Ergodic theory (nonfiction): Difference between revisions
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Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of [[Statistical physics (nonfiction)|statistical physics]]. | Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of [[Statistical physics (nonfiction)|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 (nonfiction)|Poincaré recurrence theorem]], which claims that almost all points in any subset of the phase space eventually revisit the set. More precise information is provided by various ergodic theorems which assert that, under certain conditions, the time average of a function along the trajectories exists almost everywhere and is related to the space average. Two of the most important theorems are those of [[George David Birkhoff (nonfiction)|George David Birkhoff]] (1931) and [[John von Neumann (nonfiction)|John 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. | 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 (nonfiction)|Poincaré recurrence theorem]], which claims that almost all points in any subset of the [[Phase space (nonfiction)|phase space]] eventually revisit the set. More precise information is provided by various ergodic theorems which assert that, under certain conditions, the time average of a function along the trajectories exists [[Almost everywhere (nonfiction)|almost everywhere]] and is related to the space average. Two of the most important theorems are those of [[George David Birkhoff (nonfiction)|George David Birkhoff]] (1931) and [[John von Neumann (nonfiction)|John 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 (nonfiction)|mixing]] and [[Equidistributed sequence (nonfiction)|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 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 (nonfiction)|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 (nonfiction)|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). | The concepts of [[Ergodicity (nonfiction)|ergodicity]] and the [[Ergodic hypothesis (nonfiction)|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 (nonfiction)|geometry]], methods of ergodic theory have been used to study the geodesic flow on [[Riemannian manifold (nonfiction)|Riemannian manifolds]], starting with the results of [[Eberhard Hopf (nonfiction)|Eberhard Hopf]] for Riemann surfaces of negative curvature. [[Markov chain (nonfiction)|Markov chains]] form a common context for applications in [[Probability theory (nonfiction)|probability theory]]. Ergodic theory has fruitful connections with [[Harmonic analysis (nonfiction)|harmonic analysis]], [[Lie theory (nonfiction)|Lie theory]] ([[Representation theory (nonfiction)|representation theory]], [[Lattice (discrete subgroup) (nonfiction)|lattices]] in [[Algebraic group (nonfiction)|algebraic groups]]), and [[Number theory (nonfiction)|number theory]] (the theory of [[Diophantine approximation (nonfiction)|diophantine approximations]], [[L-function (nonfiction)|L-functions]]). | ||
== In the News == | == In the News == | ||
Latest revision as of 17:03, 17 April 2020
Ergodic theory (Greek: έργον ergon "work", όδος hodos "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems. 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. 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 George David Birkhoff (1931) and John 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).
In the News
Fiction cross-reference
Nonfiction cross-reference
External links
- Ergodic theory @ Wikipedia