Ergodic theory is the branch of mathematics that studies the long-run statistical behavior of deterministic dynamical systems, asking when a system's time average along a single trajectory matches its average over the whole space. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
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Central QuestionUnder what conditions does a dynamical system, followed for a long enough time, visit every part of its space of possible states in proportion to that part's size, so that averaging over time and averaging over space agree. 1 Key DebateHow far a system must wander before its long-run behavior can be trusted to represent the whole space. Henri Poincare's recurrence theorem showed that almost every point in a bounded region eventually returns close to where it started, and John von Neumann's mean ergodic theorem later made precise, in the setting of Hilbert spaces, the sense in which time averages converge, but the recurrence theorem alone says nothing about how long a genuine return takes. 1 Classification
Pure or Applied Ergodic Theory
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Source Biographies of Women Mathematicians, Agnes Scott College: Alexandra Bellow
Source Amie Wilkinson (Wikipedia)
Source Eberhard Hopf (Wikipedia)
Source Grigory Margulis (Wikipedia)
Source Hillel Furstenberg (Wikipedia)
Source Jean Bourgain (Wikipedia)
Source Wikipedia: Ergodic Theory
Source Nelson Dunford (Wikipedia)
Source Paul Halmos (Wikipedia)
Source Robert Zimmer (Wikipedia)
Source Roy Adler (Wikipedia)
Source Spectral Gap Conjecture (Wikipedia)
Source Vladimir Rokhlin (Wikipedia)
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1. Wikipedia: Ergodic Theory
Wikimedia FoundationLead section
a branch of mathematics that studies statistical properties of deterministic dynamical systems
Ergodic theorems section, mean equality statement
But if the transformation is ergodic, and the measure is invariant, then the time average is equal to the space average almost everywhere.
Ergodic theorems section, Birkhoff and von Neumann
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.
Mean ergodic theorem
Von Neumann's mean ergodic theorem, holds in Hilbert spaces.
In Group: Dynamics and Ergodic Theory, lead paragraph
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity.
- Includes: Mean Ergodic Theorem, Lead sentence
View the Source Paul Halmos (Wikipedia)
Includes: Paul Halmos, LeadQuote, Includes: Paul Halmos, Lead
made fundamental advances in the areas of mathematical logic, probability theory, operator theory, ergodic theory, and functional analysis
Nelson Dunford (Wikipedia)
Includes: Nelson Dunford, LeadQuote, Includes: Nelson Dunford, Lead
ergodic theory, and linear operators
Robert Zimmer (Wikipedia)
Includes: Robert Zimmer, Lead, research focusQuote, Includes: Robert Zimmer, Lead, research focus
particularly ergodic theory
Hillel Furstenberg (Wikipedia)
Includes: Hillel Furstenberg, LeadQuote, Includes: Hillel Furstenberg, Lead
application of probability theory and ergodic theory methods to other areas of mathematics, including number theory and Lie groups
Roy Adler (Wikipedia)
Includes: Roy Adler, LeadQuote, Includes: Roy Adler, Lead
Adler studies dynamical systems, ergodic theory, symbolic and topological dynamics and coding theory.
Grigory Margulis (Wikipedia)
Includes: Grigory Margulis, LeadQuote, Includes: Grigory Margulis, Lead
the introduction of methods from ergodic theory into diophantine approximation
Jean Bourgain (Wikipedia)
Includes: Jean Bourgain, LeadQuote, Includes: Jean Bourgain, Lead
such as the geometry of Banach spaces, harmonic analysis, ergodic theory and nonlinear partial differential equations
Biographies of Women Mathematicians, Agnes Scott College: Alexandra Bellow
Biographies of Women Mathematicians, Agnes Scott CollegeIncludes: Alexandra Bellow, Biography, ergodic theoryQuote, Includes: Alexandra Bellow, Biography, ergodic theory
in the area of ergodic theory and probability.
Vladimir Rokhlin (Wikipedia)
Includes: Vladimir Rokhlin, Lead, contributionsQuote, Includes: Vladimir Rokhlin, Lead, contributions
made numerous contributions in algebraic topology, geometry, measure theory, probability theory, ergodic theory and entropy theory
Eberhard Hopf (Wikipedia)
Includes: Eberhard Hopf, LeadQuote, Includes: Eberhard Hopf, Lead
one of the founding fathers of ergodic theory and a pioneer of bifurcation theory
Amie Wilkinson (Wikipedia)
Includes: Amie Wilkinson, LeadQuote, Includes: Amie Wilkinson, Lead
smooth dynamical systems, ergodic theory, chaos theory
Spectral Gap Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Spectral Gap Conjecture, Lead sentenceQuote, Includes: Spectral Gap Conjecture, Lead sentence
In ergodic theory, a branch of mathematics, the spectral gap conjecture of Alexander Lubotzky, Ralph S.
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