Mixing

2020 Mathematics Subject Classification: Primary: 37A25 [MSN][ZBL]

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

Comments

For a cascade on a finite measure space the notion of weak mixing as defined above is equivalent to the property that the cascade generated by on the measure space , where denotes the product measure, is ergodic (cf. Ergodicity; Metric transitivity). See [1].

For topological dynamical systems the notions of strong and weak mixing have been defined as well [a2]. A flow on a topological space is said to be topologically weakly mixing whenever the flow on (with the usual product topology) is topologically ergodic; equivalently: whenever for every choice of four non-empty open subsets () of where exists a such that for . On compact spaces the weakly mixing minimal flows are the minimal flows that have no non-trivial equicontinuous factors; see [a1], p. 133. A flow on a space is said to be topologically strongly mixing whenever for every two non-empty open subsets and of there exists a value such that for all . For example, the geodesic flow on a complete two-dimensional Riemannian manifold of constant negative curvature is topologically strongly mixing; see [a3], 13.49. For cascades, the definitions are analogous.

References

[a1]	J. Auslander, "Minimal flows and their extensions" , North-Holland (1988)
[a2]	H. Furstenberg, "Disjointness in ergodic theory, minimal sets and a problem in diophantine approximations" Math. Systems Th. , 1 (1967) pp. 1–49
[a3]	W.H. Gottschalk, G.A. Hedlund, "Topological dynamics" , Amer. Math. Soc. (1955)