View source for Mixing

{{MSC|37A25}}

[[Category:Ergodic theory]]

A property of a dynamical system (a [[Cascade|cascade]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m0642701.png" /> or a [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m0642702.png" />) having a finite [[Invariant measure|invariant measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m0642703.png" />, in which for any two measurable subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m0642704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m0642705.png" /> of the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m0642706.png" />, the measure

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m0642707.png" /></td> </tr></table>

or, respectively,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m0642708.png" /></td> </tr></table>

tends to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064270/m0642709.png" /></td> </tr></table>

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 [[#References|[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 [[#References|[2]]]. All these properties are stronger than [[Ergodicity|ergodicity]].

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

====References====
<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>



====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 topological dynamical systems the notions of strong and weak mixing have been defined as well [[#References|[a2]]]. 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 [[#References|[a1]]], 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 [[#References|[a3]]], 13.49. For cascades, the definitions are analogous.

====References====
<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>