Difference between revisions of "Mixing"
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as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427010.png" />, or, respectively, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427011.png" />. If the transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427013.png" /> are invertible, then in the definition of mixing one may replace the pre-images of the original set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427014.png" /> with respect to these transformations by the direct images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427016.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427017.png" />, one says that the endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427018.png" /> generating it in the measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427019.png" /> also is mixing (has the property of mixing). | as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427010.png" />, or, respectively, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427011.png" />. If the transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427013.png" /> are invertible, then in the definition of mixing one may replace the pre-images of the original set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427014.png" /> with respect to these transformations by the direct images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427016.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427017.png" />, one says that the endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427018.png" /> generating it in the measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m06427019.png" /> also is mixing (has the property of mixing). | ||
− | In ergodic theory, properties related to mixing are considered: multiple mixing and weak mixing (see | + | In ergodic theory, properties related to mixing are considered: multiple mixing and weak mixing (see {{Cite|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 {{Cite|FW}}. All these properties are stronger than [[Ergodicity|ergodicity]]. |
− | There is an analogue of mixing for systems having an infinite invariant measure | + | There is an analogue of mixing for systems having an infinite invariant measure {{Cite|KS}}. |
====References==== | ====References==== | ||
− | + | {| | |
− | + | |valign="top"|{{Ref|H}}|| P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) {{MR|0097489}} {{ZBL|0073.09302}} | |
− | + | |- | |
+ | |valign="top"|{{Ref|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 {{MR|0518553}} {{ZBL|0385.28009}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|KS}}|| U. Krengel, L. Sucheston, "On mixing in infinite measure spaces" ''Z. Wahrscheinlichkeitstheor. Verw. Geb.'' , '''13''' : 2 (1969) pp. 150–164 {{MR|0254215}} {{ZBL|0176.33804}} | ||
+ | |} | ||
====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]]]. | 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]]]. | ||
− | For topological dynamical systems the notions of strong and weak mixing have been defined as well | + | 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. |
====References==== | ====References==== | ||
− | + | {| | |
+ | |valign="top"|{{Ref|A}}|| J. Auslander, "Minimal flows and their extensions" , North-Holland (1988) {{MR|0956049}} {{ZBL|0654.54027}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|F}}|| H. Furstenberg, "Disjointness in ergodic theory, minimal sets and a problem in diophantine approximations" ''Math. Systems Th.'' , '''1''' (1967) pp. 1–49 {{MR|0213508}} {{ZBL|}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|GH}}|| W.H. Gottschalk, G.A. Hedlund, "Topological dynamics" , Amer. Math. Soc. (1955) {{MR|0074810}} {{ZBL|0067.15204}} | ||
+ | |} |
Revision as of 06:02, 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 [1].
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 |
Mixing. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mixing&oldid=23637