Ergodic (adjective)

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In mathematics and physics, the adjective ergodic is used to imply that a system satisfies the ergodic hypothesis of thermodynamics or that it is a system studied in ergodic theory.

A more rigorous definition may be given as follows:

Let (X,Σ,μ) be a probability space, and T : X \to X be a measure-preserving transformation, i.e.

\mu \left( T^{-1} (E) \right) = \mu (E) for all E \in \Sigma,

so μ is an invariant measure under T. We call T an ergodic transformation (with respect to μ) and call μ an ergodic measure (with respect to T) if, whenever T(E) = E for some E \in \Sigma, then

μ(E) = 0 or μ(E) = 1.

That is, T takes "almost all sets all over the space". The only sets it "doesn't move" are some sets of measure zero and sets that are almost the entire space.

The collection of probability measures on X that are ergodic with respect to T is sometimes denoted ET(X).

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