Mixing (mathematics)

The mathematical definition of mixing aims to capture the ordinary every-day process of mixing, such as mixing paints, drinks, cooking ingredients, industrial process mixing, smoke in a smoke-filled room, and so on. To provide the mathematical rigor, such descriptions begin with the definition of a measure-preserving dynamical system, written as ${\displaystyle (X,{\mathcal {A}},\mu ,T)}$.

The set ${\displaystyle X}$ is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, etc. The measure ${\displaystyle \mu }$ is understood to define the natural volume of the space ${\displaystyle X}$ and of its subspaces. The collection of subspaces is denoted by ${\displaystyle {\mathcal {A}}}$, and the size of any given subset ${\displaystyle A\subset X}$ is ${\displaystyle \mu (A)}$; the size is its volume. Naively, one could imagine ${\displaystyle {\mathcal {A}}}$ to be the power set of ${\displaystyle X}$; this doesn't quite work, as not all subsets of a space have a volume (famously, the Banach-Tarski paradox). Thus, conventionally, ${\displaystyle {\mathcal {A}}}$ consists of the measurable subsets—the subsets that do have a volume. It is always taken to be a Borel set—the collection of subsets that can be constructed by taking intersections, unions and set complements; these can always be taken to be measurable.

The time evolution of the system is described by a map ${\displaystyle T:X\to X}$. Given some subset ${\displaystyle A\subset X}$, its map ${\displaystyle T(A)}$ will in general be a deformed version of ${\displaystyle A}$ – it is squashed or stretched, folded or cut into pieces. Mathematical examples include the baker's map and the horseshoe map, both inspired by bread-making. The set ${\displaystyle T(A)}$ must have the same volume as ${\displaystyle A}$; the squashing/stretching does not alter the volume of the space, only its distribution. Such a system is "measure-preserving" (area-preserving, volume-preserving).

A formal difficulty arises when one tries to reconcile the volume of sets with the need to preserve their size under a map. The problem arises because, in general, several different points in the domain of a function can map to the same point in its range; that is, there may be ${\displaystyle x\neq y}$ with ${\displaystyle T(x)=T(y)}$. Worse, a single point ${\displaystyle x\in X}$ has no size. These difficulties can be avoided by working with the inverse map ${\displaystyle T^{-1}:{\mathcal {A}}\to {\mathcal {A}}}$; it will map any given subset ${\displaystyle A\subset X}$ to the parts that were assembled to make it: these parts are ${\displaystyle T^{-1}(A)\in {\mathcal {A}}}$. It has the important property of not "losing track" of where things came from. More strongly, it has the important property that any (measure-preserving) map ${\displaystyle {\mathcal {A}}\to {\mathcal {A}}}$ is the inverse of some map ${\displaystyle X\to X}$. The proper definition of a volume-preserving map is one for which ${\displaystyle \mu (A)=\mu (T^{-1}(A))}$ because ${\displaystyle T^{-1}(A)}$ describes all the pieces-parts that ${\displaystyle A}$ came from.

One is now interested in studying the time evolution of the system. If a set ${\displaystyle A\in {\mathcal {A}}}$ eventually visits all of ${\displaystyle X}$ over a long period of time (that is, if ${\displaystyle \cup _{k=1}^{n}T^{k}(A)}$ approaches all of ${\displaystyle X}$ for large ${\displaystyle n}$), the system is said to be ergodic. If every set ${\displaystyle A}$ behaves in this way, the system is a conservative system, placed in contrast to a dissipative system, where some subsets ${\displaystyle A}$ wander away, never to be returned to. An example would be water running downhill -- once it's run down, it will never come back up again. The lake that forms at the bottom of this river can, however, become well-mixed. The ergodic decomposition theorem states that every ergodic system can be split into two parts: the conservative part, and the dissipative part.

Mixing is a stronger statement than ergodicity. Mixing asks for this ergodic property to hold between any two sets ${\displaystyle A,B}$, and not just between some set ${\displaystyle A}$ and ${\displaystyle X}$. Definition: Given any two nonempty sets ${\displaystyle A,B\in {\mathcal {A}}}$, a system is said to be (topologically) mixing if there is an integer ${\displaystyle N}$ such that, for all ${\displaystyle A,B}$ and ${\displaystyle n>N}$, one has that ${\displaystyle T^{n}(A)\cap B\neq \varnothing }$. Here, ${\displaystyle \cap }$ denotes set intersection and ${\displaystyle \varnothing }$ is the empty set.

The above definition of topological mixing should be enough to provide an informal idea of mixing (it is equivalent to the formal definition, given below). However, it made no mention of the volume of ${\displaystyle A}$ and ${\displaystyle B}$, and, indeed, there is another definition that explicitly works with the volume. Several, actually; one has both strong mixing and weak mixing; they are inequivalent, although a strong mixing system is always weakly mixing. The measure-based definitions are not compatible with the definition of topological mixing: there are systems which are one, but not the other. The general situation remains cloudy: for example, given three sets ${\displaystyle A,B,C\in {\mathcal {A}}}$, one can define 3-mixing. As of 2020, it is not known if 2-mixing implies 3-mixing. (If one thinks of ergodicity as "1-mixing", then it is clear that 1-mixing does not imply 2-mixing; there are systems that are ergodic but not mixing.)

The concept of strong mixing is made in reference to the volume of a pair of sets. Consider, for example, a set ${\displaystyle A}$ of colored dye that is being mixed into a cup of some sort of sticky liquid, say, corn syrup, or shampoo, or the like. Practical experience shows that mixing sticky fluids can be quite hard: there is usually some corner of the container where it is hard to get the dye mixed into. Pick as set ${\displaystyle B}$ that hard-to-reach corner. The question of mixing is then, can ${\displaystyle A}$, after a long enough period of time, not only penetrate into ${\displaystyle B}$ but also fill ${\displaystyle B}$ with the same proportion as it does elsewhere?

One phrases the definition of strong mixing as the requirement that

${\displaystyle \lim _{n\to \infty }\mu \left(T^{-n}A\cap B\right)=\mu (A)\mu (B).}$

The time parameter ${\displaystyle n}$ serves to separate ${\displaystyle A}$ and ${\displaystyle B}$ in time, so that one is mixing ${\displaystyle A}$ while holding the test volume ${\displaystyle B}$ fixed. The product ${\displaystyle \mu (A)\mu (B)}$ is a bit more subtle. Imagine that the volume ${\displaystyle B}$ is 10% of the total volume, and that the volume of dye ${\displaystyle A}$ will also be 10% of the grand total. If ${\displaystyle A}$ is uniformly distributed, then it is occupying 10% of ${\displaystyle B}$, which itself is 10% of the total, and so, in the end, after mixing, the part of ${\displaystyle A}$ that is in ${\displaystyle B}$ is 1% of the total volume. That is, ${\displaystyle \mu \left({\mbox{after-mixing}}(A)\cap B\right)=\mu (A)\mu (B).}$ This product-of-volumes has more than passing resemblance to Bayes theorem in probabilities; this is not an accident, but rather a consequence that measure theory and probability theory are the same theory: they share the same axioms (the Kolmogorov axioms), even as they use different notation.

The reason for using ${\displaystyle T^{-n}A}$ instead of ${\displaystyle T^{n}A}$ in the definition is a bit subtle, but it follows from the same reasons why ${\displaystyle T^{-1}A}$ was used to define the concept of a measure-preserving map. When looking at how much dye got mixed into the corner ${\displaystyle B}$, one wants to look at where that dye "came from" (presumably, it was poured in at the top, at some time in the past). One must be sure that every place it might have "come from" eventually gets mixed into ${\displaystyle B}$.

Let ${\displaystyle (X,{\mathcal {A}},\mu ,T)}$ 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 ${\displaystyle A,B\in {\mathcal {A}}}$, one has

${\displaystyle \lim _{n\to \infty }\mu \left(A\cap T^{-n}B\right)=\mu (A)\mu (B).}$

For shifts parametrized by a continuous variable instead of a discrete integer n, the same definition applies, with ${\displaystyle T^{-n}}$ replaced by ${\displaystyle T_{g}}$ with g being the continuous-time parameter.

A dynamical system is said to be weak mixing if one has

${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{k=0}^{n-1}\left|\mu (A\cap T^{-k}B)-\mu (A)\mu (B)\right|=0.}$

In other words, ${\displaystyle T}$ is strong mixing if ${\displaystyle \mu (A\cap T^{-n}B)-\mu (A)\mu (B)\to 0}$ in the usual sense, weak mixing if

${\displaystyle \left|\mu (A\cap T^{-n}B)-\mu (A)\mu (B)\right|\to 0,}$

in the Cesàro sense, and ergodic if ${\displaystyle \mu \left(A\cap T^{-n}B\right)\to \mu (A)\mu (B)}$ in the Cesàro sense. Hence, strong mixing implies weak mixing, which implies ergodicity. However, the converse is 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.[1]

A form of mixing may be defined without appeal to a measure, using only the topology of the system. A continuous map ${\displaystyle f:X\to X}$ is said to be topologically transitive if, for every pair of non-empty open sets ${\displaystyle A,B\subset X}$, there exists an integer n such that

${\displaystyle f^{n}(A)\cap B\neq \varnothing }$

where ${\displaystyle f^{n}}$ is the nth iterate of f. In the operator theory, a topologically transitive bounded linear operator (a continuous linear map on a topological vector space) is usually called hypercyclic operator. A related idea is expressed by the wandering set.

Lemma: If X is a complete metric space with no isolated point, then f is topologically transitive if and only if there exists a hypercyclic point ${\displaystyle x\in X}$, that is, a point x such that its orbit ${\displaystyle \{f^{n}(x):n\in \mathbb {N} \}}$ is dense in X.

A system is said to be topologically mixing if, given open sets ${\displaystyle A}$ and ${\displaystyle B}$, there exists an integer N, such that, for all ${\displaystyle n>N}$, one has

${\displaystyle f^{n}(A)\cap B\neq \varnothing .}$

For a continuous-time system, ${\displaystyle f^{n}}$ is replaced by the flow ${\displaystyle \varphi _{g}}$, with g being the continuous parameter, with the requirement that a non-empty intersection hold for all ${\displaystyle \Vert g\Vert >N}$.

A weak topological mixing is one that has no non-constant continuous (with respect to the topology) eigenfunctions of the shift operator.

Topological mixing neither implies, nor is implied by either weak or strong mixing: there are examples of systems that are weak mixing but not topologically mixing, and examples that are topologically mixing but not strong mixing.

Let ${\displaystyle (X_{t})_{-\infty <t<\infty }}$ be a stochastic process on a probability space ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}$. The sequence space into which the process maps can be endowed with a topology, the product topology. The open sets of this topology are called cylinder sets. These cylinder sets generate a σ-algebra, the Borel σ-algebra; this is the smallest σ-algebra that contains the topology.

Define a function ${\displaystyle \alpha }$, called the strong mixing coefficient, as

${\displaystyle \alpha (s)=\sup \left\{|\mathbb {P} (A\cap B)-\mathbb {P} (A)\mathbb {P} (B)|:-\infty <t<\infty ,A\in X_{-\infty }^{t},B\in X_{t+s}^{\infty }\right\}}$

for all ${\displaystyle -\infty <s<\infty }$. The symbol ${\displaystyle X_{a}^{b}}$, with ${\displaystyle -\infty \leq a\leq b\leq \infty }$ denotes a sub-σ-algebra of the σ-algebra; it is the set of cylinder sets that are specified between times a and b, i.e. the σ-algebra generated by ${\displaystyle \{X_{a},X_{a+1},\ldots ,X_{b}\}}$.

The process ${\displaystyle (X_{t})_{-\infty <t<\infty }}$ is said to be strongly mixing if ${\displaystyle \alpha (s)\to 0}$ as ${\displaystyle s\to \infty }$. That is to say, a strongly mixing process is such that, in a way that is uniform over all times ${\displaystyle t}$ and all events, the events before time ${\displaystyle t}$ and the events after time ${\displaystyle t+s}$ tend towards being independent as ${\displaystyle s\to \infty }$; more colloquially, the process, in a strong sense, forgets its history.