# Mixing

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A property of a dynamical system (a cascade or a flow (continuous-time dynamical system) ) having a finite invariant measure , in which for any two measurable subsets and of the phase space , the measure

or, respectively,

tends to

as , or, respectively, as . If the transformations and are invertible, then in the definition of mixing one may replace the pre-images of the original set with respect to these transformations by the direct images and , 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 , one says that the endomorphism generating it in the measure space also is mixing (has the property of mixing).

In ergodic theory, properties related to mixing are considered: multiple mixing and weak mixing (see [1]; 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 [2]. All these properties are stronger than ergodicity.

There is an analogue of mixing for systems having an infinite invariant measure [3].

#### References

 [1] P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) [2] H. Furstenberg, B. Weiss, "The finite multipliers of infinite ergodic transformations" N.G. Markley (ed.) J.C. Martin (ed.) W. Perrizo (ed.) , The Structure of Attractors in Dynamical Systems , Lect. notes in math. , 668 , Springer (1978) pp. 127–132 [3] U. Krengel, L. Sucheston, "On mixing in infinite measure spaces" Z. Wahrscheinlichkeitstheor. Verw. Geb. , 13 : 2 (1969) pp. 150–164