Ergodic theory · The Lumenary

MotivationH

Boltzmann's hypothesis

A litre of gas contains about $10^{22}$ molecules. Nobody can follow them, yet pressure and temperature are perfectly reproducible. In the 1870s Ludwig Boltzmann proposed an explanation: a measuring instrument records a time average of some quantity along the system's trajectory, and if the trajectory wanders through its whole energy surface, spending time in each region in proportion to its volume, then the time average equals the space average over the energy surface — which can be computed. He called such systems ergodic. Whether real gases are ergodic is still not settled, but the question created a branch of mathematics.

Ergodic theory studies transformations $T$ of a space $X$ that preserve a probability measure $\mu$ , and asks about the statistics of orbits $x, Tx, T^2x, \ldots$ . Its first theorems (Poincaré 1890; Birkhoff and von Neumann 1931–32) say that almost every orbit returns arbitrarily close to where it started, and that for ergodic systems time averages equal space averages almost everywhere. Its examples range from rotations of a circle to the continued fraction expansion of a random real number, whose digits are $1$ about $41.5\%$ of the time — a fact Gauss knew in 1812 and that the ergodic theorem explains.

IntuitionH

Time averages, space averages and indecomposability

Take $f : X \to \R$ and an orbit. The time average is $\lim_{N \to \infty} \frac1N \sum_{n=0}^{N-1} f(T^n x)$ ; the space average is $\int f \, d\mu$ . Two things can make them differ.

The space can decompose into invariant pieces. A rotation of the circle by $\frac25$ of a turn moves each point around a 5-point orbit; the time average only sees those 5 points. If $X = A \cup B$ with both pieces invariant and of positive measure, orbits starting in $A$ never visit $B$ . Ergodicity rules this out: every invariant set has measure $0$ or $1$ .
Even in an ergodic system, exceptional orbits exist: the fixed point $0$ of the doubling map $x \mapsto 2x \bmod 1$ spends all its time in $[0, \frac12)$ . The ergodic theorem promises equality only for almost every starting point.

A finer property is mixing: $\mu(T^{-n}A \cap B) \to \mu(A)\mu(B)$ , which says the event "in $A$ at time $n$ " becomes independent of "in $B$ at time $0$ " — the system forgets its past. Stirring two liquids mixes them; rotating them does not. An irrational rotation is ergodic (every orbit is dense and equidistributed) but never mixing (it is rigid: a set moves around without spreading). The doubling map is mixing: it stretches every interval by a factor $2$ each step and wraps it around.

DefinitionsH

Measure-preserving systems

Definition— measure-preserving system

A measure-preserving system $(X, \mathcal B, \mu, T)$ consists of a probability space and a measurable map $T : X \to X$ with $\mu(T^{-1}A) = \mu(A)$ for every $A \in \mathcal B$ . Equivalently $\int f \circ T \, d\mu = \int f \, d\mu$ for all $f \in L^1(\mu)$ . The Koopman operator $U_T f = f \circ T$ is then an isometry of each $L^p(\mu)$ .

Definition— ergodic, mixing

$T$ is ergodic if every measurable $A$ with $T^{-1}A = A$ (up to a null set) has $\mu(A) \in \{0, 1\}$ . It is (strongly) mixing if $\mu(T^{-n}A \cap B) \to \mu(A)\mu(B)$ as $n \to \infty$ for all measurable $A, B$ .

Definition— Birkhoff sums and averages

For $f : X \to \R$ , the Birkhoff sums are $S_N f(x) = \sum_{n=0}^{N-1} f(T^n x)$ and the Birkhoff averages $A_N f = \frac1N S_N f$ .

Definition— the standard examples

On $\mathbb T = [0, 1)$ with Lebesgue measure $\lambda$ : the rotation $R_\alpha(x) = x + \alpha \bmod 1$ and the doubling map $D(x) = 2x \bmod 1$ . On $\{0, 1\}^\N$ with the product of $(\frac12, \frac12)$ measures: the Bernoulli shift $\sigma(x_0, x_1, \ldots) = (x_1, x_2, \ldots)$ . On $[0, 1)$ : the Gauss map $G(x) = \{1/x\} = 1/x - \lfloor 1/x \rfloor$ for $x \ne 0$ , $G(0) = 0$ , with the Gauss measure $d\gamma = \frac{1}{\log 2} \frac{dx}{1 + x}$ .

The Gauss map generates continued fractions: if $x = \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cdots}}$ with digits $a_i \ge 1$ , then $a_1 = \lfloor 1/x \rfloor$ and $G(x) = [0; a_2, a_3, \ldots]$ , so $a_{n+1}(x) = a_1(G^n x)$ — the digits are read off along the orbit, just as the doubling map reads off binary digits: $D(0.b_1 b_2 b_3\ldots) = 0.b_2 b_3 \ldots$ . Through the binary expansion, $D$ on $(\mathbb T, \lambda)$ and $\sigma$ on $\{0, 1\}^\N$ are isomorphic measure-preserving systems.

Worked examplesH

Invariant measures in practice

Example— Rotations and the doubling map preserve Lebesgue measure

It suffices to check intervals, since they generate the Borel σ-algebra and form a π-system (two measures agreeing on a π-system that generates the σ-algebra agree everywhere). $R_\alpha^{-1}[a, b)$ is an interval (or two pieces wrapping around $1$ ) of the same length. $D^{-1}[a, b) = [\frac a2, \frac b2) \cup [\frac{a+1}2, \frac{b+1}2)$ : two intervals of half the length. Note that the doubling map is two-to-one, and invariance is about preimages: $D[0, \frac12) = [0, 1)$ has twice the measure of $[0, \frac12)$ .

Example— The Gauss measure is invariant

$G(x) < t$ exactly when $\frac1x - n < t$ for $n = \lfloor 1/x \rfloor$ , so $G^{-1}[0, t) = \bigcup_{n \ge 1} \big(\frac{1}{n + t}, \frac1n\big]$ (up to the point $0$ ). Since $\gamma([a, b]) = \log_2 \frac{1 + b}{1 + a}$ ,

\gamma\big(G^{-1}[0, t)\big) = \sum_{n=1}^\infty \log_2 \frac{1 + \frac1n}{1 + \frac{1}{n + t}} = \lim_{M \to \infty} \log_2 \prod_{n=1}^{M} \frac{n + 1}{n} \cdot \frac{n + t}{n + 1 + t}.

Both products telescope: $\prod_{n \le M} \frac{n+1}{n} = M + 1$ and $\prod_{n \le M} \frac{n + t}{n + 1 + t} = \frac{1 + t}{M + 1 + t}$ . The product is $\frac{(M+1)(1+t)}{M+1+t} \to 1 + t$ , so $\gamma(G^{-1}[0, t)) = \log_2(1 + t) = \gamma([0, t))$ . As the intervals $[0, t)$ form a generating π-system, $G$ preserves $\gamma$ . (Lebesgue measure is not invariant: $\lambda(G^{-1}[0, \frac12)) = \sum_n \big(\frac1n - \frac{1}{n + 1/2}\big) = 2 - 2\log 2 \approx 0.614 \ne \frac12$ .)

Example— Recurrence without finiteness fails

The translation $x \mapsto x + 1$ preserves Lebesgue measure on $\R$ , and no point of $[0, 1)$ ever returns. The finiteness of $\mu$ is essential in Poincaré's theorem below.

ProofH

Recurrence and ergodicity

Theorem— Poincaré recurrence theorem (1890)

Let $(X, \mathcal B, \mu, T)$ be measure-preserving with $\mu(X) < \infty$ , and $A \in \mathcal B$ . Then almost every point of $A$ returns to $A$ infinitely often: $\mu\big(\{x \in A : T^n x \in A \text{ for only finitely many } n\}\big) = 0$ .

Proof

Let $B = \{x \in A : T^n x \notin A \text{ for all } n \ge 1\}$ , the points of $A$ that never return; $B = A \setminus \bigcup_{n \ge 1} T^{-n}A$ is measurable. The sets $B, T^{-1}B, T^{-2}B, \ldots$ are pairwise disjoint: if $x \in T^{-m}B \cap T^{-n}B$ with $m < n$ , then $y = T^m x \in B$ and $T^{n-m} y = T^n x \in B \subseteq A$ , so $y$ returns to $A$ , contradicting $y \in B$ . They all have measure $\mu(B)$ , so $\sum_{n \ge 0} \mu(B) = \mu\big(\bigcup_n T^{-n}B\big) \le \mu(X) < \infty$ , forcing $\mu(B) = 0$ .

If $x \in A$ returns only finitely often, let $k \ge 0$ be the last time with $T^k x \in A$ ; then $T^k x \in B$ , so $x \in T^{-k}B$ . Hence the set of such $x$ lies in $\bigcup_{k \ge 0} T^{-k}B$ , a countable union of null sets.

∎

Zermelo (1896) objected that recurrence contradicts the irreversible approach to equilibrium of Boltzmann's gas. The resolution is quantitative: recurrence times for macroscopic systems are astronomically longer than the age of the universe.

Theorem— Proposition (characterisations of ergodicity)

For a measure-preserving $T$ the following are equivalent: (i) $T$ is ergodic; (ii) every measurable $f$ with $f \circ T = f$ a.e. is a.e. constant; (iii) the same for $f \in L^2(\mu)$ .

Proof

(ii) ⇒ (iii) is trivial, and (iii) ⇒ (i) by applying (iii) to $f = \mathbf 1_A$ for invariant $A$ : a constant indicator is a.e. $0$ or a.e. $1$ . (i) ⇒ (ii): for each $t$ , the set $\{f > t\}$ is invariant up to a null set, so it has measure $0$ or $1$ . The function $t \mapsto \mu(\{f > t\})$ is non-increasing, so it jumps from $1$ to $0$ at some $c \in [-\infty, \infty]$ , and $f = c$ a.e.; $c$ is finite because $f$ is real-valued.

∎

Theorem— Theorem (rotations)

$R_\alpha$ is ergodic if and only if $\alpha$ is irrational. It is never mixing.

Proof

Let $f \in L^2(\mathbb T)$ with $f \circ R_\alpha = f$ . Comparing Fourier coefficients, $\widehat{f \circ R_\alpha}(n) = e^{2\pi i n\alpha} \hat f(n) = \hat f(n)$ . If $\alpha$ is irrational, $e^{2\pi i n\alpha} \ne 1$ for $n \ne 0$ , so $\hat f(n) = 0$ for $n \ne 0$ and $f$ is a.e. constant: $R_\alpha$ is ergodic by the proposition. If $\alpha = p/q$ , the non-constant function $e^{2\pi i q x}$ is invariant.

For mixing, take $A = [0, \frac12)$ . Then $R_\alpha^{-n}A$ is a half-circle shifted by $n\alpha$ , and $\lambda(A \cap R_\alpha^{-n}A) = \frac12 - \|n\alpha\|$ , where $\|t\|$ is the distance from $t$ to the nearest integer. By Dirichlet's approximation theorem there are infinitely many $n$ with $\|n\alpha\| < 1/n$ , so along those $n$ the overlap tends to $\frac12 \ne \frac14 = \lambda(A)^2$ . (For rational $\alpha$ the overlap is periodic in $n$ .)

∎

Theorem— Theorem (the doubling map is mixing)

The doubling map is mixing, hence ergodic. More generally, mixing implies ergodicity.

Proof

Let $A = [a, b)$ be any interval and $B$ a dyadic interval $[j 2^{-\ell}, (j+1) 2^{-\ell})$ . For $n \ge \ell$ , $D^{-n}A = \bigcup_{m=0}^{2^n - 1} \big[\frac{m + a}{2^n}, \frac{m + b}{2^n}\big)$ consists of one interval of length $2^{-n}\lambda(A)$ inside each dyadic interval of level $n$ . $B$ is the union of $2^{n-\ell} = 2^n \lambda(B)$ of these, so $\lambda(D^{-n}A \cap B) = \lambda(A)\lambda(B)$ exactly for all $n \ge \ell$ . Finite disjoint unions of dyadic intervals approximate every Borel set in measure, and a standard $\varepsilon/3$ argument extends $\lambda(D^{-n}A \cap B) \to \lambda(A)\lambda(B)$ to all Borel $A, B$ .

If $T$ is mixing and $T^{-1}A = A$ , then $\mu(A) = \mu(T^{-n}A \cap A) \to \mu(A)^2$ , so $\mu(A) \in \{0, 1\}$ .

∎

The same cylinder computation shows that every Bernoulli shift is mixing: for cylinder sets depending on coordinates in a finite window, $\sigma^{-n}A$ and $B$ depend on disjoint coordinates once $n$ is large, so they are independent. The Gauss map is mixing too (indeed exponentially mixing), but the proof needs distortion estimates for the branches of $G$ (Einsiedler–Ward, Ch. 3).

ProofH

The ergodic theorems

Theorem— Mean ergodic theorem (von Neumann, 1932)

Let $U$ be a linear operator on a Hilbert space $H$ with $\|U\| \le 1$ , and $P$ the orthogonal projection onto $\mathrm{Inv} = \{h : Uh = h\}$ . Then $\frac1N \sum_{n=0}^{N-1} U^n h \to Ph$ in norm for every $h \in H$ . For a measure-preserving $T$ and $U = U_T$ on $L^2(\mu)$ : $A_N f \to Pf$ in $L^2$ , and if $T$ is ergodic, $Pf = \int f \, d\mu$ .

Proof

Step 1: $Uh = h$ iff $U^*h = h$ . If $Uh = h$ , then $\|U^*h - h\|^2 = \|U^*h\|^2 - 2\operatorname{Re}\langle U^* h, h \rangle + \|h\|^2 = \|U^*h\|^2 - 2\operatorname{Re}\langle h, Uh \rangle + \|h\|^2 = \|U^*h\|^2 - \|h\|^2 \le 0$ , because $\|U^*\| = \|U\| \le 1$ . The converse is symmetric.

Step 2: $H = \mathrm{Inv} \oplus \overline{C}$ , where $C = \{Ug - g : g \in H\}$ (the coboundaries). A vector $h$ is orthogonal to $C$ iff $\langle U^*h - h, g \rangle = \langle h, Ug - g \rangle = 0$ for all $g$ , iff $U^*h = h$ , iff $h \in \mathrm{Inv}$ by Step 1. So $C^\perp = \mathrm{Inv}$ , and $\overline C = \mathrm{Inv}^\perp$ .

Step 3: averages. On $\mathrm{Inv}$ every average equals $h$ . On $C$ the sum telescopes: $\frac1N \sum_{n < N} U^n (Ug - g) = \frac1N (U^N g - g)$ , of norm at most $\frac2N \|g\|$ . The averaging operators $\mathcal M_N = \frac1N\sum_{n < N} U^n$ have norm $\le 1$ , so $\mathcal M_N k \to 0$ for every $k \in \overline C$ as well: given $\varepsilon$ , choose $c \in C$ with $\|k - c\| < \varepsilon$ ; then $\|\mathcal M_N k\| \le \varepsilon + \|\mathcal M_N c\| < 2\varepsilon$ for large $N$ . Writing $h = Ph + (h - Ph)$ gives $\mathcal M_N h \to Ph$ .

For $U_T$ , the invariant vectors are the invariant functions; if $T$ is ergodic they are the constants, and the projection onto constants is $f \mapsto \int f \, d\mu$ .

∎

Theorem— Maximal ergodic theorem (Hopf; Garsia's proof, 1965)

Let $T$ preserve $\mu$ , $f \in L^1(\mu)$ , and $M_N = \max(0, S_1 f, \ldots, S_N f)$ . Then $\int_{\{M_N > 0\}} f \, d\mu \ge 0$ .

Proof

For $1 \le n \le N$ , $S_n f = f + (S_{n-1} f) \circ T \le f + M_N \circ T$ (with $S_0 f = 0$ ). On $E = \{M_N > 0\}$ we have $M_N = \max_{1 \le n \le N} S_n f$ , so $M_N \le f + M_N \circ T$ there. Integrating over $E$ , and using $M_N = 0$ off $E$ , $M_N \circ T \ge 0$ , and invariance of $\mu$ :

\int_E f \ge \int_E M_N - \int_E M_N \circ T \ge \int_X M_N - \int_X M_N \circ T = 0.

∎

Theorem— Pointwise ergodic theorem (Birkhoff, 1931)

Let $T$ preserve the probability measure $\mu$ and $f \in L^1(\mu)$ . Then $A_N f(x)$ converges for $\mu$ -almost every $x$ , to an invariant integrable function $f^*$ with $\int f^* = \int f$ . If $T$ is ergodic, $f^* = \int f \, d\mu$ a.e.:

\lim_{N \to \infty} \frac1N \sum_{n=0}^{N-1} f(T^n x) = \int_X f \, d\mu \quad \text{for } \mu\text{-a.e. } x.

Proof

We prove the ergodic case; the general case runs the same argument on invariant sets. Let $\bar f = \limsup_N A_N f$ . Since $A_N f(Tx) = \frac{N+1}{N} A_{N+1} f(x) - \frac{f(x)}{N}$ , we have $\bar f \circ T = \bar f$ , so for each real $a$ the set $E_a = \{\bar f > a\}$ is invariant and $\mu(E_a) \in \{0, 1\}$ .

Suppose $a > \int f$ and $\mu(E_a) = 1$ . Put $g = f - a$ . At every $x \in E_a$ some $A_n f(x) > a$ , i.e. $S_n g(x) > 0$ , so $x \in \{M_N(g) > 0\}$ for large $N$ . These sets increase with $N$ to a set of full measure, so by the maximal ergodic theorem and dominated convergence, $0 \le \int_{\{M_N(g) > 0\}} g \to \int g = \int f - a < 0$ , a contradiction. Hence $\mu(E_a) = 0$ for every rational $a > \int f$ , i.e. $\limsup A_N f \le \int f$ a.e. Applying this to $-f$ gives $\liminf A_N f \ge \int f$ a.e., so $A_N f \to \int f$ almost everywhere.

∎

For the Bernoulli shift and $f(x) = x_0$ , Birkhoff's theorem is the strong law of large numbers for fair coin tosses; via the doubling map it is Borel's theorem (1909) that almost every number is normal in base $2$ . The theorem says nothing about any particular point: $0$ (a fixed point of $D$ ) and $\frac13 = 0.0101\ldots_2$ are exceptional for suitable $f$ , and nobody knows whether $\sqrt 2$ or $\pi$ is normal in base $2$ .

ExplanationH

Equidistribution and continued fractions

Weyl's theorem. For irrational $\alpha$ , the sequence $n\alpha \bmod 1$ is equidistributed: $\frac1N \#\{n < N : n\alpha \bmod 1 \in [a, b)\} \to b - a$ for every interval — for every starting point, not just almost every one. Proof: for $f(x) = e^{2\pi i k x}$ with $k \ne 0$ , $\frac1N \sum_{n < N} e^{2\pi i k (x + n\alpha)}$ is a geometric series bounded by $\frac{2}{N |1 - e^{2\pi i k\alpha}|} \to 0$ ; by linearity averages of trigonometric polynomials converge to their integrals at every point; by Weierstrass approximation (Fejér) so do averages of continuous functions, and squeezing an interval's indicator between continuous functions finishes the proof. A system with this "every point" behaviour is uniquely ergodic: $\lambda$ is its only invariant probability measure.

Continued fraction digits. The Gauss map is ergodic for $\gamma$ , which is equivalent to Lebesgue measure (its density lies between $\frac{1}{2\log 2}$ and $\frac{1}{\log 2}$ ), so " $\gamma$ -almost every" means "Lebesgue-almost every". Apply Birkhoff to $f = \mathbf 1_{\{a_1 = k\}} = \mathbf 1_{(1/(k+1), 1/k]}$ : for almost every $x \in [0, 1)$ the digit $k$ appears with frequency

\gamma\Big(\Big(\tfrac{1}{k+1}, \tfrac1k\Big]\Big) = \log_2 \frac{1 + 1/k}{1 + 1/(k+1)} = \log_2\Big(1 + \frac{1}{k(k+2)}\Big),

the Gauss–Kuzmin law: $41.5\%$ ones, $17.0\%$ twos, $9.3\%$ threes. Applied to $f = \log a_1$ (integrable for $\gamma$ ), it gives Khinchin's theorem (1935) that the geometric mean $(a_1 \cdots a_n)^{1/n}$ tends almost everywhere to a constant $K \approx 2.685$ . Quadratic irrationals, such as the golden ratio $[0; 1, 1, 1, \ldots]$ , are exceptional: their digits are eventually periodic.

Entropy. Are the Bernoulli shifts with probabilities $(\frac12, \frac12)$ and $(\frac13, \frac13, \frac13)$ isomorphic? Both are mixing; von Neumann's spectral invariants cannot tell them apart. Kolmogorov (1958) and Sinai (1959) introduced the entropy $h(T)$ , an isomorphism invariant measuring the exponential rate at which a system generates information: $h = \log 2$ and $\log 3$ respectively, so they are not isomorphic; rotations have entropy $0$ and the doubling map entropy $\log 2$ . Ornstein (1970) proved the converse for Bernoulli shifts: equal entropy implies isomorphism.

Formal mathematicsF

Ergodic theory in Lean and Mathlib

The pigeonhole form of recurrence, proved from scratch: the sets $T^{-i}A$ , $i < N$ , all have measure $\mu(A)$ , so if $N\mu(A) > \mu(X)$ two of them meet:

recurrence_pigeonholeLean 4Proved here from basic factsFormally verified

/-- **Recurrence by pigeonhole.** Let `f` preserve a finite measure `μ` and let `μ(s) > 0`. Then
some point of `s` returns to `s`. Proof: choose `N` with `N · μ(s) > μ(X)`. The sets
`s, f⁻¹s, …, f^{-(N-1)}s` all have measure `μ(s)`, so their measures add up to more than
`μ(X)`, and two of them meet (Mathlib's measure-theoretic pigeonhole principle
`exists_nonempty_inter_of_measure_univ_lt_sum_measure`). If `y ∈ f^{-i}s ∩ f^{-j}s` with
`i < j`, then `x = fⁱ(y) ∈ s` and `f^{j-i}(x) = fʲ(y) ∈ s`. -/
theorem recurrence_pigeonhole {α : Type*} [MeasurableSpace α] {μ : Measure α}
    [IsFiniteMeasure μ] {f : α → α} (hf : MeasurePreserving f μ μ) {s : Set α}
    (hs : MeasurableSet s) (hpos : μ s ≠ 0) : ∃ x ∈ s, ∃ k, 0 < k ∧ f^[k] x ∈ s := by
  obtain ⟨N, hN⟩ := ENNReal.exists_nat_mul_gt hpos (measure_ne_top μ univ)
  have hmeas : ∀ i ∈ Finset.range N, NullMeasurableSet (f^[i] ⁻¹' s) μ :=
    fun i _ => ((hf.iterate i).measurable hs).nullMeasurableSet
  have hsum : ∑ i ∈ Finset.range N, μ (f^[i] ⁻¹' s) = N * μ s := by
    rw [Finset.sum_congr rfl fun i _ => (hf.iterate i).measure_preimage hs.nullMeasurableSet]
    simp
  obtain ⟨i, -, j, -, hij, y, hyi, hyj⟩ :=
    exists_nonempty_inter_of_measure_univ_lt_sum_measure μ hmeas (hsum ▸ hN)
  rcases lt_or_gt_of_ne hij with h | h
  · refine ⟨f^[i] y, hyi, j - i, Nat.sub_pos_of_lt h, ?_⟩
    rw [← Function.iterate_add_apply, Nat.sub_add_cancel h.le]
    exact hyj
  · refine ⟨f^[j] y, hyj, i - j, Nat.sub_pos_of_lt h, ?_⟩
    rw [← Function.iterate_add_apply, Nat.sub_add_cancel h.le]
    exact hyi