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 2 If 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
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).
for every , then
Exercise 2 Prove 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 4 If 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 5 Let 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
Theorem 6 If is a mps, is a polynomial such that , for every , , and such that , then the set
Proof: If , then from (2) the expression converges to as . Since
Theorem guarantees the conclusion.