Ergodic Theory: Foundations¶

Job: define measure-preserving systems; prove/record Birkhoff and characterizations of ergodicity; set mixing hierarchy, entropy, and unique ergodicity.

Core¶

Measure-preserving maps¶

A probability-preserving transformation is a quadruple $(X,\mathcal{B},\mu,T)$ with $(X,\mathcal{B},\mu)$ a probability space and $T:X\to X$ measurable, $\mu(T^{-1}A)=\mu(A)$ for all $A\in\mathcal{B}$. Invertible m.p.t.: $T$ bimeasurable bijection with $T^{-1}$ also measure-preserving (automorphism of the measure algebra).

Koopman operator. $U_T:L^2(\mu)\to L^2(\mu)$, $U_T f=f\circ T$ is an isometry (unitary if $T$ invertible). Spectral theory of $U_T$ organizes mixing properties.

Example	Notes
Bernoulli shift	$\sigma$ on $A^\mathbb{Z}$ with product measure
Markov shift	shift-invariant Markov measure
Torus rotation	$x\mapsto x+\alpha$ on $\mathbb{T}^d$
Torus automorphism	$x\mapsto Ax$ on $\mathbb{T}^d$, $A\in\mathrm{SL}_d(\mathbb{Z})$
Hamiltonian flow	Liouville measure on energy surface

Flows. $(T_t)_{t\in\mathbb{R}}$ one-parameter group of automorphisms; continuous-time Birkhoff averages $\frac1T\int_0^T f\circ T_t\,dt$.

Ergodicity¶

$T$ is ergodic if $T^{-1}A=A$ (mod null sets) $\Rightarrow\mu(A)\in\{0,1\}$. Equivalent formulations:

Characterization	Statement
Invariant functions	$f\circ T=f$ a.e.\ $\Rightarrow$ $f$ a.e.\ constant ($f\in L^0$ or $L^p$)
Spectral	$U_T f=f\Rightarrow f\in\mathbb{C}\cdot 1$ on $L^2$
Time average	Birkhoff averages constant a.e.\ for all $f\in L^1$
Density	$\frac1N\sum_{k=0}^{N-1}\mu(A\cap T^{-k}B)\to\mu(A)\mu(B)$ (equiv.\ form)

Ergodic decomposition. Any m.p.t.\ on a nice space is an integral of ergodic ones (Choquet / Rokhlin); reduces many questions to the ergodic case.

Birkhoff ergodic theorem¶

Theorem (Birkhoff). For $f\in L^1(\mu)$, $$ \lim_{N\to\infty}\frac1N\sum_{k=0}^{N-1}f(T^k x)=f^*(x) $$ exists a.e.\ and in $L^1$; $f^*\circ T=f^*$ a.e.; $\int f^*=\int f$. If $T$ ergodic, $f^*=\int f$ a.e.

von Neumann mean ergodic theorem. For $f\in L^2$, averages converge in $L^2$ to the orthogonal projection onto $U_T$-invariants.

Consequences. Visit frequencies to $A$ equal $\mu(A)$ a.e.\ (ergodic case); law of large numbers for stationary processes is Birkhoff in shift coordinates.

Mixing hierarchy¶

Property	Definition (for all $A,B\in\mathcal{B}$)
Ergodic	$\frac1N\sum_{k=0}^{N-1}\mu(A\cap T^{-k}B)\to\mu(A)\mu(B)$
Weak mixing	$\frac1N\sum_{k=0}^{N-1}\bigl\\|\mu(A\cap T^{-k}B)-\mu(A)\mu(B)\bigr\\|\to 0$
Strong mixing	$\mu(A\cap T^{-n}B)\to\mu(A)\mu(B)$ as $n\to\infty$
$k$-fold mixing	joint measures of $k$ sets factorize under large separations

Spectral: weak mixing $\Leftrightarrow$ $U_T$ has no non-constant eigenfunctions. Strong mixing $\Rightarrow$ weak mixing $\Rightarrow$ ergodic; converses false (rigid weakly mixing systems exist; irrational rotation: ergodic not weak mixing).

Correlation decay. For observables $f,g\in L^2$, strong mixing $\Leftrightarrow\langle U_T^n f,g\rangle\to\langle f,1\rangle\langle 1,g\rangle$. Rates (exponential mixing) need hyperbolicity / spectral gaps — link dynamical systems.

Entropy¶

For a finite measurable partition $\alpha$, $$ H_\mu(\alpha)=-\sum_{A\in\alpha}\mu(A)\log\mu(A). $$ Dynamical entropy of $\alpha$: $$ h_\mu(T,\alpha)=\lim_{N\to\infty}\frac1N H_\mu\bigl(\alpha\vee T^{-1}\alpha\vee\cdots\vee T^{-(N-1)}\alpha\bigr). $$ Measure-theoretic entropy $h_\mu(T)=\sup_\alpha h_\mu(T,\alpha)$.

Kolmogorov–Sinai. If $\alpha$ is a generating partition under $T$ (separates points mod null in the join $\bigvee_{n=-\infty}^\infty T^{-n}\alpha$ for invertible $T$), then $h_\mu(T)=h_\mu(T,\alpha)$.

System	Entropy
Bernoulli$(p)$ shift	$-\sum p_i\log p_i$
Irrational rotation	$0$
Hyperbolic toral auto	$\sum_{\\|\lambda_i\\|>1}\log\\|\lambda_i\\|$ (Pesin / Lyapunov)

Shannon–McMillan–Breiman. For ergodic $T$ and generator $\alpha$, $-\frac1N\log\mu(\alpha_N(x))\to h_\mu(T)$ a.e.\ ($\alpha_N=$ atom of the dynamical join).

Ornstein theory (pointer). Bernoulli shifts classified up to isomorphism by entropy; Kakutani equivalence, finitary isomorphisms — deeper structure theory.

Unique ergodicity¶

Topological dynamical system $(X,T)$, $X$ compact metric: uniquely ergodic if there is a unique $T$-invariant Borel probability $\mu$. Then for every continuous $f$, $$ \frac1N\sum_{k=0}^{N-1}f(T^k x)\to\int f\,d\mu $$ uniformly in $x$ (equivalent characterization).

Example	Status
Irrational rotation on $\mathbb{T}$	uniquely ergodic (Lebesgue)
Rational rotation	not ergodic; many measures
Sturmian subshifts	uniquely ergodic min.\ systems
Full shift on $2$ symbols	many Bernoulli / Markov measures

Weyl criterion / equidistribution on groups uses unique ergodicity of translations or unipotent flows (Ratner theory — advanced).

Poincaré recurrence (one line)¶

If $\mu(A)>0$, then a.e.\ $x\in A$ returns to $A$ infinitely often under invertible finite-measure-preserving $T$. Infinite measure theory needs conservative / dissipative decomposition (Hopff).

See it¶

Gauss map. $T:(0,1]\to(0,1]$, $Tx=\{1/x\}$ preserves the Gauss measure $\mu(dx)=\frac{1}{\log 2}\frac{dx}{1+x}$, is ergodic, and encodes continued fraction statistics: Birkhoff $\Rightarrow$ almost every $x$ has asymptotic frequency of partial quotients given by the Lévy–Khinchin integrals. Bridge between ergodic theory and Diophantine approximation.

Hopff argument for ergodicity of irrational rotation. If $f\circ T_\alpha=f$ continuous, then $f$ constant on the dense orbit of $0$ hence constant; approximate $L^2$ invariants by continuous functions via Fejér / density of continuous functions — spectral gap free argument.

Check¶

Prove that strong mixing implies ergodicity.

Average the mixing relation: $\frac1N\sum_{n=0}^{N-1}\mu(A\cap T^{-n}B)\to\mu(A)\mu(B)$. If $A$ invariant, left side $=\mu(A\cap B)$ for all $N$, so $\mu(A\cap B)=\mu(A)\mu(B)$ all $B$. Take $B=A$: $\mu(A)=\mu(A)^2\Rightarrow\mu(A)\in\{0,1\}$.

Why is $h_\mu(T)=0$ for any rotation on $\mathbb{T}$?

Finite partitions under irrational rotation produce dynamical joins whose element measures stay bounded below on a scale that forces $H(\alpha_N)=o(N)$ (equivalently: isometric systems / discrete spectrum have zero entropy). For rational rotations, periodic decomposition also yields zero entropy. KS: Kronecker systems are entropy zero.

Pitfalls¶

a.e.\ vs uniform: Birkhoff is a.e.; unique ergodicity upgrades continuous observables to uniform.
Endomorphisms vs automorphisms: non-invertible maps (expanding maps, Gauss) need one-sided joins for generators.
Entropy depends on $\mu$: same $T$ topological can carry measures of different entropy; variational principle $h_{\mathrm{top}}=\sup_\mu h_\mu$.
Spectral isomorphism vs metric isomorphism: same spectral measure of $U_T$ does not imply isomorphic systems (beyond Gaussian / discrete spectrum cases).

Refs¶

Einsiedler–Ward, Ergodic Theory
Walters, An Introduction to Ergodic Theory
Cornfeld–Fomin–Sinai, Ergodic Theory
Rudolph, Fundamentals of Measurable Dynamics

Next¶

Ergodic theory index · Homogeneous & Ratner · Oseledets & Pesin · Furstenberg & Joinings · Ornstein & Isomorphism · Measure theory · MT / ergodic & recurrence — the Birkhoff / the maximal / the recurrence · Dynamical systems · Probability · Information theory · Fourier analysis · Stochastic processes · Hyperbolic dynamics — Anosov / SRB / Pesin as the smooth-dynamics realization of ergodic theory

Property	Definition (for all \(A,B\in\mathcal{B}\))
Ergodic	\(\frac1N\sum_{k=0}^{N-1}\mu(A\cap T^{-k}B)\to\mu(A)\mu(B)\)
Weak mixing	\(\frac1N\sum_{k=0}^{N-1}\bigl\\|\mu(A\cap T^{-k}B)-\mu(A)\mu(B)\bigr\\|\to 0\)
Strong mixing	\(\mu(A\cap T^{-n}B)\to\mu(A)\mu(B)\) as \(n\to\infty\)
\(k\)-fold mixing	joint measures of \(k\) sets factorize under large separations