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====Comments====
 
====Comments====
For a cascade <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427020.png" /> on a finite measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427021.png" /> the notion of weak mixing as defined above is equivalent to the property that the cascade generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427022.png" /> on the measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427024.png" /> denotes the product measure, is ergodic (cf. [[Ergodicity|Ergodicity]]; [[Metric transitivity|Metric transitivity]]). See [[#References|[1]]].
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For a cascade <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427020.png" /> on a finite measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427021.png" /> the notion of weak mixing as defined above is equivalent to the property that the cascade generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427022.png" /> on the measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427024.png" /> denotes the product measure, is ergodic (cf. [[Ergodicity|Ergodicity]]; [[Metric transitivity|Metric transitivity]]). See {{Cite|H}}.
  
 
For topological dynamical systems the notions of strong and weak mixing have been defined as well {{Cite|F}}. A flow on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427025.png" /> is said to be topologically weakly mixing whenever the flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427026.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427027.png" /> (with the usual product topology) is topologically ergodic; equivalently: whenever for every choice of four non-empty open subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427028.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427029.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427030.png" /> where exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427032.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427033.png" />. On compact spaces the weakly mixing minimal flows are the minimal flows that have no non-trivial equicontinuous factors; see {{Cite|A}}, p. 133. A flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427034.png" /> on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427035.png" /> is said to be topologically strongly mixing whenever for every two non-empty open subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427037.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427038.png" /> there exists a value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427039.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427040.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427041.png" />. For example, the geodesic flow on a complete two-dimensional Riemannian manifold of constant negative curvature is topologically strongly mixing; see {{Cite|GH}}, 13.49. For cascades, the definitions are analogous.
 
For topological dynamical systems the notions of strong and weak mixing have been defined as well {{Cite|F}}. A flow on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427025.png" /> is said to be topologically weakly mixing whenever the flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427026.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427027.png" /> (with the usual product topology) is topologically ergodic; equivalently: whenever for every choice of four non-empty open subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427028.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427029.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427030.png" /> where exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427032.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427033.png" />. On compact spaces the weakly mixing minimal flows are the minimal flows that have no non-trivial equicontinuous factors; see {{Cite|A}}, p. 133. A flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427034.png" /> on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427035.png" /> is said to be topologically strongly mixing whenever for every two non-empty open subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427037.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427038.png" /> there exists a value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427039.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427040.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427041.png" />. For example, the geodesic flow on a complete two-dimensional Riemannian manifold of constant negative curvature is topologically strongly mixing; see {{Cite|GH}}, 13.49. For cascades, the definitions are analogous.

Revision as of 06:03, 15 May 2012

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 [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].

References

[H] P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) MR0097489 Zbl 0073.09302
[FW] 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 MR0518553 Zbl 0385.28009
[KS] U. Krengel, L. Sucheston, "On mixing in infinite measure spaces" Z. Wahrscheinlichkeitstheor. Verw. Geb. , 13 : 2 (1969) pp. 150–164 MR0254215 Zbl 0176.33804

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 [H].

For topological dynamical systems the notions of strong and weak mixing have been defined as well [F]. 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 [A], 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 [GH], 13.49. For cascades, the definitions are analogous.

References

[A] J. Auslander, "Minimal flows and their extensions" , North-Holland (1988) MR0956049 Zbl 0654.54027
[F] H. Furstenberg, "Disjointness in ergodic theory, minimal sets and a problem in diophantine approximations" Math. Systems Th. , 1 (1967) pp. 1–49 MR0213508
[GH] W.H. Gottschalk, G.A. Hedlund, "Topological dynamics" , Amer. Math. Soc. (1955) MR0074810 Zbl 0067.15204
How to Cite This Entry:
Mixing. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mixing&oldid=26631
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article