Medal.
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Equidistributed sequence
In mathematics, a sequence 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 interval.
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Equidistribution theorem
In mathematics, the equidistribution theorem is the statement that the sequence is uniformly distributed on the circle \mathbb/\mathbb, when a is an irrational number.
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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., that all accessible microstates are equiprobable over a long period of time.
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Ergodic process
In econometrics and signal processing, a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long, random sample of the process.
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Ergodicity
In probability theory, an ergodic dynamical system is one that, broadly speaking, has the same behavior averaged over time as averaged over the space of all the system's states in its phase space.
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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 the International Mathematical Union (IMU), a meeting that takes place every four years.
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Friederich Ignaz Mautner
Friederich Ignaz Mautner (1921–2001) was an American mathematician, known for his research on the representation theory of groups, functional analysis, and differential geometry.
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Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point: For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane and a cylinder have Gaussian curvature 0 everywhere.
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Geodesic
In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".
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Geometry
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
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George David Birkhoff
George David Birkhoff (March 21, 1884 – November 12, 1944) was an American mathematician best known for what is now called the ergodic theorem.
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Grigory 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.
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Group action
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
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Group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.
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Gustav A. Hedlund
Gustav Arnold Hedlund (May 7, 1904 – March 15, 1993), an American mathematician, was one of the founders of symbolic and topological dynamics.
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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.
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Hadamard's dynamical system
In physics and mathematics, the Hadamard dynamical system (also called Hadamard's billiard or the Hadamard-Gutzwiller model) is a chaotic dynamical system, a type of dynamical billiards.
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Hamiltonian system
A Hamiltonian system is a dynamical system governed by Hamilton's equations.
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Harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis).
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Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
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Hillel Furstenberg
Hillel (Harry) Furstenberg (הלל (הארי) פורסטנברג) (born September 29, 1935) is an American-Israeli mathematician, a member of the Israel Academy of Sciences and Humanities and U.S. National Academy of Sciences and a laureate of the Wolf Prize in Mathematics.
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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 transitively.
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Hyperbolic manifold
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension.
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Hyperbolic space
In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.
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Indefinite orthogonal group
In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an n-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature, where.
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Independent and identically distributed random variables
In probability theory and statistics, a sequence or other collection of random variables is independent and identically distributed (i.i.d. or iid or IID) if each random variable has the same probability distribution as the others and all are mutually independent.
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Indicator function
In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset.
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Integrable system
In the context of differential equations to integrate an equation means to solve it from initial conditions.
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Invariant measure
In mathematics, an invariant measure is a measure that is preserved by some function.
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Irrational number
In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.
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Irrational rotation
In the mathematical theory of dynamical systems, an irrational rotation is a map where θ is an irrational number.
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Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand (ישראל געלפֿאַנד, Изра́иль Моисе́евич Гельфа́нд; – 5 October 2009) was a prominent Soviet mathematician.
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John von Neumann
John von Neumann (Neumann János Lajos,; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, and polymath.
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Kingman's subadditive ergodic theorem
In mathematics, Kingman's subadditive ergodic theorem is one of several ergodic theorems.
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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, 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.
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L-function
In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects.
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Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure.
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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.
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Lie group
In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
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Lie theory
In mathematics, the researcher Sophus Lie initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory.
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Lindy effect
The Lindy effect is a concept that the future life expectancy of some non-perishable things like a technology or an idea is proportional to their current age, so that every additional period of survival implies a longer remaining life expectancy.
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Liouville'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.
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Lyapunov time
In mathematics, the Lyapunov time is the characteristic timescale on which a dynamical system is chaotic.
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Marina Ratner
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.
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Markov chain
A Markov chain is "a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event".
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Maximal ergodic theorem
The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics.
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Mean sojourn time
The mean sojourn time (or sometimes mean waiting time) for an object in a system is the amount of time an object is expected to spend in a system before leaving the system for good.
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Measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes.
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Measure-preserving dynamical system
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular.
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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.
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Number theory
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
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One-parameter group
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism from the real line \mathbb (as an additive group) to some other topological group G. That means that it is not in fact a group, strictly speaking; if \varphi is injective then \varphi(\mathbb), the image, will be a subgroup of G that is isomorphic to \mathbb as additive group.
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Ornstein isomorphism theorem
In mathematics, the Ornstein isomorphism theorem is a deep result for ergodic theory.
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Phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space.
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Poincaré recurrence theorem
In physics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long but finite time, return to a state very close to, if not exactly the same as, the initial state.
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Pontryagin duality
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as \R, the circle, or finite cyclic groups.
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Predictability
Predictability is the degree to which a correct prediction or forecast of a system's state can be made either qualitatively or quantitatively.
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Probability theory
Probability theory is the branch of mathematics concerned with probability.
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Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.
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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.
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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 studies modules over these abstract algebraic structures.
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Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold.
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Riemannian manifold
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.
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Rigidity (mathematics)
In mathematics, a rigid collection C of mathematical objects (for instance sets or functions) is one in which every c ∈ C is uniquely determined by less information about c than one would expect.
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Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.
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Sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds.
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Semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras \mathfrak g whose only ideals are and \mathfrak g itself.
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Sergei Fomin
Sergei Vasilyevich Fomin (Серге́й Васи́льевич Фоми́н; 9 December 1917 – 17 August 1975) was a Soviet mathematician who was co-author with Kolmogorov of Introductory real analysis, and co-author with I.M. Gelfand of Calculus of Variations (1963), both books that are widely read in Russian and in English.
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Shift operator
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation.
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SL2(R)
In mathematics, the special linear group SL(2,R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: a & b \\ c & d \end \right): a,b,c,d\in\mathbf\mboxad-bc.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Stationary process
In mathematics and statistics, a stationary process (a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time.
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Statistical mechanics
Statistical mechanics is one of the pillars of modern physics.
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Statistical physics
Statistical physics is a 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.
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Stochastic process
--> In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a collection of random variables.
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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 x varies in H. Equivalently, it is the coarsest topology such that the evaluation maps T\mapsto Tx (taking values in H) are continuous for each fixed x in H. The equivalence of these two definitions can be seen by observing that a subbase for both topologies is given by the sets U(T_0,x,\epsilon).
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Symbolic dynamics
In 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 (evolution) given by the shift operator.
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Symmetric space
In differential geometry, representation theory and harmonic analysis, a symmetric space is a pseudo-Riemannian manifold whose group of symmetries contains an inversion symmetry about every point.
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Telescoping series
In mathematics, a telescoping series is a series whose partial sums eventually only have a fixed number of terms after cancellation.
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Torus
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
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Trivial representation
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 a (Lie) algebra representation for which all elements of the algebra act as the zero linear map (endomorphism) which sends every element of V to the zero vector.
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Unimodular matrix
In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1.
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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.
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Unitary operator
In functional analysis, a branch of mathematics, a unitary operator is a surjective bounded operator on a Hilbert space preserving the inner product.
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Velocity
The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time.
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Vladimir Arnold
Vladimir Igorevich Arnold (alternative spelling Arnol'd, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician.
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Weak operator topology
In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number \langle Tx, y\rangle is continuous for any vectors x and y in the Hilbert space.
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Yakov Sinai
Yakov Grigorevich Sinai (Я́ков Григо́рьевич Сина́й; born September 21, 1935) is a mathematician known for his work on dynamical systems.
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Redirects here:
Birkhoff ergodic theorem, Birkhoff's Ergodic Theorem, Birkhoff's ergodic theorem, Birkhoff-Khinchin ergodic theorem, Birkhoff-Khinchin theorem, Birkhoff–Khinchin ergodic theorem, Birkhoff–Khinchin theorem, Ergodic Theory, Ergodic properties, Ergodic set, Ergodic system, Ergodic systems, Ergodic theorem, Ergodic theorems, Ergodic transformation, Individual ergodic theorem, Mean ergodic theorem, Metric transitivity, Occurence time, Sojourn time, Strongly ergodic, Von Neumann ergodic theorem, Von Neumann mean ergodic theorem, Von Neumann's ergodic theorem, Von Neumann's mean ergodic theorem, Weakly ergodic.
References
[1] https://en.wikipedia.org/wiki/Ergodic_theory
