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 Wikipedia: Ergodic Theory
Source Spectral Gap Conjecture (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.
- Includes: Mean Ergodic Theorem, Lead sentence
View the Source Stone-Weierstrass Theorem (Wikipedia)
Wikipedialead paragraph
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity.
Ergodic theory section
The first result in this direction is the Poincare recurrence theorem, which claims that almost all points in any subset of the phase space eventually revisit the set.
Mean ergodic theorem section
Von Neumann's mean ergodic theorem, holds in Hilbert spaces.
View the Source 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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