Ergodic theory (Greek: ἔργον ergon "work", ὁδός hodos "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expressed through the behavior of time averages 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 perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. 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. Systems for which the Poincaré recurrence theorem holds are conservative systems; 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 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, 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 processes 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 surfaces of negative curvature. Markov chains form a common context for applications in probability theory. 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). (Wikipedia).
Karma Dajani - An introduction to Ergodic Theory of Numbers (Part 2)
In this course we give an introduction to the ergodic theory behind common number expansions, like expansions to integer and non-integer bases, Luroth series and continued fraction expansion. Starting with basic ideas in ergodic theory such as ergodicity, the ergodic theorem and natural ex
From playlist École d’été 2013 - Théorie des nombres et dynamique
Karma Dajani - An introduction to Ergodic Theory of Numbers (Part 3)
In this course we give an introduction to the ergodic theory behind common number expansions, like expansions to integer and non-integer bases, Luroth series and continued fraction expansion. Starting with basic ideas in ergodic theory such as ergodicity, the ergodic theorem and natural ex
From playlist École d’été 2013 - Théorie des nombres et dynamique
Karma Dajani - An introduction to Ergodic Theory of Numbers (Part 1)
In this course we give an introduction to the ergodic theory behind common number expansions, like expansions to integer and non-integer bases, Luroth series and continued fraction expansion. Starting with basic ideas in ergodic theory such as ergodicity, the ergodic theorem and natural ex
From playlist École d’été 2013 - Théorie des nombres et dynamique
Uri Bader - 2/4 Algebraic Representations of Ergodic Actions
Ergodic Theory is a powerful tool in the study of linear groups. When trying to crystallize its role, emerges the theory of AREAs, that is Algebraic Representations of Ergodic Actions, which provides a categorical framework for various previously studied concepts and methods. Roughly, this
From playlist Uri Bader - Algebraic Representations of Ergodic Actions
Uri Bader - 4/4 Algebraic Representations of Ergodic Actions
Ergodic Theory is a powerful tool in the study of linear groups. When trying to crystallize its role, emerges the theory of AREAs, that is Algebraic Representations of Ergodic Actions, which provides a categorical framework for various previously studied concepts and methods. Roughly, this
From playlist Uri Bader - Algebraic Representations of Ergodic Actions
Quantum Ergodicity for the Uninitiated - Zeev Rudnick
Zeev Rudnick Tel Aviv University; Member, School of Mathematics October 26, 2015 https://www.math.ias.edu/seminars/abstract?event=47561 A key result in spectral theory linking classical and quantum mechanics is the Quantum Ergodicity theorem, which states that in a system in which the cl
From playlist Members Seminar
Uri Bader - 1/4 Algebraic Representations of Ergodic Actions
Ergodic Theory is a powerful tool in the study of linear groups. When trying to crystallize its role, emerges the theory of AREAs, that is Algebraic Representations of Ergodic Actions, which provides a categorical framework for various previously studied concepts and methods. Roughly, this
From playlist Uri Bader - Algebraic Representations of Ergodic Actions
Amos Nevo: Representation theory, effective ergodic theorems, and applications - Lecture 1
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Dynamical Systems and Ordinary Differential Equations
Interview at CIRM: Mariusz Lemanczyk
Chaire Jean-Morlet Cirm-Luminy 1 August 2016 - 30 January 2017 Semester on 'Ergodic Theory and Dynamical Systems in their Interactions with Arithmetic and Combinatorics' General themes During the last decades, ergodic theory, in spite of its rather physical origins, has found deep conn
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Amine Marrakchi: Ergodic theory of affine isometric actions on Hilbert spaces
The Gaussian functor associates to every orthogonal representation of a group G on a Hilbert space, a probability measure preserving action of G called a Gaussian action. This construction is a fundamental tool in ergodic theory and is the source of a large and interesting class of probabi
From playlist Probability and Statistics
Amos Nevo: Representation theory, effective ergodic theorems, and applications - Lecture 3
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Dynamical Systems and Ordinary Differential Equations
Uri Bader - 3/4 Algebraic Representations of Ergodic Actions
Ergodic Theory is a powerful tool in the study of linear groups. When trying to crystallize its role, emerges the theory of AREAs, that is Algebraic Representations of Ergodic Actions, which provides a categorical framework for various previously studied concepts and methods. Roughly, this
From playlist Uri Bader - Algebraic Representations of Ergodic Actions
Amos Nevo: Representation theory, effective ergodic theorems, and applications - Lecture 2
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Dynamical Systems and Ordinary Differential Equations
Vitaly Bergelson : Potpourri of open problems and conjectures in linear dynamics and ergodic theory
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From playlist Dynamical Systems and Ordinary Differential Equations
An ergodic approach towards an equidistribution result of Ferrero–Washington by Bharathwaj Palvannan
PROGRAM ELLIPTIC CURVES AND THE SPECIAL VALUES OF L-FUNCTIONS (HYBRID) ORGANIZERS: Ashay Burungale (CalTech/UT Austin, USA), Haruzo Hida (UCLA), Somnath Jha (IIT Kanpur) and Ye Tian (MCM, CAS) DATE: 08 August 2022 to 19 August 2022 VENUE: Ramanujan Lecture Hall and online The program pla
From playlist ELLIPTIC CURVES AND THE SPECIAL VALUES OF L-FUNCTIONS (2022)
Amos Nevo: Representation theory, effective ergodic theorems, and applications - Lecture 4
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Dynamical Systems and Ordinary Differential Equations