My co-author George Costakis and I have recently uploaded to arxiv our paper “Szemerédi’s theorem, frequent hypercyclicity and multiple recurrence”. As I’m invited to talk about this subject next month, I will try to give here a general overview of the paper, the notions therein and the main ideas involved in the proofs. Our main objective in this paper is to relate some notions in linear dynamics to more classical notions from topological dynamics. In particular we show that frequently Cesàro hypercyclic operators are necessarily topologically multiply recurrent. The main tool we use to prove this result is Szemerédi’s theorem on arithmetic progressions in sets of positive density. In order to motivate this theorem, I will have to define many standard notions from linear dynamics as well as corresponding notions from topological dynamics. Before discussing the main result and (some of) its applications, I will try to give a picture of hypercyclic operators and their properties, as well as examples of `natural’ operators which are hypercyclic.

** — 1. Introduction: notions of hypercyclicity — **

First of all, I will review some basic notions from linear dynamics that will be quite central throughout the exposition. I refer the reader to the excellent book of Bayart and Matheron (Bayart and Matheron, 2009) where most of this material is drawn from anyways. We will state several classical results here omitting the proof. If no other reference is given, this means the proof can be found in (Bayart and Matheron, 2009).

** — 1.1. Hypercyclic operators — **

We will work on *a separable Banach space* over or . We will always use the symbol to denote a *bounded linear operator* acting on . In what follows I will just write , , without any further comment, assuming always that these symbols have the meaning described above.

The most central notion in linear dynamics is that of hypercyclicity.

Definition 1Theorbitof a vector under (or the -orbit) is the setThe operator T is said to be

hypercyclicif there is some vector such that the set is dense in . Such a vector will be called ahypercyclic vector for(or a -hypercyclic vector).

Some remarks are in order. First of all let us point out that these definitions only make sense if the space is *separable*. On the other hand, hypercyclicity is an infinite dimensional phenomenon; there are no hypercyclic operators on a finite-dimensional space To see this quickly think of a square matrix in its Jordan normal form.

An easy consequence of these definitions is that whenever an operator is hypercyclic, we must have . Moreover, whenever is an invertible operator, is hypercyclic if and only if is hypercyclic. These facts will be used in the discussion below .

The definition of hypercyclicity does not require any linear structure. It makes sense for an arbitrary *continuous* map acting on a topological space .

The most general setup *linear dynamics* is that of an arbitrary separable topological vector space . We will stick however to the case of a Banach space to simplify the exposition, the generalizations being mostly of a technical nature.

The notion of hypercyclicity is strictly stronger (though relevant) than that of *cyclicity*. Recall from classical operator theory that an operator is called *cyclic* if there exists a vector (a *cyclic vector for *) such that the linear span of

is dense in . This notion is related to the *invariant subspace problem*; the operator lacks (non-trivial) invariant closed subspaces if and only if every non-zero vector is cyclic for .

Likewise, the notion of hypercyclicity is closely related to the *invariant subset problem*. It is an easy observation that an operator lacks non-trivial invariant subsets if and only if every non-zero vector is hypercyclic for . P. Enflo first answered the question in the negative for a constructing a rather peculiar Banach space. After that, C.J. Read has proved that there is an operator on for which every non-zero vector is hypercyclic. So the invariant subspace problem has a negative solution on . However the problem remains open in the case of Hilbert spaces.

** — 1.2. Universal sequences of operators — **

We will be interested in the following generalization of hypercyclicity to *families* of continuous linear operators , where each and are two topological spaces.

Definition 2The family is calleduniversalif there exists a such that the set is dense in .

Of course hypercyclicity is a special case of universality, where the family of operators is defined as the *iterates* of a fixed operator and is a topological vector space.

** — 1.3. Cesàro Hypercyclicity — **

In (León-Saavedra, 2002), F. León-Saavedra introduced the notion of *Cesàro hypercyclicity*.

Definition 3An operator is calledCesàro hypercyclicif itsCesàro orbit, that is the setis dense in . Such a vector will be called

Cesàro hypercyclicfor .

Saavedra showed in (León-Saavedra, 2002) that is Cesàro hypercyclic if and only if there is a vector such that the set

is dense in . Observe that this means that the family of operators is universal. We stress here that, in general, the notions of hypercyclicity and Cesàro hypercyclicity are not `ordered’; hypercyclicity does not imply Cesàro hypercyclicity and vice versa.

** — 1.4. How to prove that an operator is hypercyclic — **

This first characterization of hypercyclicity comes from topological dynamics and is often referred to as `Birkhoff’s transitivity theorem’.

Theorem 4 (Brkhoff’s transitivity theorem)Let be a continuous linear operator on a separable Banach space . Then is hypercyclic if and only if it istopologically transitive; that is, for every pair of open sets , there exists such that .

A byproduct of the proof of Theorem 4 is that the set of -hypercyclic vectors, , is a dense subset of .

Actually Birkhoff’s theorem is true in a much more general context but I won’t pursue that here. It is important however that no linearity is necessary in Theorem 4. As a result, when one adds linearity, the following handy criterion becomes available.

Definition 5 (Hypercyclicity criterion)Let be a separable Banach space and a bounded linear operator. We say that satisfies thehypercyclicity criterionif there exists an increasing sequence of positive integers , two dense sets and a sequence of maps such that:(i) for any ,

(ii) for any ,

(iii) for any .

Using Theorem 4 one can prove the following:

Theorem 6Let be a continuous linear operator on a separable Banach space . Suppose that satisfies the hypercyclicity criterion 5. Then is hypercyclic.

Definition 5 and Theorem 6 are originally due to Kitai (Kitai, 1982), in the case that and . The criterion was then evolved by R.Gethner and J. H. Shapiro in (Gethner and Shapiro, 1987) and J. Bès (Bès, 1998).

It was a long-standing question whether *every* hypercyclic operator satisfies the hypercyclicity criterion. This problem was recently resolved in the negative by M. De La Rosa and C.J. Read. It is not hard to show (and it was known) that the hypercyclicity criterion is equivalent to the operator being hypercyclic. In topological dynamics this property is referred to as being *weakly mixing*. This problem was recently resolved in the negative in (de la Rosa and Read, 2009) and later in (Bayart and Matheron, 2007) for all classical Banach spaces.

A consequence of the hypercyclicity criterion 5 and Theorem 6 is the following result, which highlights the connection between linear dynamics and spectral theory. Roughly speaking, the following *Godefroy-Shapiro criterion* states that an operator which has a `large supply’ of eigenvectors is hypercyclic. See (Godefroy and Shapiro, 1991).

Theorem 7 (Godefroy-Shapiro criterion)Let be a continuous linear operator on a separable Banach space . Suppose that and both span a dense subspace of . Then is hypercyclic.

** — 1.5. Examples of hypercyclic operators — **

We will now use the previous hypercyclicity criteria to show that some very natural operators are hypercyclic. We will also take the chance to define some classes of operators which I want to discuss later on, in relevance to our main theorem.

Example 1Let denote the space of all entire functions on endowed with the topology of uniform convergence on compact sets. Now is not a Banach space but it is a separable Frèchet space so all the notions and theorems discussed above go through. We consider thederivative operator. To see this, apply the hypercyclicity criterion with andNow the operator in the hypercyclicity criterion needs to be defined as a sort of (asymptotic) right inverse of the derivative operator so it is natural to define and . Then we have that as for every monomial so that takes care of

(i)in the hypercyclicity criterion. Condition(iii)is trivial to verify since on . Finally, in order to check the validity of condition(ii)in the hypercyclicity criterion we need to see that as for every positive integer . However, we readily see thatfrom which we easily conclude that uniformly on compact subsets of .

Example 2Let us now consider the Hilbert space . Thebackward shift operatoris defined by . Observe that this operator can never be hypercyclic since so the orbit of any vector under stays inside the unit ball. However, the operator is hypercyclic for every with . Again it is an easy exercise to check the validity of the hypercyclicity criterion with and , where is the space of all finitely supported sequences. Again where is the natural candidate, the right inverse of which in this case is theforward shiftoperator defined as .

Our last example one the one hand illustrates the Godefroy-Shapiro criterion and on the other hand gives an introduction to a class of operators I would like to consider later on in the discussion.

Example 3Here we consider a Hilbert space of analytic functions , where is the open unit disk of the complex plane. The space is pretty general but we require the following two conditions:

- , and
- for every , the point evaluation functionals are bounded.
The second condition assures that convergence in implies pointwise convergence on . By the boundedness of holomorphic functions on compact sets and the uniform boundedness principle the second condition amounts to requiring that convergence in implies uniform convergence on compact subsets of . The reader is thus encouraged to think of the Hardy space or the Bergman space in the place of , keeping in mind however that interesting phenomena occur outside these two particular cases.

A feature of that we will use is the existence of a

reproducing kernel. In particular, For each , the boundedness of the point evaluation functionals and the Riesz representation theorem provide a unique function , thereproducing kernelof at , such thatRecall that a function is called a

multiplierof if for every . Such a defines amultiplication operatorin terms of the formulaBy the boundedness of point evaluation functionals and the closed graph theorem it follows that is a bounded linear operator on . Moreover, every multiplier is a bounded holomorphic function, this is,

Observe that for every and every we have that

Remembering that there is at least one which is not identically we conclude that . Thus every multiplier is a bounded holomorphic function with . The opposite is not always true under our assumptions as can be seen by considering for example the Dirichlet space of holomorphic functions on , that is the space of all functions such that

Here denotes area measure. In the Dirichlet space not every bounded holomorphic function is a multiplier.

In general it is not difficult to see that a multiplication operator is

neverhypercyclic. The situation is quite different for theadjoints of multiplication operators. In order to make the statement of the following theorem more clear we require the extra assumption thateveryholomorphic function is a multiplier of such that . This extra assumption is automatically satisfied in the case of the Hardy space or the Bergman space but not in the Dirichlet space. The following theorem is from (Godefroy and Shapiro, 1991).

Theorem 8 (Godefroy, Shapiro)Assume that is a Hilbert space of holomorphic functions as above. Furthermore assume that every bounded holomorphic function is a multiplier of such that . Then the adjoint multiplication operator is hypercyclic if and only if is non-constant and .

Proof:We first prove that if then is hypercyclic. For we consider the reproducing kernel . Sincefor every , we conclude that for every . That is, for every , is an eigenvector of with corresponding eigenvalue . Now let and . Since is non-constant and we have that both are non-empty open sets (by the open-mapping theorem for analytic functions is an open set). By the Godefroy Shapiro criterion, in order to show that is hypercyclic it suffices to show that and both span a dense subset of . Indeed, assume that there exists a function which is orthogonal to all either for all or for all . In either case vanishes on a non-empty open set and thus is identically zero.

In order to prove the other direction first observe that whenever is hypercyclic, is non-constant. Moreover we have that is connected so it either lies entirely inside, or entirely outside the unit disk. In the first case we have that , thus cannot be hypercyclic. In the complementary case, the function is a bounded holomorphic function and . By the first case, is not hypercyclic, and since , neither is .

Example 4We finish this short list of examples by giving another typical class of hypercyclic operators, namely unilateral and bilateral weighted shifts. Let be the Hilbert space of square summable sequences . Consider the canonical basis of and let be a (bounded) sequence of positive numbers. The operator is aunilateral (backward) weighted shiftwith weight sequence if for every and .Let be the Hilbert space of square summable sequences endowed with the usual norm. That is, if . Let be a (bounded) sequence of positive numbers. The operator is a

bilateral (backward) weighted shiftwith weight sequence if for every . Here is the canonical basis of .

Theorem 9Let be defined as above, with weight sequences respectively.(i) is hypercyclic if and only if

(ii) is hypercyclic if and only if, for any

and

** — 2. Recurrence, multiple recurrence and hypercyclicity — **

Let us consider a bounded linear operator on a separable Banach space . We have already seen that saying that an operator is *hypercyclic* is equivalent to saying that an operator is topologically transitive, that is that for every pair of open sets , there is some positive integer such that . In what follows I will introduce some notions that come from topological dynamical systems.

** — 2.1. Recurrence and Multiple recurrence — **

A somewhat weaker notion in topological dynamics is that of *recurrence*.

Definition 10The operator is calledrecurrentif for every open set there is a such that .

Clearly every hypercyclic operator is recurrent. Unlike hypercyclicity which is a purely infinite dimensional phenomenon, there are recurrent operators in finite dimensions (consider for example a rotation on the plane).

A recurrent operator has many points whose orbit under asymptotically `returns’ to the point. To make this more precise, let us call a vector *recurrent vector for * if there exists an increasing sequence of positive integers such that as . It turns out that a recurrent operator has a dense set of recurrent vectors.

Proposition 11An operator is recurrent if and only if the set of recurrent vectors for is dense in . In this case the set of recurrent vectors for is a subset of .

*Proof:* Let us first prove the easy implication. That is we assume that has a dense set of recurrent points and let be an open set in . Since the recurrent points of are dense, there is a which is recurrent for . Take such that . Since is recurrent, there is a such that . Thus . That is we have that . Let us now assume that is recurrent. We fix an open ball for some and . We need to show that there is a recurrent vector in . Since is recurrent there exists a positive integer such that , for some .That is we have that and . Since is continuous, there exists such that and . Now since is recurrent, there is a such that for some . By continuity again there is an such that and . Continuing inductively we construct a sequence , a strictly increasing sequence of positive integers and a sequence of positive real numbers , such that

Since is complete we conclude by Cantor’s theorem that

for some . We also have that , for all . Thus we have that for every , which means that in . That is, is a recurrent point in the original ball .

Finally, let us write for the set of -recurrent vectors. Observe that

which shows that the set of -recurrent vectors is a -set.

After (simple) recurrence, let’s now consider multiple recurrence. An operator is called *topologically multiply recurrent* if for every non-empty open set and every there is a such that

Of course a hypercyclic operator is always recurrent. However, there is no reason why a hypercyclic operator should be topologically multiply recurrent in general. This is illustrated in the following proposition.

Proposition 12 (Costakis and Parissis, 2010)There exists a hypercyclic bilateral weighted shift on which is not topologically multiply recurrent.

** — 2.2. Frequent hypercyclicity and Szemerédi’s theorem — **

Recently, Bayart and Grivaux introduced in (Bayart and Grivaux, 2005) and (Bayart and Grivaux, 2006) a notion that examines how frequently the orbit of a hypercyclic operator visits a non-empty open set.

Definition 13An operator is calledfrequently hypercyclicif there exists a vector such that, for every non-empty open set , the sethas positive lower density.

This is the strongest form of this definition, using the `weakest’ density. There are variations where the lower density is replaced for example by the upper density. Recall that the lower density of a set is defined as

while the upper density of is

In (Bayart and Grivaux, 2006) a `frequent hypercyclicity criterion’ was established. We won’t describe this here but point out one of its applications. Going back to adjoints of multiplication operators, an application of the Bayart-Grivaux frequent hypercyclicity criterion yields the following result:

Example 5Recall that is a non-trivial Hilbert space of holomorphic functions with bounded point evaluation functionals. We consider multiplier operators with symbol . We have the following result which is a corollary of the Bayart-Grivaux criterion

Proposition 14 (Bayart, Grivaux)Assume that is a Hilbert space of holomorphic functions as above. Furthermore assume that every bounded holomorphic function is a multiplier of such that . The following are equivalent:(i) The adjoint multiplication operator is hypercyclic.

(ii) The adjoint multiplication operator is frequently hypercyclic.

(iii) The function is non-constant and .

The notion of frequent hypercyclicity seems to be the right one in relevance to topological multiple recurrence. In order to illustrate this connection we need Szemerédi’s theorem on arithmetic progressions.

Theorem 15 (Szemerédi)Let be a subset of with positive upper density. Then contains arbitrarily long arithmetic progressions.

The following proposition is just an easy application of Szemerédi’s theorem:

Proposition 16Let be a frequently hypercyclic operator. Then is topologically multiple recurrent.

*Proof:* Let be an open set and let . Since is frequently hypercyclic, there exists a such that the set

has positive lower density. By Szemerédi’s theorem, contains an arithmetic progression of length , that is we have that

This means that

that is, is topologically multiply recurrent.

** — 2.3. Frequently Cesàro hypercyclic operators — **

As we have seen earlier, an operator is Cesàro hypercyclic if and only if there exists a such that the set

is dense in . In accordance to frequently hypercyclicity, Costakis and Ruzsa introduced in (Costakis and Ruzsa, 2010) the notion of a *frequently Cesàro hypercyclic* operator in the obvious way.

Definition 17An operator is calledfrequently Cesàro hypercyclicif there is a vector such that, for every open set , the sethas positive lower density.

In contrast with Cesàro hypercyclic operators, frequently Cesàro hypercyclic operators are always hypercyclic:

Theorem 18 (Costakis and Ruzsa, 2010)Let be a frequently Cesàro hypercyclic operator. Then is hypercyclic.

As in the case of frequently hypercyclic operators, frequently Cesàro hypercyclic operators are always topologically multiply recurrent. However, this is not so obvious any more.

Theorem 19 (Costakis and Parissis, 2010)Let be a frequently Cesàro hypercyclic operator. Then is topologically multiply recurrent.

The hypothesis of the previous theorem is optimal in the sense that a Cesàro hypercyclic is not in general topologically multiply recurrent.

Proposition 20 (Costakis and Parissis, 2010)There exists a Cesàro hypercyclic bilateral weighted shift on which is not recurrent, and hence not topologically multiply recurrent.

Before giving the actual proof of Theorem 19, let us try to repeat the simple argument used in the proof of Proposition 16. We begin by fixing a positive integer and an open set . We will assume that is a ball, say . We need to show that there exists some vector with

or, in other words, that there is a such that

By the hypothesis and Szemerédi’s theorem there is a vector and an arithmetic progression of length

such that

In this case it is not obvious which is the natural candidate for the vector but let’s take . We then have for

where we know that all the ‘s are in . We can then naively estimate

There are two problems here. The first is that we cannot control the factor . The second is that even if we could, say we had , this estimate would give us that which is one too large. The second problem is easy to deal with. We just start with a smaller ball inside our original set and carry out this reasoning for the smaller ball. In the proof given below we will consider two cases. In the first we will just assume that is small. In the complementary case, we will appropriately use the information that is large!

*Proof of Theorem 19:* Let be any non-empty open set in . We fix a non-zero vector and take a positive number such that . Without loss of generality we may assume that . Consider the ball with

Observe that . Since is a frequently Ces\`{a}ro hypercyclic operator there exists such that the set

has positive lower density. By Szemerédi’s theorem the set contains an arithmetic progression of length , i.e. there exist positive integers such that

Therefore the vectors

belong to .

As promised, we will consider two cases depending on the values of the ratio of the step over the first term of the arithmetic progression provided by Szemerédi’s theorem:

**Case 1. .**

We define the vector as

Then we have

for every . Since

we conclude that

and therefore

as we wanted to show.

**Case 2. .**

Here we first need to specify a number such that

for every . Indeed, solving the above equation for we get

We now define the vector as

Then we have

that is . On the other hand,

for every . The last equality and the above estimates imply

for every . Let . Since

we conclude that

Therefore

This completes the proof of the theorem.

** — 3. Back to adjoints of multiplication operators. — **

We can now give a full characterization of frequent hypercyclicity and multiple recurrence in the case of adjoints of multiplication operators on a non-trivial Hilbert space of holomorphic functions. It turns out that the weaker property of being recurrent is equivalent to frequent hypercyclicity and thus to every other property we have discussed here.

Proposition 21 (Costakis and Parissis, 2010)Assume that is a Hilbert space of holomorphic functions as above. Furthermore assume that every bounded holomorphic function is a multiplier of such that . The following are equivalent:(i) is recurrent.

(ii) The adjoint multiplication operator is hypercyclic.

(iii) The adjoint multiplication operator is frequently hypercyclic.

(iv) The adjoint multiplication operator is topologically multiply recurrent.

(v) The function is non-constant and .

*Proof:* We have already seen in Theorem 8 and Proposition 14 that conditions *(ii), (iii)* and *(v)* are equivalent. Also, by Proposition 16, *(iii)* implies *(iv)* and obviously *(iv)* implies *(i)*. So the proof will be complete if we show for example that *(i)* implies *(v)*.

Indeed, assume that is recurrent. Suppose, for the sake of contradiction, that . Since is connected, so is ; thus, we either have that or .

**Case 1. .**

Then we have . We will consider two complementary cases. Assume that there exist and a recurrent vector for such that

The above inequality and the fact that imply that for every positive integer

On the other hand for some strictly increasing sequence of positive integers we have . Using the last inequality we arrive at , a contradiction. In the complementary case we must have for every vector which is recurrent for . Since the set of recurrent vectors for is dense in we get that for every . Hence for every . Take now and consider the reproducing kernel of . We have already seen in the proof of Theorem 8 that where is the reproducing kernel at . We conclude that

However, this is clearly impossible since is an isometry.

**Case 2. .**

Here is a bounded holomorphic function satisfying ; therefore, is invertible. It is easy to see that if an operator is invertible, then is recurrent if and only if is recurrent. Thus the operator is recurrent and the proof follows by Case 1.

Remark 22It is easy to see that under the hypotheses of Proposition 21, is never recurrent. On the other hand, suppose that is a constant function with for some and every . Then we have that (or equivalently ) is recurrent if and only if is topologically multiply recurrent if and only if . In order to prove this it is enough to notice that for every non-zero complex number , with , and every positive integer , there exists an increasing sequence of positive integers such that

** — 4. Some open questions — **

I will close this post by suggesting a couple of open problems. For more information you can check the actual paper.

** — 4.1. Multipliers on the Dirichlet space. — **

First of all, let me come back to the adjoints of multiplication operators. Recall that the Dirichlet space is defined as the space of holomorphic functions such that

The reader might have noticed that throughout the discussion here, I have assumed that the multipliers of the Hilbert space are exactly the bounded holomorphic functions and that . Although this is actually the case on the Hardy space or the Bergman space , things are quite different on the Dirichlet space defined before. On the Dirichlet space, not all bounded holomorphic functions are multipliers. In fact the characterization of multipliers on the Dirichlet space is a bit more technical and is due to Stegenga (Stegenga 1980):

Theorem 23 (Stegenga)The function is a multiplier for the Dirichlet space if and only if and the measure is a Carleson measure for the Dirichlet space .

Of course this theorem doesn’t tell us much if we can’t understand which are the Carleson measures for the Dirichlet space. Here I will just give the definition as the characterization of these measures is completely beyond the scope of this post.

Definition 24A positive Borel measure on is a Carleson measure for the Dirichlet space if for some positive constantfor every .

Due to the more involved characterization of the multipliers on the Dirichlet space, characterizing when adjoints of multiplication operators on are hypercyclic is an open question. It is however known that the condition is no longer necessary, though it is sufficient. An example is provided by the function on . On the other hand it is known that is necessary. For this, see for example the PhD thesis of Irina Seceleanu.

** — 4.2. Frequently universal sequences of operators. — **

Remember that a family of operators on is called *universal* if there exists a such that the set

is dense in . The following definition is the natural extension of frequent hypercyclicity to universal families

Definition 25The family of operators is calledfrequently universalif there exists a such that for every open set the sethas positive lower density.

Thus saying that an operator is frequently Cesàro hypercyclic amounts to saying that the family is frequently universal. Theorem 19 says that if the family is frequently universal then is topologically multiply recurrent. However, there is nothing too special about the sequence . One can consider the family of operators where is an appropriate sequence of complex numbers.

Under what condition on the sequence of complex numbers one may conclude that is topologically multiply recurrent from the hypothesis that the family is frequently universal?

** — 5. Bibliography — **

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Bayart, Frédéric and Étienne Matheron. 2009. *Dynamics of linear operators, Cambridge Tracts in Mathematics*, vol. 179, Cambridge University Press, Cambridge. MR2533318.

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Bès, Juan, P. 1998. *Three problems on hypercyclic operators.*, PhD. Thesis.

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Costakis, George and Imre Z. Ruzsa. 2010. *Frequently Cesàro hypercylic operators are hypercyclic*, preprint.

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