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Difference between revisions of "Mixing"

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(MSC|37A25 Category:Ergodic theory)
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.R. Halmos,   "Lectures on ergodic theory" , Math. Soc. Japan (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> U. Krengel,   L. Sucheston,   "On mixing in infinite measure spaces" ''Z. Wahrscheinlichkeitstheor. Verw. Geb.'' , '''13''' : 2 (1969) pp. 150–164</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) {{MR|0097489}} {{ZBL|0073.09302}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> 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}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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}} </TD></TR></table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Auslander,   "Minimal flows and their extensions" , North-Holland (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Furstenberg,   "Disjointness in ergodic theory, minimal sets and a problem in diophantine approximations" ''Math. Systems Th.'' , '''1''' (1967) pp. 1–49</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W.H. Gottschalk,   G.A. Hedlund,   "Topological dynamics" , Amer. Math. Soc. (1955)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Auslander, "Minimal flows and their extensions" , North-Holland (1988) {{MR|0956049}} {{ZBL|0654.54027}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> 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|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W.H. Gottschalk, G.A. Hedlund, "Topological dynamics" , Amer. Math. Soc. (1955) {{MR|0074810}} {{ZBL|0067.15204}} </TD></TR></table>

Revision as of 10:31, 27 March 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 [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) MR0097489 Zbl 0073.09302
[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 MR0518553 Zbl 0385.28009
[3] 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 [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) MR0956049 Zbl 0654.54027
[a2] H. Furstenberg, "Disjointness in ergodic theory, minimal sets and a problem in diophantine approximations" Math. Systems Th. , 1 (1967) pp. 1–49 MR0213508
[a3] 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=21775
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article