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{{MSC|37A25}}
 
{{MSC|37A25}}
  
 
[[Category:Ergodic theory]]
 
[[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
+
A property of a dynamical system (a [[Cascade|cascade]] $  \{ S  ^ {n} \} $
 +
or a [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] $  \{ S _ {t} \} $)  
 +
having a finite [[Invariant measure|invariant measure]] $  \mu $,  
 +
in which for any two measurable subsets $  A $
 +
and $  B $
 +
of the phase space $  W $,  
 +
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>
+
$$
 +
\mu (( S  ^ {n} )  ^ {-} 1 A \cap B),
 +
$$
  
 
or, respectively,
 
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>
+
$$
 +
\mu (( S _ {t} )  ^ {-} 1 A \cap B),
 +
$$
  
 
tends to
 
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>
+
$$
 +
 
 +
\frac{\mu ( A) \mu ( B) }{\mu ( W) }
 +
 
 +
$$
  
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 $  n \rightarrow \infty $,  
 +
or, respectively, as $  t \rightarrow \infty $.  
 +
If the transformations $  S $
 +
and $  S _ {t} $
 +
are invertible, then in the definition of mixing one may replace the pre-images of the original set $  A $
 +
with respect to these transformations by the direct images $  S  ^ {n} A $
 +
and $  S _ {t} A $,  
 +
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 $  \{ S  ^ {n} \} $,  
 +
one says that the endomorphism $  S $
 +
generating it in the measure space $  ( W, \mu ) $
 +
also is mixing (has the property of mixing).
  
 
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]].
 
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]].
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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 {{Cite|H}}.
+
For a cascade $  \{ S  ^ {n} \} $
 +
on a finite measure space $  ( W , \mu ) $
 +
the notion of weak mixing as defined above is equivalent to the property that the cascade generated by $  S \times S $
 +
on the measure space $  ( W \times W , \mu \otimes \mu ) $,  
 +
where $  \mu \otimes \mu $
 +
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 $  W $
 +
is said to be topologically weakly mixing whenever the flow $  \{ S _ {t} \times S _ {t} \} $
 +
on $  W \times W $(
 +
with the usual product topology) is topologically ergodic; equivalently: whenever for every choice of four non-empty open subsets $  U _ {i} , V _ {i} $(
 +
$  i = 1 , 2 $)  
 +
of $  W $
 +
where exists a $  t $
 +
such that $  S _ {t} U _ {i} \cap V _ {i} \neq \emptyset $
 +
for $  i = 1 , 2 $.  
 +
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 $  \{ S _ {t} \} $
 +
on a space $  W $
 +
is said to be topologically strongly mixing whenever for every two non-empty open subsets $  U $
 +
and $  V $
 +
of $  W $
 +
there exists a value $  t _ {0} $
 +
such that $  S _ {t} U \cap V \neq \emptyset $
 +
for all $  | t | \geq  t _ {0} $.  
 +
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====

Latest revision as of 08:01, 6 June 2020


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

A property of a dynamical system (a cascade $ \{ S ^ {n} \} $ or a flow (continuous-time dynamical system) $ \{ S _ {t} \} $) having a finite invariant measure $ \mu $, in which for any two measurable subsets $ A $ and $ B $ of the phase space $ W $, the measure

$$ \mu (( S ^ {n} ) ^ {-} 1 A \cap B), $$

or, respectively,

$$ \mu (( S _ {t} ) ^ {-} 1 A \cap B), $$

tends to

$$ \frac{\mu ( A) \mu ( B) }{\mu ( W) } $$

as $ n \rightarrow \infty $, or, respectively, as $ t \rightarrow \infty $. If the transformations $ S $ and $ S _ {t} $ are invertible, then in the definition of mixing one may replace the pre-images of the original set $ A $ with respect to these transformations by the direct images $ S ^ {n} A $ and $ S _ {t} A $, 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 $ \{ S ^ {n} \} $, one says that the endomorphism $ S $ generating it in the measure space $ ( W, \mu ) $ 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 $ \{ S ^ {n} \} $ on a finite measure space $ ( W , \mu ) $ the notion of weak mixing as defined above is equivalent to the property that the cascade generated by $ S \times S $ on the measure space $ ( W \times W , \mu \otimes \mu ) $, where $ \mu \otimes \mu $ 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 $ W $ is said to be topologically weakly mixing whenever the flow $ \{ S _ {t} \times S _ {t} \} $ on $ W \times W $( with the usual product topology) is topologically ergodic; equivalently: whenever for every choice of four non-empty open subsets $ U _ {i} , V _ {i} $( $ i = 1 , 2 $) of $ W $ where exists a $ t $ such that $ S _ {t} U _ {i} \cap V _ {i} \neq \emptyset $ for $ i = 1 , 2 $. 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 $ \{ S _ {t} \} $ on a space $ W $ is said to be topologically strongly mixing whenever for every two non-empty open subsets $ U $ and $ V $ of $ W $ there exists a value $ t _ {0} $ such that $ S _ {t} U \cap V \neq \emptyset $ for all $ | t | \geq t _ {0} $. 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=47863
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article