Ergodicity

Any precise definition of the phenomenon of ergodicity from a mathematical viewpoint requires measure theory. A more intuitive description, from a physical viewpoint, is the ergodic hypothesis. Ergodic processes give a more probabilistic formulation for certain cases.

For a discrete dynamical system ${\displaystyle (X,T)}$, where the space ${\displaystyle X}$ is endowed with the additional structure of a probability measure space which is invariant under the transformation ${\displaystyle T}$, ergodicity means that there is no way to measurably isolate a nontrivial part of ${\displaystyle X}$ which is invariant under ${\displaystyle T}$. (Here "trivial" means that the subset or its complement has measure 0.) Figuratively one could say that the transformation mashes the space together in a (measurably) intractable way; a stronger, quantitative notion is that of mixing.

The relation of this dynamical notion with the physical notion of ergodicity is made via Birkhoff's ergodic theorem. The relation with ergodic processes is discussed below.

Let ${\displaystyle (X,{\mathcal {B}})}$ be a measurable space. If ${\displaystyle T}$ is a measurable function from ${\displaystyle X}$ to itself and ${\displaystyle \mu }$ a probability measure on ${\displaystyle (X,{\mathcal {B}})}$ then we say that ${\displaystyle T}$ is ${\displaystyle \mu }$-ergodic or ${\displaystyle \mu }$ is an ergodic measure for ${\displaystyle T}$ if ${\displaystyle T}$ preserves ${\displaystyle \mu }$ and the following condition holds:

For any ${\displaystyle A\in {\mathcal {B}}}$ such that ${\displaystyle T^{-1}(A)\subset A}$ either ${\displaystyle \mu (A)=0}$ or ${\displaystyle \mu (A)=1}$.

In other words there are no ${\displaystyle T}$-invariant subsets up to measure 0 (with respect to ${\displaystyle \mu }$). Recall that ${\displaystyle T}$ preserving ${\displaystyle \mu }$ (or ${\displaystyle \mu }$ being ${\displaystyle T}$-invariant) means that ${\displaystyle \mu (T^{-1}(A))=\mu (A)}$ for all ${\displaystyle A\in {\mathcal {B}}}$ (see also Measure-preserving dynamical system).

The simplest example is when ${\displaystyle X}$ is a finite set and ${\displaystyle \mu }$ the counting measure. Then a self-map of ${\displaystyle X}$ preserves ${\displaystyle \mu }$ if and only if it is a bijection, and it is ergodic if and only if ${\displaystyle T}$ has only one orbit (that is, for every ${\displaystyle x,y\in X}$ there exists ${\displaystyle k\in \mathbb {N} }$ such that ${\displaystyle y=T^{k}(x)}$). For example, if ${\displaystyle X=\{1,2,\ldots ,n\}}$ then the cycle ${\displaystyle (1\,2\,\cdots \,n)}$ is ergodic, but the permutation ${\displaystyle (1\,2)(3\,4\,\cdots n)}$ is not (it has the two invariant subsets ${\displaystyle \{1,2\}}$ and ${\displaystyle \{3,4,\ldots ,n\}}$).

The definition given above admits the following immediate reformulations:

• for every ${\displaystyle A\in {\mathcal {B}}}$ with ${\displaystyle \mu (T^{-1}(A)\bigtriangleup A)=0}$ we have ${\displaystyle \mu (A)=0}$ or ${\displaystyle \mu (A)=1\,}$ (where ${\displaystyle \bigtriangleup }$ denotes the symmetric difference);
• for every ${\displaystyle A\in {\mathcal {B}}}$ with positive measure we have ${\displaystyle \mu \left(\bigcup _{n=1}^{\infty }T^{-n}(A)\right)=1}$;
• for every two sets ${\displaystyle A,B\in {\mathcal {B}}}$ of positive measure, there exists ${\displaystyle n>0}$ such that ${\displaystyle \mu ((T^{-n}(A))\cap B)>0}$;
• Every measurable function ${\displaystyle f:X\to \mathbb {R} }$ with ${\displaystyle f\circ T=f}$ is constant on a subset of full measure.

Importantly for applications, the condition in the last characterisation can be restricted to square-integrable functions only:

• If ${\displaystyle f\in L^{2}(X,\mu )}$ and ${\displaystyle f\circ T=f}$ then ${\displaystyle f}$ is constant almost everywhere.

Let ${\displaystyle S}$ be a finite set and ${\displaystyle X=S^{\mathbb {Z} }}$ with ${\displaystyle \mu }$ the product measure (each factor ${\displaystyle S}$ being endowed with its counting measure). Then the shift operator ${\displaystyle T}$ defined by ${\displaystyle T\left((s_{k})_{k\in \mathbb {Z} })\right)=(s_{k+1})_{k\in \mathbb {Z} }}$ is ${\displaystyle \mu }$-ergodic[1].

There are many more ergodic measures for the shift map ${\displaystyle T}$ on ${\displaystyle X}$. Periodic sequences give finitely supported measures. More interestingly, there are infinitely-supported ones which are subshifts of finite type.

Let ${\displaystyle X}$ be the unit circle ${\displaystyle \{z\in \mathbb {C} ,\,|z|=1\}}$, with its Lebesgue measure ${\displaystyle \mu }$. For any ${\displaystyle \theta \in \mathbb {R} }$ the rotation of ${\displaystyle X}$ of angle ${\displaystyle \theta }$ is given by ${\displaystyle R_{\theta }(z)=e^{2i\pi \theta }z}$. If ${\displaystyle \theta \in \mathbb {Q} }$ then ${\displaystyle T_{\theta }}$ is not ergodic for the Lebesgue measure as it has infinitely many finite orbits. On the other hand, if ${\displaystyle \theta }$ is irrational then ${\displaystyle T_{\theta }}$ is ergodic[2].

Let ${\displaystyle X=\mathbb {R} ^{2}/\mathbb {Z} ^{2}}$ be the 2-torus. Then any element ${\displaystyle g\in \mathrm {SL} _{2}(\mathbb {Z} )}$ defines a self-map of ${\displaystyle X}$ since ${\displaystyle g(\mathbb {Z} ^{2})=\mathbb {Z} ^{2}}$. When ${\displaystyle g=\left({\begin{array}{cc}2&1\\1&1\end{array}}\right)}$ one obtains the so-called Arnold's cat map, which is ergodic for the Lebesgue measure on the torus.

If ${\displaystyle \mu }$ is a probability measure on a space ${\displaystyle X}$ which is ergodic for a transformation ${\displaystyle T}$ the pointwise ergodic theorem of G. Birkhoff states that for every measurable functions ${\displaystyle f:X\to \mathbb {R} }$ and for ${\displaystyle \mu }$-almost every point ${\displaystyle x\in X}$ the time average on the orbit of ${\displaystyle x}$ converges to the space average of ${\displaystyle f}$. Formally this means that

$${\displaystyle \lim _{k\to +\infty }\left({\frac {1}{k+1}}\sum _{i=0}^{k}f(T^{i}(x))\right)=\int _{X}fd\mu .}$$

The mean ergodic theorem of J. von Neumann is a similar, weaker statement about averaged translates of square-integrable functions.

An immediate consequence of the definition of ergodicity is that on a topological space ${\displaystyle X}$, and if ${\displaystyle {\mathcal {B}}}$ is the σ-algebra of Borel sets, if ${\displaystyle T}$ is ${\displaystyle \mu }$-ergodic then ${\displaystyle \mu }$-almost every orbit of ${\displaystyle T}$ is dense in the support of ${\displaystyle \mu }$.

This is not an equivalence since for a transformation which is not uniquely ergodic, but for which there is an ergodic measure with full support ${\displaystyle \mu _{0}}$, for any other ergodic measure ${\displaystyle \mu _{1}}$ the measure ${\textstyle {\frac {1}{2}}(\mu _{0}+\mu _{1})}$ is not ergodic for ${\displaystyle T}$ but its orbits are dense in the support. Explicit examples can be constructed with shift-invariant measures[3].

A transformation ${\displaystyle T}$ of a probability measure space ${\displaystyle (X,\mu )}$ is said to be mixing for the measure ${\displaystyle \mu }$ if for any measurable sets ${\displaystyle A,B\subset X}$ the following holds:

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

It is immediate that a mixing transformation is also ergodic (taking ${\displaystyle A}$ to be a ${\displaystyle T}$-stable subset and ${\displaystyle B}$ its complement). The converse is not true, for example a rotation with irrational angle on the circle (which is ergodic per the examples above) is not mixing (for a sufficiently small interval its successive images will not intersect itself most of the time). Bernoulli shifts are mixing, and so is Arnold's cat map.

This notion of mixing is sometimes called strong mixing, as opposed to weak mixing which means that

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

The transformation ${\displaystyle T}$ is said to be properly ergodic if it does not have an orbit of full measure. In the discrete case this means that the measure ${\displaystyle \mu }$ is not supported on a finite orbit of ${\displaystyle T}$.

As in the discrete case the simplest example is that of a transitive action, for instance the action on the circle given by ${\displaystyle T_{t}(z)=e^{2i\pi t}z}$ is ergodic for Lebesgue measure.

An example with infinitely many orbits is given by the flow along an irrational slope on the torus: let ${\displaystyle X=\mathbb {S} ^{1}\times \mathbb {S} ^{1}}$ and ${\displaystyle \alpha \in \mathbb {R} }$. Let ${\displaystyle T_{t}(z_{1},z_{2})=(e^{2i\pi t}z_{1},e^{2\alpha i\pi t}z_{2})}$; then if ${\displaystyle \alpha \not \in \mathbb {Q} }$ this is ergodic for the Lebesgue measure.

Further examples of ergodic flows are:

• Billiards in convex Euclidean domains;
• the geodesic flow of a negatively curved Riemannian manifold of finite volume is ergodic (for the normalised volume measure);
• the horocycle flow on a hyperbolic manifold of finite volume is ergodic (for the normalised volume measure)

A very powerful alternate definition of ergodic measures can be given using the theory of Banach spaces. Radon measures on ${\displaystyle X}$ form a Banach space of which the set ${\displaystyle {\mathcal {P}}(X)}$ of probability measures on ${\displaystyle X}$ is a convex subset. Given a continuous transformation ${\displaystyle T}$ of ${\displaystyle X}$ the subset ${\displaystyle {\mathcal {P}}(X)^{T}}$ of ${\displaystyle T}$-invariant measures is a closed convex subset, and a measure is ergodic for ${\displaystyle T}$ if and only if it is an extreme point of this convex[4].

In the setting above it follows from the Banach-Alaoglu theorem that there always exists extremal points in ${\displaystyle {\mathcal {P}}(X)^{T}}$. Hence a transformation of a compact metric space always admits ergodic measures.

In general an invariant measure need not be ergodic, but as a consequence of Choquet theory it can always be expressed as the barycenter of a probability measure on the set of ergodic measures. This is referred to as the ergodic decomposition of the measure[5].

In the case of ${\displaystyle X=\{1,\ldots ,n\}}$ and ${\displaystyle T=(1\,2)(3\,4\,\cdots n)}$ the counting measure is not ergodic. The ergodic measures for ${\displaystyle T}$ are the uniform measures ${\displaystyle \mu _{1},\mu _{2}}$ supported on the subsets ${\displaystyle \{1,2\}}$ and ${\displaystyle \{3,\ldots ,n\}}$ and every ${\displaystyle T}$-invariant probability measure can be written in the form ${\displaystyle t\mu _{1}+(1-t)\mu _{2}}$ for some ${\displaystyle t\in [0,1]}$. In particular ${\textstyle {\frac {2}{n}}\mu _{1}+{\frac {n-2}{n}}\mu _{2}}$ is the ergodic decomposition of the counting measure.

Everything in this section transfers verbatim to continous actions of ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {R} _{+}}$ on compact metric spaces.

The transformation ${\displaystyle T}$ is said to be uniquely ergodic if there is a unique Borel probability measure ${\displaystyle \mu }$ on ${\displaystyle X}$ which is ergodic for ${\displaystyle T}$.

In the examples considered above, irrational rotations of the circle are uniquely ergodic[6]; shift maps are not.

Let ${\displaystyle S}$ be a finite set. A Markov chain on ${\displaystyle S}$ is defined by a matrix ${\displaystyle P\in [0,1]^{S\times S}}$, where ${\displaystyle P(s_{1},s_{2})}$ is the transition probability from ${\displaystyle s_{1}}$ to ${\displaystyle s_{2}}$, so ${\displaystyle \sum _{s'\in s}P(s,s')=1}$. A stationary measure for ${\displaystyle P}$ is a probability measure ${\displaystyle \nu }$ on ${\displaystyle S}$ such that ${\displaystyle \nu P=\nu }$ ; that is ${\displaystyle \sum _{s'\in S}\nu (s)P(s',s)=\nu (s)}$ for all ${\displaystyle s\in S}$.

Using this data we can define a probability measure ${\displaystyle \mu _{\nu }}$ on the set ${\displaystyle X=S^{\mathbb {Z} }}$ with its product σ-algebra by giving the measures of the cylinders as follows:

$${\displaystyle \mu _{\nu }(\cdots \times S\times \{(s_{n},\ldots ,s_{m})\}\times S\times \cdots )=\nu (s_{n})P(s_{n},s_{n+1})\cdots P(s_{m-1},s_{m}).}$$

Stationarity of ${\displaystyle \nu }$ then means that the measure ${\displaystyle \mu _{\nu }}$ is invariant under the shift map ${\displaystyle T\left((s_{k})_{k\in \mathbb {Z} })\right)=(s_{k+1})_{k\in \mathbb {Z} }}$.

The measure ${\displaystyle \mu _{\nu }}$ is always ergodic for the shift map if the associated Markov chain is irreducible (any state can be reached with positive probability form any other state in a finite number of steps)[7].

The hypotheses above imply that there is a unique stationary measure for the Markov chain. In terms of the matrix ${\displaystyle P}$ a sufficient condition for this is that 1 be a simple eigenvalue of the matrix ${\displaystyle P}$ and all other eigenvalues of ${\displaystyle P}$ (in ${\displaystyle \mathbb {C} }$) are of modulus <1.

Note that in probability theory the Markov chain is called ergodic if in addition each state is aperiodic (the times where the return probability is positive are not multiples of a single integer >1). This is not necessary for the invariant measure to be ergodic; hence the notions of "ergodicity" for a Markov chain and the associated shift-invariant measure are different (the one for the chain is stricty stronger)[8].

Moreover the criterion is an "if and only if" if all communicating classes in the chain are recurrent and we consider all stationary measures.

If ${\displaystyle P(s,s')=1/|S|}$ for all ${\displaystyle s,s'\in S}$ then the stationary measure is the counting measure, the measure ${\displaystyle \mu _{P}}$ is the product of counting measures. The Markov chain is ergodic, so the shift example from above is a special case of the criterion.

Markov chains with recurring communicating classes are not irreducible are not ergodic, and this can be seen immediately as follows. If ${\displaystyle S_{1}\subsetneq S}$ are two distinct recurrent communicating classes there are nonzero stationary measures ${\displaystyle \nu _{1},\nu _{2}}$ supported on ${\displaystyle S_{1},S_{2}}$ respectively and the subsets ${\displaystyle S_{1}^{\mathbb {Z} }}$ and ${\displaystyle S_{2}^{\mathbb {Z} }}$ are both shift-invariant and of measure 1.2 for the invariant probability measure ${\displaystyle {\frac {1}{2}}(\nu _{1}+\nu _{2})}$. A very simple example of that is the chain on ${\displaystyle S=\{1,2\}}$ given by the matrix ${\textstyle \left({\begin{array}{cc}1&0\\0&1\end{array}}\right)}$ (both states are stationary).

The Markov chain on ${\displaystyle S=\{1,2\}}$ given by the matrix ${\textstyle \left({\begin{array}{cc}0&1\\1&0\end{array}}\right)}$ is irreducible but periodic. Thus it is not ergodic in the sense of Markov chain though the associated measure ${\displaystyle \mu }$ on ${\displaystyle \{1,2\}^{\mathbb {Z} }}$ is ergodic for the shift map. However the shift is not mixing for this measure, as for the sets

$${\displaystyle A=\cdots \times \{1,2\}\times 1\times \{1,2\}\times 1\times \{1,2\}\cdots }$$

and

$${\displaystyle B=\cdots \times \{1,2\}\times 2\times \{1,2\}\times 2\times \{1,2\}\cdots }$$

we have ${\displaystyle \mu (A)=1/2=\mu (B)}$ but

$${\displaystyle \mu (T^{-n}A\cap B)={\begin{cases}1/2{\text{ if }}n{\text{ is odd}}\\0{\text{ if }}n{\text{ is even.}}\end{cases}}}$$

The definition of ergodicity also makes sense for group actions. The classical theory (for invertible transformations) corresponds to actions of ${\displaystyle \mathbb {Z} }$ or ${\displaystyle \mathbb {R} }$.

For non-abelian groups there might not be invariant measures even on compact metric spaces. However the definition of ergodicity carries over unchanged if one replaces invariant measures by quasi-invariant measures.

Important examples are the action of a semisimple Lie group (or a lattice therein) on its Furstenberg boundary.

A measurable equivalence relation it is said to be ergodic if all saturated subsets are either null or conull.