rigidityFeb2011(1).pdf

RIGIDITY AND NON-RECURRENCE ALONG SEQUENCES

V. BERGELSON, A. DEL JUNCO, M. LEMA NCZYK, J. ROSENBLATT

Abstract. Two properties of a dynamical system, rigidity and non-recurrence, are examined

in detail. The ultimate aim is to characterize the sequences along which these properties do

or do not occur for dierent classes of transformations. The main focus in this article is to

characterize explicitly the structural properties of sequences which can be rigidity sequences or

non-recurrent sequences for some weakly mixing dynamical system. For ergodic transformations

generally and for weakly mixing transformations in particular there are both parallels and

distinctions between the class of rigid sequences and the class of non-recurrent sequences. A

variety of classes of sequences with various properties are considered showing the complicated

and rich structure of rigid and non-recurrent sequences.

1. Introduction

Let (X;B;p;T) be a dynamical system: that is, we have a non-atomic probability space

(X;B;p) and an invertible measure-preserving transformation T of (X;B;p). We consider here

two properties of the dynamical system (X;B;p;T), rigidity and non-recurrence. Ultimately we

would like to characterize the sequences along which these properties do, or do not occur, for

dierent classes of transformations. The main focus here is to characterize which subsequences

(n m ) inZ

+ can be a sequence for rigidity, and which can be a sequence for non-recurrence, for

some weakly mixing dynamical system. In the process of doing this, we will see that there are

parallels and distinctions between the class of rigid sequences and the class of non-recurrent

sequences, both for ergodic transformations in general and for weakly mixing transformations

in particular.

The properties of rigidity and non-recurrence along a given sequence (n m ) are opposites of

one another. By rigidity along the sequence (n m ) we mean that the powers (T n m ) are converging

in the strong operator topology to the identity; that is, kf T n m fk 2 ! 0 as m ! 1, for

all f 2 L 2 (X;p). So rigidity along (n m ) means that p(T n m A \ A) ! p(A) as m ! 1 for

all A 2 B. On the other hand, non-recurrence along the sequence (n m ) means that for some

A 2 B with p(A) > 0, we have p(T n m A \ A) = 0 for all m 1. Nonetheless, there are

structural parallels between these two properties of a sequence (n m ). For example, neither

property can occur for an ergodic transformation unless the sequence is sparse. Also, these two

properties cannot occur without the sequence (n m ) having (or avoiding) various combinatorial

or algebraic structures. These properties can occur simultaneously for a given transformation if

the sequences are disjoint. For example, we are able to use Baire category results to show that

the generic transformation T is weak mixing and rigid along some sequence (n m ) such that it is

also non-recurrent along (n m 1). In proving this, one sees a connection between rigidity and

non-recurrence. The non-recurrence along (n m 1) is created by rst using rigidity to take a

rigid sequence (n m ) for T and a set A;p(A) > 0, such that

p(T n m AA) 100 p(A). This

allows one to prove that T is non-recurrent along (n m 1) for some subset C of TA. One can

extend this argument somewhat and show that for every whole number K, there is a weakly

m=1

Date: February, 2011.

V. BERGELSON, A. DEL JUNCO, M. LEMA NCZYK, J. ROSENBLATT

mixing transformation T that is rigid along a sequence (n m ), such that also for some set C,

p(C) > 0, T is non-recurrent for C along (n m + k) for all k 6= 0;jkj K.

First, in Section 2, we discuss some generalities about rigidity and weak mixing. We also

consider the more restrictive property of IP-rigidity. We will see that both rigidity and IP-

rigidity can be viewed as a spectral property and therefore characterized in terms of the behavior

of the Fourier transforms b of the positive Borel measures onTthat are the spectral measures

of the dynamical system. We will see that rigidity sequences must be sparse, but later in

Section 3.1.3, it is made clear that they are not necessarily very sparse. In addition, we show

that rigidity sequences cannot have certain types of algebraic structure for rigidity to occur

even for an ergodic transformations, let alone a weakly mixing one.

After this in Section 3, we prove a variety of results about rigidity that serve to demonstrate

how rich and complex is the structure of rigid sequences. Here is a sample of what we prove:

n m+1

a) In Proposition 3.5 we show that if lim

m!1

n m = 1, then (n m ) is a rigidity sequence for some

weakly mixing transformation T. This result uses the Gaussian measure space construction.

Also, by a cutting and stacking construction, we construct an innite measure preserving rank

one transformation S for which (n m ) is a rigidity sequence. Under some additional assumptions

on (n m ), we can use the cutting and stacking construction to produce a weakly mixing rank one

transformation T on a probability space for which (n m ) is a rigidity sequence. See specically

Proposition 3.10 and generally Section 3.1.2.

b) In contrast, we show that sequences like (a m : m 1), and a 2N;a 6= 1, are also rigidity

sequences for weakly mixing transformations. However, perturbations of them, like (a m +p(m) :

m 1) with p 2Z[x];p 6= 0, are never rigidity sequences for ergodic transformations, let alone

weakly mixing transformations. See Proposition 3.27 and Remark 2.25 c).

c) We prove a number of results in Section 3.1.3 that show that rigidity sequences do not

necessarily have to grow quickly, but rather can have their density decreasing to zero innitely

often slower than any given rate. One consequence may illustrate what this tells us: we show

that there are rigidity sequences for weakly mixing transformations that are not Sidon sets.

See Proposition 3.16 and Corollary 3.21.

d) In Section 3.3.1, we show that there is no universal rigid sequence. That is, we show that

given a weakly mixing transformation T that is rigid along some sequence, there is another

weakly mixing transformation S which is rigid along some other sequence such that T S is

not rigid along any sequence.

e) In Section 3.4, we show how cocycle construction can be used to construct rigidity sequences

for weakly mixing transformations. One particular result is Corollary 3.51: if ( p n

q n : n 1) are

the convergents associated with the continued fraction expansion of an irrational number, then

(q n ) is a rigidity sequence for a weakly mixing transformation.

We then consider non-recurrence in Section 4. We show that the sequences exhibiting non-

recurrence must be sparse and cannot have certain types of algebraic structure for there to be

non-recurrence even for ergodic transformation, let alone a weakly mixing one. We conjecture

that any lacunary sequence is a sequence of non-recurrence for some weakly mixing transfor-

mation, but we have not been able to prove this result at this time.

Here are some specic

results on non-recurrence that we prove:

a) It is well-known that the generic transformation T is weakly mixing and rigid. We show

that in addition, there is a rigidity sequence (n m ) for such a generic T, so that for any whole

RIGIDITY AND NON-RECURRENCE ALONG SEQUENCES

number K, each (n m + k); 0 < jkj K, is a non-recurrent sequence for T. See Proposition 4.4

and Remark 4.5.

b) We observe in Proposition 4.6 that some weakly mixing transformations, like Chacon's

transformation, are non-recurrent along a lacunary sequence with bounded ratios.

n m

c) We also show that for any increasing sequence (n m ) with

n m+1 < 1, and a whole

number K, there is a weakly mixing transformation T and a set C;p(C) > 0, such that (n m ) is

a rigidity sequence for T and T non-recurrent for C along (n m + k) for all k; 0 < jkj K. See

Proposition 4.8 and Remark 4.9.

m=1

When considering both rigidity and non-recurrence of measure-preserving transformations,

there are often unitary versions of the results that are either almost identical in statement and

proof, or worth more consideration. When possible, we will take note of this. See Krengel [35]

for a general reference on this and other aspects of ergodic theory used in this article.

There is also a larger issue of considering both rigidity and non-recurrence for general groups

of invertible measure-preserving transformations. This will require a careful look at general

spectral issues, including the irreducible representations of the groups. We plan to pursue this

in a later paper.

2. Generalities on Rigidity and Weak Mixing

Suppose we consider a dynamical system (X;B;p;T). Unless it is noted otherwise, we will be

assuming that (X;B;p) is a standard Lebesgue probability space i.e. it is measure theoretically

isomorphic to [0; 1] in Lebesgue measure. In particular it is non-atomic and L 2 (X;p) has a

countable dense subset in the norm topology. We say that the dynamical system is separable in

this case. We also assume that T is an invertible measure-preserving transformation (X;B;p).

This section provides the background information needed in this article. First, in Section 2.1

we deal with the basic properties of rigidity and weak mixing in order to give a general ver-

sion of the well-known fact that the generic transformation is both weakly mixing and rigid

along some sequence. Second, in Section 2.2 we look at weak mixing and aspects of it that are

important to this article. Third, in Section 2.3 we consider rigidity itself in somewhat more

detail. See Furstenberg and Weiss [18] and Queelec [55], especially Section 3.2.2, for back-

ground information about rigidity as we consider it, and other types of rigidity that have been

considered by other authors.

2.1. Rigidity and Weak Mixing in General. Given an increasing sequence (n m ) of integers

we consider the family

A(n m ) = fA 2B : p (T n m A4A) ! 0g:

We now recall some basic and well-known facts about A(n m ). See Walters [68] for the following

result.

Proposition 2.1. A(n m ) B is a sub--algebra which is also T-invariant. A(n m ) is the

maximal -algebra AB such that

T n m j A ! Idj A as m !1:

Moreover

L 2 (X;A(n m );p) = ff 2 L 2 (X;B;p) : f T n m ! f in L 2 (X;B;p)g:

(2.1)

V. BERGELSON, A. DEL JUNCO, M. LEMA NCZYK, J. ROSENBLATT

Remark 2.2. An approach to the above result dierent than in [68] begins by observing that

ff 2 L 1 (X;p) : kf T n m fk 2 ! 0g is an algebra. So there is a corresponding factor map of

(X;B;p;T) for which there is an associated T invariant sub--algebra, namely A(n m ) B.

If A(n m ) = B then one says that (n m ) is a rigidity sequence for (X;B;p;T). Systems

possessing rigidity sequences are called rigid. The fact that (n m ) is a rigidity sequence for T is

a spectral property; that is, it is a unitary invariant of the associated Koopman operator U T on

L 2 (X;p) given by the formula U T (f) = f T. The following discussion should make this clear.

First, recall some basic notions of spectral theory (see e.g. [7], [30], [52]). For each f 2

L 2 (X;p), the function (n) = hf T n ;fi is a positive-denite function and hence, by the

Herglotz Theorem, is the Fourier transform of a positive Borel measure on the circleT. So for

each f 2 L 2 (X;p), there is a unique positive Borel measure f onT, called the spectral measure

for T corresponding to f which is determined by c f (n) = hf T n ;fi for all n 2Z. Spectral

measures are non- negative and have f (T) = kfk 2 . We will also need to use t he adj oint

given by (E) = (E 1 ) for all Borel sets E T.

The adjoint has b (n) = b (n) for all

n 2Z.

Absolute continuity of measures is important here: given two positive Borel measures 1 and

2 onT, we say 1 is absolutely continuous with respect to 2 , denoted by 1 2 , if 1 (E) = 0

for all Borel sets E such that 2 (E) = 0. Now, among all spectral measures there exist measures

F such that all other spectral measures are absolutely continuous with respect to F . Any one

of these is called a maximal spectral measure of T. These measures are all mutually absolutely

continuous with respect to one another. The equivalence class of the maximal spectral measures

is denoted by T . By abuse of notation, we refer to T as a measure too. Recall that the type

of a nite positive measure (e.g. whether the measure is singular, absolutely continuous with

respect to Lebesgue measure, etc.) is a property of the equivalence class of all nite positive

measures ! such that ! and !. The type of T (that is, of a maximal spectral

measure F ) has a special role in the structure of the transformation. For this reason the type

of T is called the maximal spectral type of T. For example, rigid transformations must have

singular maximal spectral type; see Remark 2.8. Also, Bernoulli transformations must have

Lebesgue type i.e. their maximal spectral measures are equivalent to Lebesgue measure. In

general, a strongly mixing transformation does not need to be of Lebesgue type. It could be

of singular type (this occurs when every maximal spectral measure is singular but yet has the

Fourier transform tending to zero at innity).

For a given sequence (n m ), a transformation T and a function f 2 L 2 (X;p), we say (n m ) is

a rigidity sequence of T for f if f T n m ! f in L 2 -norm. Recall that c f (n m ) = hf T n m ;fi.

Proposition 2.3. Fix the transformation T. The following are equivalent for f 2 L 2 (X;p):

(1) The sequence (n m ) is a rigi di ty sequence for the function f.

(2) hf T n m ;fi = R X f T n m f dp !kfk 2 .

(3) c f (n m ) !kfk 2 .

(4) z n m ! 1 in L 2 (T; f ).

(5) z n m ! 1 in measure with respect to f .

Proof. We have kf T n m fk 2 = 2kfk 2 2Rehf T n m ;fi. Since jhf T n m ;fij kfk 2 , we

see that (1) is equivalent to (2). Now (2) is equivalent to (3) by the denition of the spectral

measure f . We also have R jz n m 1j 2 d f (z) = 2kfk 2 2Re( c f (n m )). Since j c f (n m )jkfk 2 ,

RIGIDITY AND NON-RECURRENCE ALONG SEQUENCES

we see that (3) is equivalent to (4). It is clear that (4) is equivalent to (5) because j1z n m j 2

and f is a nite, positive measure.

Remark 2.4. This result is really a fact about a unitary operator U on a Hilbert space H.

That is, a sequence (n m ) and vector v 2 H satisfy

m!1 jjU n m v vk H = 0 if and only if

the spectral measure v determined by c v (k) = hU k v;vi for all k 2Zhas the property that

lim

m!1 c v (n m ) = kvk 2 H .

Proposition 2.3 shows that if (n m ) is a rigidity sequence for T for a given function F, then

for any spectral measure f F , we would also have z n m ! 1 in measure with respect to f .

Hence, (n m ) would be a rigidity sequence for T for the function f too. It follows then easily

that T is rigid and has a rigidity sequence (n m ) if and only if (n m ) is a rigidity sequence for F

where F is a maximal spectral measure for T.

Corollary 2.5. T is rigid if and only if for each function f 2 L 2 (X;p) there exists (n m ) =

(n m (f)) such that f T n m ! f in L 2 (X;p).

Remark 2.6. It is clear that an argument like this works equally well for a unitary transformation

U of a separable Hilbert space H. That is, there is one sequence (n m ) such that for all v 2 H,

kU n m v vk H ! 0 as m ! 1 if and only if for every vector v 2 H, there exists a sequence

(n m ) such that kU n m vvk H ! 0 as m !1

Remark 2.7. J.-P. Thouvenot was the rst to observe that T is rigid if and only if for each

f 2 L 2 (X;p) (or just for each characteristic function f = 1 A ;A 2 B), there exists (n m )

depending on f such that kf T n m fk 2 ! 0 as m ! 1. There are a number of dierent

ways to prove this. We have given one such argument above. Another argument would use the

characterization up to isomorphism of unitary operators as multiplication operators. Here is an

interesting approach via Krieger's Generator Theorem; see Krieger [37]. It is sucient to prove

rigidity holds assuming that one has the weaker condition of there being rigidity sequences

for each characteristic function. Suppose that an automorphism T has the property that for

each set A 2 B there exists (n m ) = (n m (A)) such that p (T n m A4A) ! 0. Then all spectral

measures of functions of the form 1 A , A 2B are singular, and since the family of such functions

is linearly dense, the maximal spectral type of T is singular. It follows that T has zero entropy;

see Remark 2.8 for an explanation of this point. Hence, by Krieger's Generator Theorem, there

exists a two element partition P = fA;A c g which generates B. Now, let (n m ) = (n m (A)) and

notice that for each k 1 and for each B 2 W k1

i=0 T i P we have p(T n m B4B) ! 0. Hence by

approximating the L 2 (X;p) functions by simple functions, (n m ) is a rigidity sequence for T.

Remark 2.8. From Proposition 2.3, we see that a maximal spectral measure T of a rigid

transformation is a Dirichlet measure. This means that for some increasing sequence (n m ), we

have n m ! 1 in measure with respect to T as m !1. Hence, as in Proposition 2.3, we have

c T (n m ) ! T (T) as m ! 1. A measure with this property is also sometimes called a rigid

measure. Note that a measure absolutely continuous with respect to a Dirichlet measure is a

Dirichlet measure. So by the Riemann-Lebesgue Lemma, there is no non-zero positive measure

which is absolutely continuous with respect to Lebesgue measure such that f for a non-

zero spectral measure f of a rigid transformation. Therefore, for a rigid transformation, all

spectral measures, and T itself, are Dirichlet measures and hence singular measures. So T has

singular maximal spectral type. Rokhlin shows in his classical paper [57] that if T has positive

entropy, then for every maximal spectral measure F , there is a non-zero spectral measure

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