Practice: Ergodic theory

Ergodic theory

Warm-upHuman

Which maps of $[0, 1)$ preserve Lebesgue measure $\lambda$ (that is, $\lambda(T^{-1}A) = \lambda(A)$ for all Borel $A$ )? Select all that apply.

Choose all that apply

T(x) = 2x \bmod 1

T(x) = x^2

T(x) = x + \sqrt 2 \bmod 1

T(x) = 1 - x

(for

x \ne 0

),

T(0) = 0

Ergodic theory

Warm-upHuman

Iterate the Gauss map $G(x) = 1/x - \lfloor 1/x \rfloor$ on $x_0 = \frac{7}{30}$ until you reach $0$ , recording the digits $a_{n+1} = \lfloor 1/x_n \rfloor$ . What is the sum of the digits?

Ergodic theory

CoreHuman

The translation $T(x) = x + 1$ on $\R$ preserves Lebesgue measure, yet no point of $[0, 1)$ ever returns to $[0, 1)$ . Which hypothesis of the Poincaré recurrence theorem fails?

Choose one

The measure of the whole space is finite.

T

is measure-preserving.

T

is invertible.

[0, 1)

has positive measure.

Ergodic theory

CoreHuman

Which of these measure-preserving maps of $([0, 1), \lambda)$ is ergodic but not mixing?

Choose one

x \mapsto x + \frac{\sqrt 5 - 1}{2} \bmod 1

x \mapsto 2x \bmod 1

x \mapsto x + \frac25 \bmod 1

x \mapsto x

Ergodic theory

CoreHuman

The doubling map $D$ is ergodic for Lebesgue measure. Let $f = \mathbf 1_{[0, 1/2)}$ . Which statement is correct?

Choose one

For Lebesgue-almost every

x

,

\frac1N\sum_{n<N} f(D^n x) \to \frac12

, but there are points (for example

x = 0

) where the average is

1

.For every

x

, the average tends to

\frac12

, because

D

is ergodic.The average converges to

\frac12

only at rational

x

.The averages need not converge anywhere, since

D

is not invertible.

Ergodic theory

Warm-upFormal

Which Lean statement formalises "rotation of the circle $\R/\Z$ by $\alpha$ is ergodic exactly when $\alpha$ is irrational"? (AddCircle (1 : ℝ) is $\R/\Z$ with Haar measure, and Ergodic f includes measure preservation.)

Choose one

theorem t (α : ℝ) : Ergodic (fun y : AddCircle (1 : ℝ) => (α : AddCircle (1 : ℝ)) + y) ↔ Irrational αtheorem t (α : ℝ) : Ergodic (fun y : AddCircle (1 : ℝ) => (α : AddCircle (1 : ℝ)) + y)theorem t (α : ℝ) (h : Irrational α) : Ergodic (fun y : AddCircle (1 : ℝ) => 2 • y + α)theorem t (α : ℝ) : Ergodic (fun y : AddCircle (1 : ℝ) => (α : AddCircle (1 : ℝ)) + y) ↔ α ≠ 0

Ergodic theory

StretchHuman

In the proof of von Neumann's mean ergodic theorem, why do the averages $\frac1N\sum_{n<N} U^n h$ tend to $0$ for $h = Ug - g$ ?

Choose one

The sum telescopes to

\frac1N(U^N g - g)

, whose norm is at most

\frac{2\|g\|}{N}

because

\|U\| \le 1

.Because

U^n h \to 0

for every

h

.Because

Ug - g

is orthogonal to itself.By Birkhoff's pointwise ergodic theorem.

Ergodic theory

CoreComputational

Write cf_digits(p, q) returning the list of continued fraction digits $[a_1, a_2, \ldots]$ of the rational $p/q \in (0, 1)$ (with $0 < p < q$ ), by iterating the Gauss map with exact integer arithmetic (Euclid's algorithm). Then digit_frequency(digits, k) should return the fraction of the digits equal to k.

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