# Mixing

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2010 Mathematics Subject Classification: Primary: 37A25 [MSN][ZBL]

A property of a dynamical system (a cascade $\{ S ^ {n} \}$ or a flow (continuous-time dynamical system) $\{ S _ {t} \}$) having a finite invariant measure $\mu$, in which for any two measurable subsets $A$ and $B$ of the phase space $W$, the measure

$$\mu (( S ^ {n} ) ^ {-} 1 A \cap B),$$

or, respectively,

$$\mu (( S _ {t} ) ^ {-} 1 A \cap B),$$

tends to

$$\frac{\mu ( A) \mu ( B) }{\mu ( W) }$$

as $n \rightarrow \infty$, or, respectively, as $t \rightarrow \infty$. If the transformations $S$ and $S _ {t}$ are invertible, then in the definition of mixing one may replace the pre-images of the original set $A$ with respect to these transformations by the direct images $S ^ {n} A$ and $S _ {t} A$, which are easier to visualize. If a system has the mixing property, one also says that the system is mixing, while in the case of a mixing cascade $\{ S ^ {n} \}$, one says that the endomorphism $S$ generating it in the measure space $( W, \mu )$ also is mixing (has the property of mixing).

In ergodic theory, properties related to mixing are considered: multiple mixing and weak mixing (see [H]; in the old literature the latter is often called mixing in the wide sense or simply mixing, while mixing was called mixing in the strong sense). A property intermediate between mixing and weak mixing has also been discussed [FW]. All these properties are stronger than ergodicity.

There is an analogue of mixing for systems having an infinite invariant measure [KS].

How to Cite This Entry:
Mixing. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mixing&oldid=47863
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article