Editing Mixing (mathematics) (section)

== Mixing in dynamical systems ==
Let <math>(X, \mathcal{A}, \mu, T)</math> be a [[measure-preserving dynamical system]], with ''T'' being the time-evolution or [[shift operator]]. The system is said to be '''strong mixing''' if, for any <math>A,B \in \mathcal{A}</math>, one has
: <math>\lim_{n\to\infty} \mu \left  (A \cap T^{-n}B \right ) = \mu(A)\mu(B).</math>

For shifts parametrized by a continuous variable instead of a discrete integer ''n'', the same definition applies, with <math>T^{-n}</math> replaced by <math>T_g</math> with ''g'' being the continuous-time parameter.

A dynamical system is said to be '''weak mixing''' if one has
: <math>\lim_{n\to\infty} \frac 1 n \sum_{k=0}^{n-1} \left  |\mu(A \cap T^{-k}B) - \mu(A)\mu(B) \right | = 0.</math>

In other words, <math>T</math> is strong mixing if <math>\mu (A \cap T^{-n}B) - \mu(A)\mu(B) \to 0</math> in the usual sense, weak mixing if
: <math> \left |\mu(A \cap T^{-n} B) - \mu(A)\mu(B) \right | \to 0,</math>

in the [[Cesàro mean|Cesàro]] sense, and ergodic if <math>\mu \left (A \cap T^{-n}B \right ) \to \mu(A)\mu(B)</math> in the Cesàro sense. Hence, strong mixing implies weak mixing, which implies ergodicity. However, the converses are not true: There exist ergodic dynamical systems which are not weakly mixing, and weakly mixing dynamical systems which are not strongly mixing. The [[Chacon system]] was historically the first example given of a system that is weak mixing but not strong mixing.<ref name="nicol">
Matthew Nicol and Karl Petersen, (2009) "[https://www.math.uh.edu/~nicol/pdffiles/petersen.pdf Ergodic Theory: Basic Examples and Constructions]",
''Encyclopedia of Complexity and Systems Science'', Springer https://doi.org/10.1007/978-0-387-30440-3_177
</ref>

'''Theorem.''' Weak mixing implies ergodicity.

'''Proof.''' If the action of the map decomposes into two components {{tmath|1= A, B }}, then we have {{tmath|1= \mu(T^{-n}(A) \cap B) = \mu(A \cap B) = \mu(\emptyset) = 0 }}, so weak mixing implies {{tmath|1= \vert \mu(A\cap B) - \mu(A)\mu(B) \vert = 0 }}, so one of <math>A, B</math> has zero measure, and the other one has full measure.

=== Covering families ===
Given a topological space, such as the unit interval (whether it has its end points or not), we can construct a measure on it by taking the open sets, then take their unions, complements, unions, complements, and so on to [[First uncountable ordinal|infinity]], to obtain all the [[Borel set]]s. Next, we define a measure <math>\mu </math> on the Borel sets, then add in all the subsets of measure-zero ("negligible sets"). This is how we obtain the [[Lebesgue measure]] and the Lebesgue measurable sets.

In most applications of ergodic theory, the underlying space is almost-everywhere isomorphic to an open subset of some <math>\R^n</math>, and so it is a Lebesgue measure space. Verifying strong-mixing can be simplified if we only need to check a smaller set of measurable sets.

A covering family <math>\mathcal C</math> is a set of measurable sets, such that any open set is a ''disjoint'' union of sets in it. Compare this with [[Base (topology)|base in topology]], which is less restrictive as it allows non-disjoint unions.

'''Theorem.''' For Lebesgue measure spaces, if <math>T</math> is measure-preserving, and <math>\lim_n \mu(T^{-n}(A)\cap B) = \mu(A)\mu (B)</math> for all <math>A, B</math> in a covering family, then <math>T</math> is strong mixing.

'''Proof.''' Extend the mixing equation from all <math>A, B</math> in the covering family, to all open sets by disjoint union, to all closed sets by taking the complement, to all measurable sets by using the regularity of Lebesgue measure to approximate any set with open and closed sets. Thus, <math>\lim_n \mu(T^{-n}(A)\cap B) = \mu(A)\mu (B)</math> for all measurable {{tmath|1= A, B }}.

=== ''L''<sup>2</sup> formulation ===

The properties of ergodicity, weak mixing and strong mixing of a measure-preserving dynamical system can also be characterized by the average of observables. By von Neumann's ergodic theorem, ergodicity of a dynamical system <math>(X, \mathcal{A}, \mu, T)</math> is equivalent to the property that, for any function <math>f \in L^2 (X, \mu)</math>, the sequence <math>(f \circ T^n)_{n \ge 0}</math> converges strongly and in the sense of Cesàro to {{tmath|1= \int_X f \, d \mu }}, i.e.,
: <math> \lim_{N \to \infty} \left \| {1 \over N} \sum_{n=0}^{N-1} f \circ T^n - \int_X f \, d \mu \right \|_{L^2 (X, \mu)}= 0.</math>

A dynamical system <math>(X, \mathcal{A}, \mu, T)</math> is weakly mixing if, for any functions <math>f</math> and <math>g \in L^2 (X, \mu),</math>
: <math> \lim_{N \to \infty} {1 \over N} \sum_{n=0}^{N-1} \left  | \int_X f \circ T^n \cdot g \, d \mu- \int_X f \, d \mu \cdot \int_X g \, d \mu \right |= 0.</math>

A dynamical system <math>(X, \mathcal{A}, \mu, T)</math> is strongly mixing if, for any function {{tmath|1= f \in L^2 (X, \mu) }}, the sequence <math>(f \circ T^n)_{n \ge 0}</math> converges weakly to {{tmath|1= \int_X f \, d \mu }}, i.e., for any function <math>g \in L^2 (X, \mu),</math>
: <math> \lim_{n \to \infty} \int_X f \circ T^n \cdot g \, d \mu = \int_X f \, d \mu \cdot \int_X g \, d \mu.</math>

Since the system is assumed to be measure preserving, this last line is equivalent to saying that the [[covariance]] {{tmath|1= \lim_{n \to \infty} \operatorname{Cov} (f \circ T^n, g) = 0 }}, so that the random variables <math>f \circ T^n</math> and <math>g</math> become orthogonal as <math>n</math> grows. Actually, since this works for any function {{tmath|1= g }}, one can informally see mixing as the property that the random variables <math>f \circ T^n</math> and <math>g</math> become independent as <math>n</math> grows.

=== Products of dynamical systems ===

Given two measured dynamical systems <math>(X, \mu, T)</math> and <math>(Y, \nu, S),</math> one can construct a dynamical system <math>(X \times Y, \mu \otimes \nu, T \times S)</math> on the Cartesian product by defining <math>(T \times S) (x,y) = (T(x), S(y)).</math> We then have the following characterizations of weak mixing:<ref name="EinsiedlerWard"> Theorem 2.36, Manfred Einsiedler and Thomas Ward, ''Ergodic theory with a view towards number theory'', (2011) Springer {{isbn|978-0-85729-020-5}}</ref>

: '''Proposition.''' A dynamical system <math>(X, \mu, T)</math> is weakly mixing if and only if, for any ergodic dynamical system {{tmath|1= (Y, \nu, S) }}, the system <math>(X \times Y, \mu \otimes \nu, T \times S)</math> is also ergodic.

: '''Proposition.''' A dynamical system <math>(X, \mu, T)</math> is weakly mixing if and only if <math>(X^2, \mu \otimes \mu, T \times T)</math> is also ergodic. If this is the case, then <math>(X^2, \mu \otimes \mu, T \times T)</math> is also weakly mixing.

=== Generalizations ===

The definition given above is sometimes called '''strong 2-mixing''', to distinguish it from higher orders of mixing.  A '''strong 3-mixing system''' may be defined as a system for which
: <math>\lim_{m,n\to\infty} \mu (A \cap T^{-m}B \cap T^{-m-n}C) = \mu(A)\mu(B)\mu(C)</math>
holds for all measurable sets ''A'', ''B'', ''C''.  We can define '''strong k-mixing''' similarly.  A system which is '''strong''' ''k''-'''mixing''' for all ''k''&nbsp;=&nbsp;2,3,4,... is called '''mixing of all orders'''.

It is unknown whether strong 2-mixing implies strong 3-mixing.  It is known that strong ''m''-mixing implies [[ergodicity]].

=== Examples ===

[[Irrational rotation]]s of the circle, and more generally irreducible translations on a torus, are ergodic but neither strongly nor weakly mixing with respect to the Lebesgue measure.

Many maps considered as chaotic are strongly mixing for some well-chosen invariant measure, including: the [[dyadic map]], [[Arnold's cat map]], [[horseshoe map]]s, [[Kolmogorov automorphism]]s, and the [[Anosov flow]] (the [[geodesic flow]] on the unit [[tangent bundle]] of [[compact manifold]]s of [[negative curvature]].)

The dyadic map is "shift to left in binary". In general, for any <math>n \in \{2, 3, \dots\}</math>, the "shift to left in base {{tmath|1= n }}" map <math>T(x) = nx \bmod 1</math> is strongly mixing on the covering family {{tmath|1= \left\{ \left( \tfrac{k}{n^s}, \tfrac{k+1}{n^s} \right) \smallsetminus \Q: s \geq 0, \leq k < n^s \right\} }}, therefore it is strongly mixing on {{tmath|1= (0, 1) \smallsetminus \Q }}, and therefore it is strongly mixing on {{tmath|1= [0, 1] }}.

Similarly, for any finite or countable alphabet {{tmath|1= \Sigma }}, we can impose a discrete probability distribution on it, then consider the probability distribution on the "coin flip" space, where each "coin flip" can take results from {{tmath|1= \Sigma }}. We can either construct the singly-infinite space <math>\Sigma^\N</math> or the doubly-infinite space {{tmath|1= \Sigma^\Z }}. In both cases, the '''shift map''' (one letter to the left) is strongly mixing, since it is strongly mixing on the covering family of cylinder sets. The [[Baker's map]] is isomorphic to a shift map, so it is strongly mixing.