The Poincaré Recurrence Theorem

Recurrence in phenomena, physical or otherwise, has piqued the curiosity of humans for a long time. Without going into a historical detour on the study of periodic phenomena (of which the author is painfully ignorant), we cut to the chase and introduce one of the foundational results of ergodic theory: the Poincaré Recurrence Theorem. The theorem essentially states that under certain transformations of a space (to be made precise shortly), the system will almost return to its initial state under repeated iterations of the transformation.

Preliminaries

In order to state the theorem, we recall the definition of a probability space and a measure-preserving transformation here.
Definition.[Probability Space] A probability space $(X,\mathcal{B},\mu)$ is a nonempty set $X$, a $\sigma$-algebra $\mathcal{B}$ on $X$, and a measure $\mu$ on $(X,\mathcal{B})$ with $\mu(X)=1$. Assume further that the space $(X,\mathcal{B},\mu)$ is complete, in the sense that all subsets of measurable subsets with measure zero are measurable themselves.
Definition.[Measure-Preserving Transformation] A function $T:X\rightarrow X$ is said to be a measure-preserving transformation if for every $A\in\mathcal{B}$,

1. $T^{-1}(A)\in\mathcal{B}$, i.e., $T$ is measurable, and
2. $\mu(T^{-1}(A))=\mu(A)$, i.e., $T$ is measure-preserving.

Some Examples

We now present some examples of probability spaces and measure-preserving transformations on them.

1. The space $([0,1),\mathcal{M},\mu)$, where $\mathcal{M}$ is the collection of Lebesgue measurable subsets of $[0,1)$ and $\mu$ is the usual Lebesgue measure. A measure-preserving transformation of this space is the Bernoulli map $T:[0,1)\rightarrow [0,1)$ given by \[T(x)=2x\mod 1.\]
2. Another example is the space $([0,1],\mathcal{M},\mu)$ together with the irrational rotation map $T_{\theta}:[0,1]\rightarrow [0,1]$ defined as \[T_{\theta}(x)=(x+\theta)\mod 1\]where $\theta$ is some fixed irrational.
3. The $d$-dimensional torus $\mathbb{T}^d$ is defined as $\mathbb{R}^d/\mathbb{Z}^d$. The space ($\mathbb{T^d},\mathcal{B},\mu)$ where $\mu$ is the standard Lebesgue measure (identifying $\mathbb{T}^d$ with $[0,1)^d$) and $\mathcal{B}$ is the collection of Lebesgue measurable subsets is a probability space. A large class of measure-preserving transformations of this space consists of toral autmorphisms, defined by \[(x_1,\dots,x_d)\mapsto A(x_1,\dots,x_d)^T\]for all $(x_1,\dots,x_d)\in\mathbb{T}$, where $A$ is an invertible $d\times d$ matrix with integer entries.

The Theorem

Without further ado, we now state our main result.
Theorem.[Poincaré Recurrence] Let $(X,\mathcal{B},\mu)$ be a probability space, and let $T:X\rightarrow X$ be a measure-preserving transformation. For each $E\in\mathcal{B}$, there exists $F(\subseteq E)\in\mathcal{B}$ with $\mu(F)=\mu(E)$ such that for every $x\in F$, $T^n(x)\in E$ for infinitely many $n\in\mathbb{N}$.
Essentially, given any measurable subset $E$ of $X$, the set of points of $E$ which eventually escape $E$ (in other words, never return to $E$ after some finite number of iterations of $T$) is ``small'' (measure zero).
The proof of the theorem is fairly straightforward. We present the argument from [EinWar, Theorem 2.11]. Let $B=\{x\in E:T^n(x)\not\in E\text{ for any }n\geq 1\}$. Then \[B=E\cap T^{-1}(X\setminus E)\cap T^{-2}(X\setminus E)\cap,...,\]so $B$ is measurable since $T$ is a measurable function. Now, for any $n\geq 1$, \[T^{-n}(B)=T^{-n}(E)\cap T^{-n-1}(X\setminus E)\cap...,\]so the sets $B,T^{-1}(B),T^{-2}(B),...$ are disjoint and all have measure $\mu(B)$ since $T$ is measure-preserving. As $X$ is a probability space, we must have $\mu(X)<\infty$, and consequently $\mu(B)=0$. Thus, there is a set $F_1\subseteq E$ with $\mu(F_1)=\mu(E)$ such that for every $x\in F_1$, there exists some $n\in\mathbb{N}$ such that $T^n(x)\in E$.
Applying the same argument to $T^2,T^3,...$ yields subsets $F_2,F_3,...$ of $E$ with $\mu(F_n)=\mu(E)$ with every point of $F_n$ returning to $E$ under $T^n$ for $n\geq 1$. Thus, the set \[F=\bigcap_{n\geq 1}F_n\subseteq E\]has $\mu(F)=\mu(E)$, and the claim is proved since for each $x\in F$, $T^n(x)\in E$ for infinitely many $n$.

Some generalisations and applications

One of the most important generalisations of this result is the following. Theorem[Furstenberg's Multiple Recurrence] Let $(X,\mathcal{B},\mu)$ be a probability space, and let $T:X\to X$ be a measure-preserving bijection such that $T$ and $T^{-1}$ are both measurable. Let $E$ be a measurable subset of positive measure. Then for any $k\geq 1$, there exists $n>0$ such that \[E\cap T^{-n}(E)\cap\cdots\cap T^{-(k-1)n}(E)\neq\emptyset.\]Equivalently, there exists $n>0$ and $x\in X$ such that \[x,T^n(x),...,T^{(k-1)n}(x)\in E.\] This theorem was the pivotal step in Furstenberg's proof of Szemerédi's theorem, originally proved by Szemerédi using an intricate combinatorial argument. Furstenberg's proof introduced ergodic-theoretic methods in number theory, and provided much stimulus to research in the area. See [EinWar] for details and more.
A generalisation going in a different direction is the following theorem.
Theorem. [Kac] Let $T$ be an ergodic (a measure-preserving dynamical system $(X,\mathcal{B},\mu,T)$ is said to be ergodic if for every $A\in\mathcal{B}$, $T^{-1}(A)=A$ implies $\mu(A)=0$ or $\mu(A)=\mu(X)$) measure-preserving transformation of the probability space $(X,\mathcal{B},\mu)$. Let $E$ be a measurable set of positive measure. Then \[\int_A n_A\mathrm{d}\mu=1,\]where \[n_A(x):=\inf\{n\geq 1:T^n(x)\in A\}\]is defined for a.e. $x\in A$. Intuitively, this theorem encodes the idea that the larger a set is, the shorter the "return time" should be for points of that set.

References

[EinWar]: Einseidler, M., Ward, T., Ergodic Theory with a view towards Number Theory, Graduate Texts in Mathematics (259), Springer