Branches of Mathematics
Ergodic Theory
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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.
Facts
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 Cross-Tradition Connections
Sources
1. Wikipedia: Ergodic Theory
Wikimedia FoundationLead sectionQuote, Lead section
a branch of mathematics that studies statistical properties of deterministic dynamical systems
1. Wikipedia: Ergodic Theory
Wikimedia FoundationErgodic theorems section, mean equality statementQuote, 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.
1. Wikipedia: Ergodic Theory
Wikimedia FoundationErgodic theorems section, Birkhoff and von NeumannQuote, 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.
Stone-Weierstrass Theorem (Wikipedia)
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Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity.
Stone-Weierstrass Theorem (Wikipedia)
WikipediaErgodic theory sectionQuote, 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.
Stone-Weierstrass Theorem (Wikipedia)
WikipediaMean ergodic theorem sectionQuote, Mean ergodic theorem section
Von Neumann's mean ergodic theorem, holds in Hilbert spaces.
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