Posted by: yglima | November 1, 2009

## ERT3: Other Polynomial Ergodic Averages

Continuing ERT2, we’ll discuss other results related to the convergence of polynomial ergodic averages. Given a probability space ${(X,\mathcal B,\mu)}$, there are two notions of convergence of functions defined on ${(X,\mathcal B,\mu)}$. The first one is norm convergence. Given ${1\le p\le\infty}$, let ${L^p}$ denote the space of functions ${f:X\rightarrow\mathbb C}$ such that

$\displaystyle \left\|f\right\|_p\doteq\left(\int_X|f|^p\right)^{1/p}<+\infty\,.$

We say that the sequence ${(f_n)_{n\in\mathbb N}\subset L^p}$ converges in the ${L^p}$-norm if there exists ${f\in L^p}$ such that

$\displaystyle \lim_{n\rightarrow+\infty}\left\|f-f_n\right\|_p=0.$

The other is pointwise convergence: a sequence ${(f_n)_{n\in\mathbb N}\subset L^p}$ converges pointwise if there are ${f\in L^p}$ and a set ${A\in\mathcal B}$ such that ${\mu(A)=1}$ and

$\displaystyle \lim_{n\rightarrow+\infty}f_n(x)=f(x),\ \forall\,x\in A.$

These notions relate to each other in the following way.

Theorem 1 If ${1\le p\le\infty}$ and ${(f_n)_{n\in\mathbb N}\subset L^p}$ converges in the ${L^p}$-norm to ${f\in L^p}$, then there is a subsequence ${(f_{n_k})_{k\in\mathbb N}}$ which converges pointwise to ${f}$.

Proof: Look at Rudin’s book Real and Complex Analysis. $\Box$

Theorem 1 might infer that norm convergence is stronger than pointwise convergence. This is not the case. In fact, we are interested in proving convergence along the whole sequence, so that norm convergence does not guarantee pointwise convergence. These are two distinct notions and neither of them is stronger than the other: they are just different!

Exercise 1 Consider the sequence of functions ${f_n:[0,1]\rightarrow[0,1]}$ , ${n\in\mathbb N}$, defined in the following way: given ${n=2^k+a}$ , ${0\le a<2^k}$ ,$\displaystyle \begin{array}{rclcl} f_n(x)&=&0 &,&x\not\in\left[\dfrac{a}{2^k}\,,\,\dfrac{a+1}{2^k}\right],\\ &&&&\\ &=&2^{k+1}\cdot x-2a&,&x\in\left[\dfrac{a}{2^k}\,,\,\dfrac{a+1/2}{2^k}\right],\\ &&&&\\ &=&-2^{k+1}\cdot x+2a+2&,&x\in\left[\dfrac{a+1/2}{2^k}\,,\,\dfrac{a+1}{2^k}\right]. \end{array}$

Prove that, for any ${p\ge 1}$, ${(f_n)_{n\in\mathbb N}}$ converges in the ${L^p}$ norm to the zero function but does not converge pointwise.

Given a mps ${(X,\mathcal B,\mu,T)}$ and ${f\in L^p}$, ${p\ge 1}$, consider the sequence of functions

$\displaystyle f_N=\dfrac{1}{N}\sum_{n=1}^{N}T^nf,\ \ N\ge 1.$

Von Neumann’s Theorem proves ${L^2}$-norm convergence. A more general and deeper result is Birkhoff’s Ergodic Theorem, which is one of the main pilars in Ergodic Theory.

Theorem 2 (Birkhoff, 1931) Given any ${f\in L^1}$, the sequence ${(f_N)_{N\ge 1}}$ converges pointwise to a ${T}$-invariant function ${\tilde f\in L^1}$.

Furstenberg’s Theorem on ${L^2}$-norm convergence of polynomial ergodic averages (see ERT2) induces a natural question: does norm/pointwise convergence holds for polynomial ergodic averages of ${L^1}$-functions? This was solved partially (and in a very satisfactory way) by the Fields medalist Jean Bourgain.

Theorem 3 (Bourgain, 1988) Let ${(X,\mathcal B,\mu,T)}$ be a mps and ${p(x)\in\mathbb Z[x]}$ a polynomial such that ${p(n)\ge 0}$, for every ${n\ge 0}$, and ${p(0)=0}$. Then, for every ${f\in L^p}$, ${p>1}$, the limit

$\displaystyle \lim_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^{N}T^{p(n)}f$

converges pointwise to a function ${\tilde f\in L^p}$.

The case ${p=1}$ remained open until 2005, when Daniel Mauldin and Zoltan Buczolich published the paper Divergent Square Averages in which they construct a mps ${(X,\mathcal B,\mu,T)}$ and a function ${f\in L^1}$ for which the quadratic ergodic averages ${N^{-1}\cdot\sum_{n=1}^{N}T^{n^2}f}$ do not converge pointwise.

After Bourgain’s paper, other beautiful results were published using his sharp estimate methods. One of them proves pointwise convergence along the prime numbers.

Theorem 4 (Wierdl, 1988) Let ${(X,\mathcal B,\mu,T)}$ be a mps and $\{p_1 represent the set of prime numbers. For every $f\in L^p$, $p>1$, the limit

$\displaystyle \lim_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^{N}T^{p_n}f$

converges pointwise to a function ${\tilde f\in L^p}$.

At first sight, this theorem is very restrictive: the prime numbers form a very special sequence. The fact is that, in constrast to polynomials, prime numbers have a randomic distribution, so that methods used to prove prime number’s ergodic theorems may be applied to other non-structured situations. In addition, they help to better understand the dichotomy between structure and ramdomness (see ERT2). The work of Ben Green and Terence Tao about the existence of arbitrarily long arithmetic progressions in the prime numbers is an important example of this situation (Terence Tao received his Fields medal in part because of this result).

We end this note with a recent result of Elon Lindenstrauss about the convergence of ergodic averages on amenable groups, which generalizes Birkhoff’s Ergodic Theorem.

Theorem 5 (Lindenstrauss, 2001) Let ${G}$ be a locally compact amenable group acting on a probability space ${(X,\mathcal B,\mu)}$ and ${(F_n)_{n\in\mathbb N}}$ a tempered F${\phi}$lner sequence. For any ${f\in L^1}$, there is a ${G}$-invariant ${\tilde f\in L^1}$ such that

$\displaystyle \lim_{n\rightarrow+\infty}\dfrac{1}{|F_n|}\int_{F_n}f(gx)dm_L(g)=\tilde f(x)$

for ${\mu}$-almost every ${x\in X}$, where ${m_L}$ denotes the (unique) left Haar probability on ${G}$.

Remark 1 The interested reader not used to the above notions is encouraged to see this note of Alexander Gorodnik and Vitaly Bergelson.

Previous posts: ERT0, ERT1, ERT2.