As proved in ERT1, the existence of ergodic averages implies recurrence results for measure-preserving system (from now on, denoted by **mps**). A natural question is to ask about some kind of **generalized ergodic averages** and its implications in recurrence. By generalized ergodic averages we mean expressions like

where is a sequence of positive integers. Von Neumann’s Theorem shows that convergence holds if , where are positive integers: just apply the result to the transformation and the function . In this post, we prove that the same result holds if , where is a polynomial such that , for every . We can assume, without lost of generality, that . In fact, and satisfies the required condition.

Theorem 1(H. Furstenberg) If is a mps and is a polynomial such that , for every , and , then the limit

converges in for every .

Again, this theorem is *Hilbertian* in nature and follows from a more general version for unitary operators.

Theorem 2If is a unitary operator on a Hilbert space and is a polynomial such that , for every , and , then the sequence of operators

converges pointwise in norm.

*Proof:* The idea is the same as in Von Neumann’s Theorem: we look for an orthogonal decomposition such that the behaviour of is understood in each component. will represent the **structured **component of and the **randomic** one in the following sense:

- the long-time behaviour of elements is (almost-)periodic.
- the long-time behaviour of elements of self-amortizes and converges to zero.

Unfortunately, the decomposition of Von Neumann’s Theorem does not work here. In fact, let be periodic with respect to , say , . If we write , ,

converges to , because

This means that every periodic point of has a structured behaviour with respect to . For this reason, let

Exercise 1Prove that the set of for which the sequence converges is a closed subspace of .

By Exercise 1, the sequence converges whenever . By linearity, it remains to prove convergence for . Such subspace is characterized by

This follows from Von Neumann’s Theorem: if is the decomposition with respect to , , then is equal to and its orthogonal complement is given by

which proves (1). This means that for every and for every degree-one polynomial , . The proof will be complete if we show that the same happens for larger-degree polynomials. By induction, suppose that

for every polynomial such that , and . Consider such that , and . We wish to reduce the convergence to one of the form , with (which we know to be true, by the induction hypothesis). This is done with the use of Van der Corput’s Trick (see this lecture of Terry Tao for a broader discussion on this trick).

Theorem 3(Van der Corput Trick) If is a bounded sequence such that

for every , then

Exercise 2Prove the above theorem. (Hint: this is Theorem 2.2 of this survey of Vitaly Bergelson.)

We’re done if the sequence , , satisfies the conditions of Theorem 3. In fact, as is unitary,

where is a polynomial of smaller degree, and so (2) is satisfied. This concludes the proof of Theorem 2.

The method used above is one of the main principles Ergodic Ramsey Theory: the **dichotomy between structure and randomness**, decomposing the object of study into these two components. Usually, we first define the structured one, in terms of the desired ergodic averages, so that convergence follows almost directly from the definition. Its orthogonal complement is the randomic component and convergence along it is proved using Van der Corput like theorems. For a further discussion on this dichotomy, the reader is referred to this paper of Terence Tao. Observe that the same method applies to prove the following

Theorem 4If is a unitary operator on a Hilbert space and is a polynomial such that , for every , and , then the sequence of operators

converges pointwise in norm.

Now it’s time to obtain the recurrence consequences (which, as expected, will be stronger than those in ERT1). Let be the orthogonal projection. We’ll proceed exactly as in Proposition 6 of ERT1, except that the notation will be heavier.

Proposition 5Let be such that . Then and .

*Proof:* Consider the subspaces , . By approximation, if each projection of into satisfies and , the same happens to . Fix and consider the function . Then (**Exercise 3**) and . Because minimizes the distance of to , we have . In addition, if we had , then

implying that for some . Integrating, we conclude

a contradiction.

Theorem 6If is a mps, is a polynomial such that , for every , , and such that , then the set

is syndetic.

*Proof:* If , then from (2) the expression converges to as . Since

Theorem guarantees the conclusion.

It’s actually not that much harder to prove ( maybe a good exercise for the students – using the van der Corput trick ) that in fact convergence in the polynomial von Neumann theorem takes place almost everywhere with respect to any totally ergodic ( meaning that there are no rational eigenvalues ) system and for any function in L^2 ( or bounded to be on the

safe side ) In this case the limit equals the integral.

It’s a much deeper theorem due to Bourgain that

almost sure convergence always holds in any ergodic system for L^2-functions. This is known

to fail for L^1-functions. Note however that the

limit in this case might fail to be invariant.

It’s also plain that the mean convergence can be

extended to cover linear contractions ( not necessarily unitary ) by a simple lifting argument ( going back to Sz. Nagy I believe ). I don’t think

the corresponding theorem ( however the exact phrasing is not quite clear to me ) is known for

non-linear contractions.

By:

Anonymouson October 24, 2009at 10:10 pm

Dear Reader,

Thanks for the informations. In fact, Bourgain’s and other related results and conjectures will be the topic for the next post. With respect to contractions, you are absolutely right. This follows from the two facts below: if T is a linear contraction (meaning that |Tf| <= |f|) operator on a Hilbert space H, then

(a) |T^*f| <= |f| for every f in H;

(b) Ker(T^* – I) = Ker(T – I), where T^* stands for the adjoint operator of T,

so that the proof presented in ERT1 also holds.

By:

yglimaon October 27, 2009at 4:43 am

Hi,

This was a very nice article.

Could you give a reference for Furstenberg’s theorem?

By:

asdfon February 18, 2010at 1:07 am

actually, i realized that the neatest proof (but maybe not extendible) is from the spectral theorem + weyl’s theorem on equidistribution.

but a reference to furstenberg’s paper where the argument you mentioned above would be nice.

By:

asdfon February 18, 2010at 2:57 pm

Dear asdf, please see below Yuri’s comments (sent to me by email):

Dear asdf,

The original reference is Theorem 3.5 of Furstenberg’s paper “Poincare recurrence and number theory”, published in Bulletin of the American Mathematical Society, 1981, no. 3, 211–234. His argument is just like you said: spectral theorem and Weyl’s equidistribution theorem.

If I am not mistaken, the proof in the post is due to V. Bergelson. Actually, the reference I used was Bergelson’s survey Ergodic Ramsey theory – and update.

Best, Yuri.

By:

matheuscmsson February 19, 2010at 1:40 pm