## Convergence and Recurrence of Z actions

Ergodic Ramsey Theory started with Furstenberg’s proof of Szemeredi’s theorem in arithmetic progressions in 1977. Through a correspondence principle, Furstenberg realized that Szemeredi’s theorem follows from a dynamical statement:

$\displaystyle \liminf_{N\rightarrow\infty}\frac1N\sum_{n=1}^N\mu(A\cap T^nA\cap T^{2n}A\cap\dots\cap T^{kn}A)>0 \ \ \ \ \ (1)$

for every invertible, ergodic measure preserving transformation ${T}$ of a probability space ${(X,\mu)}$ and every set ${A\subset X}$ with positive measure.
Equation (1) can be rephrased as

$\displaystyle \liminf_{N\rightarrow\infty}\frac1N\sum_{n=1}^N\int_X\prod_{i=0}^k1_A(T^{in}x)d\mu>0\ \ \ \ \ (2)$

and it is natural to question whether the limit exists.
Since Furstenberg’s seminal paper, many versions of results on recurrence and convergence of non-conventional ergodic averages were established by several people. In this post I will list the most important developments in this area, including some recent results. I will restrict to ${{\mathbb Z}}$-actions and averaging over the classical F\o lner sequence ${F_n=\{1,2\dots,n\}}$. There are many developments for other group actions which have interesting combinatorial applications, but that’s another story.

— 1. First example: Poincare’s recurrence and von Neumann’s mean ergodic theorem —

In this section I will use the example of Poincare’s recurrence theorem and the mean ergodic theorem to illustrate the relation between convergence and recurrence results. Essentially neither convergence nor recurrence imply the other result, although convergence plus some information about the limit (i.e. if it is an orthogonal projection) is enough to give a proof of the recurrence result. Despite this fact, it is usually (at least historically) harder to establish convergence results than recurrence results. One reason for this is the fact that if a convergence result gives enough information about the limit to deduce the corresponding recurrence result, then one usually obtains additional information about the recurrence result (for instance, the set of return times will be “large” in certain senses).
One of the many versions of Poincare recurrence theorem can be stated as

$\displaystyle \limsup_{N\rightarrow\infty}\frac1N\sum_{n=1}^N\mu(A\cap T^{-n}A)=\limsup_{N\rightarrow\infty}\frac1N\sum_{n=1}^N\int_X1_A.(U^n1_A)d\mu>0$

whenever ${A}$ is a set of positive measure in a measure preserving system ${(X,\mu,T)}$.
A formulation that seems more general at a first look (but is ultimately equivalent) is the following:

$\displaystyle \limsup_{N\rightarrow\infty}\frac1N\sum_{n=1}^N\int_Xf.U^nfd\mu>0 \ \ \ \ \ (3)$

for any non-negative function ${f\in L^\infty}$ such that ${\int_Xfd\mu>0}$.
The convergence result that deals with the same ergodic averages (i.e. the convergence statement that contains the expression ${fU^nf}$) is von Neumann’s mean ergodic theorem. A version of this theorem can be formulated the following way: for any ${f\in L^2(X)}$ (where again ${(X,\mu,T)}$ is a ${{\mathbb Z}}$-system) the limit

$\displaystyle Pf:=\lim_{N\rightarrow\infty}\frac1N\sum_{n=1}^NU^nf\ \ \ \ \ (4)$

exists in the strong ${L^2}$ norm. Notice that (because strong convergence implies weak convergence in ${L^2}$) this statement implies that for any ${f_1,f_2\in L^2(X)}$, the limit

$\displaystyle \lim_{N\rightarrow\infty}\frac1N\sum_{n=1}^N\int_Xf_1U^nf_2d\mu$

exists. In particular putting ${f_1=f_2=f}$ we conclude that in (3) we can replace ${\limsup}$ with ${\lim}$. However, this is not enough to conclude that the limit is positive when ${f\geq0}$ and ${\int_Xfd\mu>0}$. Luckily, the mean ergodic theorem gives some information about the limit ${Pf}$ in (4). Namely, we have that ${Pf}$ is an orthogonal projection onto the subspace formed by the functions ${g\in L^2(X)}$ such that ${Ug=g}$. This is enough to deduce recurrence from convergence (i.e. to deduce equation (3) from equation (4)): In the following equation, the norm and inner product are of the Hilbert space ${L^2(X)}$ and we use the symbol ${1}$ to denote the constant function equal to ${1}$ in ${X}$.

$\displaystyle \begin{array}{rcl} \lim_{N\rightarrow\infty}\frac1N\sum_{n=1}^N\int_XfU^nfd\mu&=&\displaystyle\left\langle f,\lim_{N\rightarrow\infty}\frac1N\sum_{n=1}^NU^nf\right\rangle=\langle f,Pf\rangle\\&=&\displaystyle\langle Pf,Pf\rangle+\langle f-Pf,Pf\rangle=\|Pf\|^2\\&=&\displaystyle\|Pf\|^2\|1\|^2\geq\left|\langle Pf,1\rangle\right|^2=\left|\langle f,1\rangle-\langle f-Pf,1\rangle\right|^2\\&=&\displaystyle\langle f,1\rangle^2=\left(\int_Xfd\mu\right)^2 \end{array}$

where on the first line we use the continuity of the inner-product, in the second line we use the fact that ${Pf}$ is an orthogonal projection (and hence ${f-Pf}$ is orthogonal to the image of ${L^2(X)}$ under ${P}$), in the third line we use the Cauchy-Schwartz inequality with the functions ${Pf}$ and ${1}$ and in the last line we use again the fact that ${P}$ is a projection and ${1=P1}$ is in its image.

— 2. Non-conventional ergodic averages for a single action —

The first non-trivial recurrence result is the ergodic version of Roth’s theorem. This result is equivalent to Roth’s theorem in arithmetic progressions (Szemeredi’s theorem for progressions of length ${3}$). The ergodic version states that whenever ${A}$ is a set of positive measure in a ${{\mathbb Z}}$-system, then ${\mu(A\cap T^{-n}A\cap T^{-2n}A)>0}$ for some ${n\in{\mathbb Z}}$. More generally, Furstenberg’s multiple recurrence theorem, which is equivalent to Szemeredi’s theorem, states that under the same conditions, for every ${k\in{\mathbb N}}$ there exists some ${n\in{\mathbb N}}$ such that

$\displaystyle \mu(A\cap T^{-n}A\cap T^{-2n}A\cap\dots\cap T^{-kn}A)>0\ \ \ \ \ (5)$

This was proved by Furstenberg in his seminal paper in 1977. In fact he showed that

$\displaystyle \liminf_{N\rightarrow\infty}\frac1N\sum_{n=1}^N\mu(A\cap T^{-n}A\cap T^{-2n}A\cap\dots\cap T^{-kn}A)>0$

and more generally, for any non-negative function ${f\in L^\infty(X)}$ such that ${\int_Xfd\mu>0}$ we have

$\displaystyle \liminf_{N\rightarrow\infty}\frac1N\sum_{n=1}^N\int_XfU^nfU^{2n}f\dots U^{kn}fd\mu>0\ \ \ \ \ (6)$

The convergence result associated with this result is the following: for any ${f_1,\dots,f_k\in L^\infty(X)}$, the limit

$\displaystyle \lim_{N\rightarrow\infty}\frac1N\sum_{n=1}^NU^nf_1U^{2n}f_2\dots U^{kn}f_k\ \ \ \ \ (7)$

exits in the ${L^2(X)}$ norm. Observe that the case ${k=1}$ of this result is von Neumann’s mean ergodic theorem. This result was establish when ${k=2}$ in Furstenberg’s 1977 paper, and the general case was established by Host and Kra in 2005.
Observe that, while the recurrence result in equation (6) was proved in 1977, only in 2005 the convergence result (7) was deduced. Moreover, by itself, the convergence result does not imply recurrence. However, as in the mean ergodic theorem, the convergence result (7) of Host and Kra gives some information about the limit obtained (roughly speaking, the dynamics of the limit in (7) behaves as a function on a nil-system. For comparison, the dynamics of the limit in (4) behaves as a function on a trivial one-point system). This information can be used to deduce the recurrence result (6), it seems this has been done implicitly in many places (and maybe explicitly somewhere), see this post of Terry Tao (and the comments in particular).
There is one more type of results about a single ${{\mathbb Z}}$-action, and this concerns polynomial sequences.

Theorem 1 (Bergelson-Leibman) Let ${(X,\mu,T)}$ be an invertible measure preserving system and let ${p_1,\dots,p_k:{\mathbb Z}\rightarrow{\mathbb Z}}$ be polynomials such that ${p_i(0)=0}$ for all ${i=1,\dots,k}$. Let ${A\subset X}$ with ${\mu(A)>0}$. Then

$\displaystyle \liminf_{N\rightarrow\infty}\frac1N\sum_{n=1}^N\mu\left(A\cap T^{-p_1(n)}A\cap\dots\cap T^{-p_k(n)}A\right)>0$

This result was later generalized by Bergelson, Leibman and Lesigne to include all families of jointly divisible polynomials, i.e. ${p_1,\dots,p_k}$ such that for every ${d\in{\mathbb N}}$ there is some ${n\in{\mathbb Z}}$ (dependent on ${d}$ but not on ${i}$) such that ${d|p_i(n)}$ for all ${i=1,\dots,k}$. It is easy to check that for the polynomial ${p(x)=x^2-2}$ in a ${3}$-point rotation, no point returns to itself after an iteration of the form ${p(x)}$. In fact, the condition that the polynomials are jointly divisible is a necessary and sufficient for recurrence to occur.
The convergence result associated with this result also holds (and for any polynomials). This was first established by Host and Kra and independently by Leibman. It also follows from the significantly more general Theorem 3 below.

— 3. Several ${{\mathbb Z}}$-actions —

In this section I still deal with more than one measure preserving transformation. For commuting actions, this is in principle the same as studying a single ${{\mathbb Z}^d}$ action, but the averages are taken with respect to a parameter living in ${{\mathbb Z}}$, so even in the case when the actions commute, our interest is in the individual ${{\mathbb Z}}$-actions, rather than the ${{\mathbb Z}^d}$ action they induce.

— 3.1. Commuting ${{\mathbb Z}}$-actions —

The first result in this category is the multidimensional Szemeredi theorem. This result was deduced by Furstenberg and Katznelson in 1978 through its ergodic version. It states that given any ${k}$ commuting measure preserving transformations ${T_1,\dots,T_k}$ of a probability space ${(X,\mu)}$ and any set ${A\subset X}$ with positive measure there exists ${n\in{\mathbb N}}$ such that

$\displaystyle \mu(A\cap T_1^{-n}A\cap T_2^{-n}A\cap\dots\cap T_k^{-n}A)>0$

Notice that when ${T_i=T^i}$ this implies Furstenberg’s multiple recurrence theorem (5). This result was established in 1978 and was deduced from the following averaging result

$\displaystyle \liminf_{N\rightarrow\infty}\frac1N\sum_{n=1}^N\int_XfU_1^nfU_2^nf\dots U^n_kfd\mu>0\ \ \ \ \ (8)$

for any ${f\in L^\infty(X)}$ such that ${\int_Xfd\mu>0}$.
The convergence result associated with this result was establish by Tao in 2007, after some particular cases by Conze and Lesigne, and states that the limit

$\displaystyle \lim_{N\rightarrow\infty}\frac1N\sum_{n=1}^NU_1^nf_1U_2^nf_2\dots U_k^nf_k\ \ \ \ \ (9)$

exists in ${L^2}$. This proof does not give any information about the limit in (9), and is therefore essentially independent from the recurrence result (8). A second proof that the limit in (9) exists was obtained by Austin in 2008 and gives some information about the limit. Austin then used this approach with some additional ideas to give a new proof of the recurrence result (8).
The ergodic multidimensional polynomial Szemeredi theorem of Bergelson and Leibman is a generalization of Theorem 1 (obtained in the same paper as Theorem 1) to the case of many commuting actions. It states that for any polynomials ${p_1,\dots,p_k\in{\mathbb Z}[x]}$, with ${p_i(0)=0}$ for all ${i=1,\dots,k}$, for any commuting measure preserving transformations ${T_1,\dots, T_k}$ of a probability space ${(X,\mu)}$ and any non-negative function ${f\in L^\infty(X)}$ such that ${\int_Xfd\mu>0}$ we have

$\displaystyle \liminf_{N\rightarrow\infty}\frac1N\sum_{n=1}^N\int_XfU_1^{p_1(n)}fU_2^{p_2(n)}f\dots U_k^{p_k(n)}fd\mu>0\ \ \ \ \ (10)$

Observe that when ${p_i(x)=x}$ this result reduces to (8). There is currently no generalization of this result to more general polynomials (in the spirit of the Bergelson-Leibman-Lesigne result mentioned above), although there have been some generalizations for other functions, culminating in this recent paper by Zorin-Kranich about generalized polynomials.
The convergence of the averages in (10) follows from the more general Theorem 3 by Walsh.

— 3.2. Non-commuting ${{\mathbb Z}}$-actions —

To say that ${T_1,\dots,T_k}$ are commuting invertible measure preserving transformations is a short way to say that the group generated by ${T_1,\dots, T_k}$ is abelian. The next simplest case is when the transformations ${T_1,\dots,T_k}$ generate a nilpotent group. We will conveniently refer to this situation by saying that the transformations “nilpot” or “are nilpoting”.

Theorem 2(Leibman, 1998) Let ${k\in{\mathbb N}}$ and ${T_1,\dots,T_k}$ be nilpoting invertible measure preserving transformations of a probability space ${(X,\mu)}$. Let ${p_1,\dots,p_k\in{\mathbb Z}[x]}$ be polynomials such that ${p_i(0)=0}$. Then for any ${A\subset X}$ with ${\mu(A)>0}$ we have

$\displaystyle \liminf_{N\rightarrow\infty}\frac1N\sum_{n=1}^N\mu\left(A\cap T_1^{-p_1(n)}A\cap\dots,\cap T_k^{-p_k(n)}A\right)>0$

The convergence result associated with this result is due to Walsh:

Theorem 3 (Walsh, 2012)Let ${k\in{\mathbb N}}$ and ${T_1,\dots,T_k}$ be nilpoting invertible measure preserving transformations of a probability space ${(X,\mu)}$. Let ${p_1,\dots,p_k\in{\mathbb Z}[x]}$ be polynomials and let ${f_0,f_1,\dots,f_k\in L^\infty(X)}$. Then the limit

$\displaystyle \lim_{N\rightarrow\infty}\frac1N\sum_{n=1}^Nf_0T_1^{p_1(n)}f_1T_2^{p_2(n)}f_2\dots T_k^{p_k(n)}f_k$

exists in ${L^2(X)}$.

Walsh’s result gives no information about the limit, in the same spirit as Tao’s proof of (9). Observe that in Walsh’s result there are no restriction on the polynomials, unlike the recurrence results such as Theorem 1 or even the Bergelson-Leibman-Lesigne result.
Finally if ${T}$ and ${S}$ are measure preserving transformations of the probability space ${(X,\mu)}$ that do not nilpot, both convergence and recurrence may fail, even when the group generated by ${T}$ and ${S}$ is solvable. This was discovered by Bergelson and Leibman. This means that for ${{\mathbb Z}}$-actions, the convergence result of Walsh is (in a certain sense) the best possible.