Mixing

2020 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].

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For a cascade $ \{ S ^ {n} \} $ on a finite measure space $ ( W , \mu ) $ the notion of weak mixing as defined above is equivalent to the property that the cascade generated by $ S \times S $ on the measure space $ ( W \times W , \mu \otimes \mu ) $, where $ \mu \otimes \mu $ denotes the product measure, is ergodic (cf. Ergodicity; Metric transitivity). See [H].

For topological dynamical systems the notions of strong and weak mixing have been defined as well [F]. A flow on a topological space $ W $ is said to be topologically weakly mixing whenever the flow $ \{ S _ {t} \times S _ {t} \} $ on $ W \times W $( with the usual product topology) is topologically ergodic; equivalently: whenever for every choice of four non-empty open subsets $ U _ {i} , V _ {i} $( $ i = 1 , 2 $) of $ W $ where exists a $ t $ such that $ S _ {t} U _ {i} \cap V _ {i} \neq \emptyset $ for $ i = 1 , 2 $. On compact spaces the weakly mixing minimal flows are the minimal flows that have no non-trivial equicontinuous factors; see [A], p. 133. A flow $ \{ S _ {t} \} $ on a space $ W $ is said to be topologically strongly mixing whenever for every two non-empty open subsets $ U $ and $ V $ of $ W $ there exists a value $ t _ {0} $ such that $ S _ {t} U \cap V \neq \emptyset $ for all $ | t | \geq t _ {0} $. For example, the geodesic flow on a complete two-dimensional Riemannian manifold of constant negative curvature is topologically strongly mixing; see [GH], 13.49. For cascades, the definitions are analogous.

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