Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 108 (Topics in Ergodic Theory)

Work with a probability measure-preserving system, so $\mu (X)=1$. PutThe maximal ergodic theorem, also called the maximal ergodic lemma, says that for real $f\in L^1(\mu )$,No ergodic transformation or invertibility assumption is needed for this maximal ergodic lemma.

Here is a proof of the maximal ergodic lemma. Set $M_0=0$ and $M_N=\max (0,S_{1}f,\dots ,S_{N}f)$. These are integrable: $0\le M_N\le {\sum }_{j=0}^{N-1}|f|\circ T^j$. Splitting off the first summand givesOn $E_N=\{M_N>0\}$, this implies $f=M_N-M_{N-1}\circ T\ge M_N-M_N\circ T$. Outside $E_N$ the right side is $-M_N\circ T\le 0$. ConsequentlyIntegrating and using the measure-preserving transformation property gives ${\int }_{E_N}fd\mu \ge 0$. Since $E_N$ increases to $E$, the dominated convergence theorem, with dominating function $|f|$, proves the infinite-supremum assertion of the maximal ergodic lemma.

For example, applying the maximal ergodic lemma to $|f|-\epsilon$ yields the useful maximal boundIndeed on the latter set $G$, the maximal ergodic lemma gives $\epsilon \mu (G)\le {\int }_G|f|d\mu$.

The pointwise ergodic theorem states that for a probability measure-preserving system and $f\in L^1(\mu )$,where $\mathcal{I}$ is the invariant sigma-algebra. The conditional expectation is invariant and has integral $\int fd\mu$. If $T$ is an ergodic transformation, this conditional expectation is the constant $\int fd\mu$. The assertion applies to real or complex integrable functions; it is the Birkhoff ergodic theorem.

For a nonnegative measurable $f$, if $\int fd\mu <\infty$, the pointwise ergodic theorem already proves the required conclusion. For the remaining case, let $f_K=\min (f,K)$, $K=1,2,\dots$. Each $f_K$ is integrable on the probability measure-preserving system. The pointwise ergodic theorem and ergodic transformation property give, on a single set of full measure obtained by a countable intersection,Since $f\ge f_K$, every such $x$ satisfies ${\mathrm{liminf}}_n{A}_{n}f(x)\ge \int f_{K}d\mu$ for all $K$. The monotone convergence theorem gives $\int f_{K}d\mu \uparrow \int fd\mu =+\infty$. Thus the nonnegative ergodic averages with infinite integral tend to $+\infty$, rather than merely having an unbounded subsequence. Combining the two cases,Nonnegativity is what makes truncation a lower bound. The probability normalization is also essential to the displayed value: for a finite measure of total mass $c>0$, the corresponding constant is $c^{-1}\int fd\mu$.

The Szemerédi theorem says that for every integer $k\ge 2$ and every $\delta >0$, there is $N_0(k,\delta )$ such that, for $N\ge N_0$, every $A\subseteq \{1,\dots ,N\}$ with $|A|\ge \delta N$ contains a nonconstant arithmetic progressionEquivalently, every subset of the positive integers with positive upper asymptotic density contains arithmetic progressions of every finite length. Length one is immediate.

The Furstenberg multiple recurrence theorem states that for every probability measure-preserving system, every measurable $B$ with $\mu (B)>0$, and every integer $k\ge 2$, there is $n\ge 1$ withHere $T^{-jn}B$ means a preimage, so invertibility is unnecessary. The Furstenberg multiple recurrence theorem also does not assume an ergodic transformation.

We prove the implication to the finite Szemerédi theorem by constructing the Furstenberg correspondence principle explicitly. Suppose, to the contrary, that for fixed $k$ and $\delta >0$ there are $N_{\ell }\to \infty$ and $A_{\ell }\subseteq \{1,\dots ,N_{\ell }\}$ with $|A_{\ell }|\ge \delta N_{\ell }$, but with no length-$k$ arithmetic progression of positive common difference. In the binary full shift $\Omega =\{0,1{\}}^{\mathbb{Z}}$, encode $A_{\ell }$ as $x_j^{(\ell )}=1_{A_{\ell }}(j)$, taking all coordinates outside $A_{\ell }$ to be zero. Let $S$ be the left shift, $(S\omega {)}_j={\omega }_{j+1}$, and setThe cylinder set $C$ is a clopen set, and ${\nu }_{\ell }(C)=|A_{\ell }|/N_{\ell }\ge \delta$. The binary full shift is a compact metric space, so compactness of probability measures on a compact metric space gives a subsequence of these empirical measures with weak convergence of probability measures to a Borel probability measure $\nu$.

For every continuous $h$ on the full shift,Passing to the weak limit shows that $\nu$ is an invariant measure for the continuous left shift. Thus $(\Omega ,\mathcal{B},\nu ,S)$ is a probability measure-preserving system. Since the indicator function of $C$ is continuous, $\nu (C)\ge \delta$.

Apply the Furstenberg multiple recurrence theorem to $C$. For some $n\ge 1$, the clopen sethas $\nu (D)>0$. Its continuous indicator function gives ${\nu }_{\ell }(D)\to \nu (D)>0$. For large $\ell$, at least one $S^a{x}^{(\ell )}$ in the defining empirical measure therefore belongs to $D$. The choice of left shift impliesso $a,a+n,\dots ,a+(k-1)n$ lies in $A_{\ell }$. This contradicts the assumed absence of arithmetic progressions and proves the finite Szemerédi theorem. Applying the finite Szemerédi theorem on intervals where an infinite set has density bounded below proves the stated upper asymptotic density formulation.

For multiple recurrence for circle rotations, write $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and $R_{\alpha }x=x+\alpha$, with normalized Lebesgue measure $m$, which is the Haar measure of the circle group. Fix a measurable $B\subseteq \mathbb{T}$ with $m(B)>0$ and $k\ge 2$. If $\alpha$ is rational, there is $q\ge 1$ with $R_{\alpha }^q$ the identity, and $n=q$ gives an intersection of measure $m(B)$.

For irrational $\alpha$, the pigeonhole principle applied to $0,\alpha ,\dots ,Q\alpha$ in $Q$ equal arcs gives $1\le n\le Q$ with $\Vert n\alpha {\Vert }_{\mathbb{T}}\le 1/Q$. In particular there are arbitrarily small nonzero returns to zero for the irrational rotation of the circle.

We also need translation continuity in L1 on the circle. Given a measurable $B$, approximate its indicator function in the L1 norm by a continuous $g$ on the circle group. Such approximation follows from regularity of Lebesgue measure, or approximation by finite unions of intervals. Translation invariance and uniform continuity givewhere the approximation error is first made arbitrarily small. Equivalently, $m(B△(B-t))\to 0$.

Choose a return time $n\ge 1$ so small modulo one that for each $1\le j<k$,This is possible because multiplication by each fixed $j$ is continuous on the circle group and translation continuity in L1 on the circle applies to the finitely many translates. The union bound now givesThis proves the Furstenberg multiple recurrence theorem for every circle rotation with its normalized Lebesgue measure, including both rational and irrational angles.

A continuous map $F$ of a compact metric space is uniquely ergodic if it has exactly one invariant measure that is a Borel probability measure. The uniqueness requirement ranges over all invariant Borel probability measures.

For an irrational rotation of the circle, let $R_{\alpha }x=x+\alpha$ on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$, and let $\nu$ be any invariant Borel probability measure. For each integer $r$, set ${\chi }_r(x)=e^{2\pi irx}$. Invariance givesFor $r\ne 0$, irrationality forces $e^{2\pi ir\alpha }\ne 1$, so the corresponding Fourier coefficient is zero. For $r=0$ it is one. These are exactly the Fourier coefficients of normalized Lebesgue measure $m$. By the Stone-Weierstrass theorem, trigonometric polynomials are uniformly dense in the continuous functions on the circle group; hence $\nu$ and $m$ integrate every continuous function equally and are the same Borel probability measure. Since $m$ is invariant, the irrational rotation of the circle is uniquely ergodic, with unique measure $m$.

For the irrational skew shift on the two-dimensional torus, writeThe map is invertible, with $T^{-1}(x,y)=(x-\alpha ,y-x+\alpha )$ modulo one, and it preserves $m_2$ as allowed in the question. We first prove the ergodic transformation property by the invariant-function characterization of ergodicity.

Let $F\in L^2(m_2)$ satisfy $F\circ T=F$. Its expansion in the Fourier basis is $F={\sum }_{r,s}{c}_{r,s}{e}_{r,s}$ in $L^2$. Direct calculation of the Koopman operator givesUniqueness of the Fourier coefficients therefore impliesFor $s\ne 0$, the magnitudes of the Fourier coefficients along all distinct indices $(r+js,s)$, $j\in \mathbb{Z}$, are equal. By the Bessel inequality they are square summable, so every such coefficient must be zero. When $s=0$, the relation becomes $c_{r,0}=e^{2\pi ir\alpha }{c}_{r,0}$, which forces $c_{r,0}=0$ for $r\ne 0$. Only $c_{0,0}$ remains. Thus every invariant $L^2$ function is constant, and

To prove unique ergodicity, we will establish uniform averages for every continuous function directly. No theorem on unique ergodicity of skew products is needed. Induction on $n$, using the old first coordinate in the second coordinate of $T$, yieldsIn particular, for a Fourier basis element,If $s=0$ and $r\ne 0$, this is a geometric series in $n$, anduniformly in $(x,y)$.

For $s\ne 0$, we give the finite Van der Corput inequality for finite scalar sequences and its proof. For $|u_n|\le 1$, extend $u_n$ by zero outside $0\le n<N$. Fix an integer $1\le H\le N$ and set $C_h={\sum }_{n=0}^{N-h-1}{u}_{n+h}\overline{u_n}$. Every original summand appears $H$ times in the identityThe Cauchy-Schwarz inequality, followed by expansion of the squared window sums, givesConsequently the Van der Corput inequality for finite scalar sequences isThe finite prefactor is important; the order of limits will be $N\to \infty$ with $H$ fixed, followed by $H\to \infty$.

For the irrational skew shift character sequence above, differencing cancels the quadratic term:For every fixed $h\ge 1$, $s\alpha h$ is irrational, so another geometric series estimate givesuniformly in $(x,y)$. With fixed $H$, the Van der Corput inequality for finite scalar sequences therefore impliesLetting $H\to \infty$ proves that the averages of every nonconstant Fourier basis element converge uniformly to zero. The constant character has average one. This establishes uniform equidistribution of an irrational skew shift on all trigonometric polynomials.

The Stone-Weierstrass theorem makes these trigonometric polynomials uniformly dense in $C({\mathbb{T}}^2)$. If $p$ approximates a continuous $g$ with $\Vert g-p{\Vert }_{\infty }<\epsilon$, thenTaking $N\to \infty$ and then $\epsilon \downarrow 0$ proves uniform convergence to $\int gd{m}_2$ for every continuous $g$.

Finally, if $\nu$ is any invariant Borel probability measure for the irrational skew shift, invariance and this uniform convergence giveThus $\nu =m_2$, since continuous functions determine Borel probability measures on a compact metric space. We concludeThe uniform-average proof also shows that every starting point has the same limiting continuous-function averages, a stronger conclusion than the almost-everywhere assertion provided by the pointwise ergodic theorem.