ABSTRACT

We will present some restrictions for a rigidity sequence of a nontrivial topological dynamical system. For instance, any finite linear combination of a rigidity sequence by integers has upper Banach density zero. However, there are rigidity sequences for some uniformly rigid systems whose reciprocal sums are infinite. We also show that if is a family of subsets of natural numbers whose dual is filter, then a minimal -mixing system does not have -rigid factor for.

1. Introduction

A topological dynamical system (TDS) is a pair such that X is a compact metric space and T is a homeomorphism. Our main concern is rigidity. This notion was first introduced by Furstenberg and Wiess for measure theoretical dynamical system (MDS); then Glasner and Maon defined the topological version of this notion [1].

A comprehensive study for rigidity in MDS has been done in [2]. In MDS, these are interesting; because, it is well-known that a generic transformation is rigid [3]. In this respect and in TDS, Glasner and Maon [1] established examples to show that even in minimal weakly mixing systems, there are plenty of examples with uniformly rigidity.

Let us recall the main definitions. An MDS

is rigid along if, as for all. A TDS is called rigid if there exists a sequence, called the rigidity sequence, such that for any; it is called uniformly rigid if uniformly on X.

Let, then

(1.1)

is called the upper density of A and it is called lower density or density if we replace limsup in (1.1) with liminf or lim respectively. We call  

the upper Banach density of a set.

In a TDS, the return time set is defined to be

where U and V are opene (nonempty and open) sets. A TDS is transitive if for any two opene sets and, we have; and it is weak mixing if the product system is transitive. A TDS is mild mixing if for any transitive, the product system is transitive; and it is strong mixing if is cofinite for opene sets.

A collection of subsets of integers is called family if it is hereditary upward: if and, then.

It is well-known that mild mixing systems do not have uniformly rigid factors while minimal equicontinuous systems have comparatively large rigidity sequences. Therefore, one expects to have rigidity along large sequences is system with low complexity. In this note, we define some other classes of mixings. These are defined when generates a certain family of integers. In particular, we use this concept and define -mixings and we show that minimal - mixings do not have any rigidity factor.

2. Main Results

It is well known that in a transitive TDS, any almost equicontinuous is uniformly rigid [1]. In [4] the authors showed that a uniformly rigid mild mixing dynamical system is trivial. Also in [1], Glasner and Maon constructed a generic minimal uniformly rigid weakly mixing. On the other hand, any system with rigidity sequence has zero entropy [1]. Therefore, a uniformly rigid TDS with zero entropy is generic. However, there are some restrictions for a sequence to be a rigidity sequence. The following shows some of these restrictions which are compatible with the rigidity sequences in MDS [5, Proposition 2.20 (b), 2.24 and 2.26].

Theorem 2.1. Let be an increasing sequence in and suppose that for any,.

1) If T is rigid along A, then A has gaps tending to infinity.

2) Suppose where. If for each, there is such that, then A is not a rigidity sequence for any TDS.

3) Suppose A has the property that for some integers. Then A cannot be a rigidity sequence for a TDS. In particular,.

Proof. We prove only (1) and the two others follow similarly. Let be rigid along. Then for every. By the dominated convergence theorem for every invariant measure and in particular for ergodic measure and any we have. This shows that is rigid along in measure theoretical sense. By [5, Proposition 2.20(b)], if is a rigidity sequence for ergodic, then has gaps tending to infinity.

Note that the second part of the conclusion in (2) follows from the fact that sets having positive upper Banach density have a certain distance appearing infinitely many times.

If, then and so it has positive density. Therefore, cannot be a uniformly rigidity sequence for any . This is also true for sequence of prime numbers and polynomial sequence with integer coefficients.

Let

. (2.1)

From largeness point of view, is next to the family of positive upper density, that is, if then [6]. This family has many interesting properties and it is a long standing conjecture by Erdös that any member of this family has arbitrary long arithmetic progression. In the following example we show that there are some uniformly rigid TDS whose rigidity sequence is in.

Example 2.2. Let and T the irrational rotation on (or consider any equicontinuous minimal system). Note that for any and any opene set U, the return time set is syndetic and if rigidity is established for a point, that is if there exists such that, then rigidity is established for all points. Also rigidity and uniform rigidity are equivalent for our system.

First we construct a rigidity sequence and then we will show that which is trivially a rigidity sequence is in. So let and let be a decreasing sequence to zero and set to be the sequence with the maximum gap for.

Set, and pick consecutive

. For any we have and

Use induction argument and let

such that. So  and thus. But for any which implies.

Remark 2.3.

1) Let. Then and in general can be defined for any. In [7], for any, an explicit subset of such as depending on n is given such that. Now the existence of such sets is established by the above example and Theorem 2.1(3). In fact, we have more: there is such that for any,.

2) If along a subsequence, then is rigid along and it is uniform rigid if is uniform. If is also rigid along, then is rigid and it is uniform if both and are uniformly rigid along.

Theorem 2.4. Suppose is rigid along and a TDS. Then is rigid if there exists and such that is rigid along.

Proof. If is rigid along, then it is also rigid along for any and.

Corollary 2.5. Let be a rotation and a rigid (resp. uniformly rigid) system. Then is rigid (resp. uniformly rigid).

Proof. Suppose is rigid along some sequence. Let be the rotation map. For any, by passing to a subsequence if necessary, we have for some. This means that

where. Hence uniformly and.

Theorem 2.6. Suppose is rigid. Then any factor is rigid.

Proof. This is clearly true for the trivial factor. So let be rigid along and a nontrivial factor with factor map. We show that is rigid along a subsequence of. To this end, let be an arbitrary point in, and an opene set containing. Since is rigid there exists such that for any. Thus and so.

Let be a family of nonempty subsets of. The dual of, denoted by, is defined to be all subsets of meeting all sets in:

A family is called partition regular if is partitioned to finite sets, then there is such that. An example of a family with partition regularity is the family defined as (2.1). A nonempty family closed under finite intersections is called a filter. It is known that if is partition regular, then is a filter. A filter which is partition regular is called an ultrafilter.

Now let be an increasing sequence of integers. Then is the finite sums of A. A set is called an IP-set if it contains the finite sums of some sequence of integers. A set is called a ∆-set if a sequence of integers exists such that the difference set

. Let ∆ be the family of all

∆-sets. Any IP-set is a ∆-set for let

. Let (resp. ∆) be the family of all IP-sets (resp. ∆-sets). It is known that the families and are filters [8].

Definition 2.7. A TDS is called -transitive if for any two opene sets we have , and it is called -mixing if the product system is -transitive.

Theorem 2.8. [9] Let be a TDS. The following conditions are equivalent:

1) is -mixing;

2) is weak mixing and -transitive;

3) for any opene sets U, V.

For a family and, the shifted family is defined as where

. If for any, then is called a shift invariant family. For instance, if, then both and are shift invariant families. But not all families are shift invariant. There are two ways to build a shift invariant family from a given [8]. These are and where

We have and both and are shift invariant families with [8]. Also, if

then and which implies that. If is a filter so is any shift of and since the finite intersections of filters are again filters is a filter.

Theorem 2.9.

1) is -transitive if and only if it is -transitive.

2) Suppose is a filter. Then is -mixing if and only if it is -mixing.

Proof. 1) We have, so -transitive is - transitive. Conversely, suppose is -transitive. Then for any opene U and V, we have. Since is opene for, . This means that for, which in turn implies .

2) This is a direct consequence of the first part and Theorem 2.8.

Theorem 2.10. Let be a filter and an - mixing system. Then any non-trivial factor of is also -mixing.

Proof. Suppose is -mixing and a non-trivial factor and the factor map:. For any two opene sets U, , . We will show that this will hold for as well.

Let be two opene sets in Y and let such that. Then

Since is a family, so. Also, since is a filter which implies is -mixing.

Let be a family of subsets of integers closed under finite intersections (in general like a filter). Then we say that a sequence is -convergent to if for any neighborhood of we have

and we write.

A family is called an family if any member contains the difference set of an IP-set.

Theorem 2.11. Let be uniformly rigid along

. Then.

Proof. First we prove that id. Let be uniformly rigid along. Fix and let be an increasing sequence and each sufficiently large so that

Hence for any we have

(2.2)

Now set. Then contains B and so is an IP-set.

To see that, note that if is a rigidity sequence so is. Then an inequality such as (2.2) implies that where.

Now we investigate the existence of rigidity (not necessarily uniform) in minimal systems with some sort of mixings. Recall that a minimal system is mild mixing if and only if it is -mixing if and only if it is

-transitive [10].

A pair is said to be a proximal pair if

and is proximal system if any pair of is a proximal pair. A TDS is called distal if for every and . In [1], the authors showed that any minimal strong mixing system admits only trivial rigid factors. An extension of that result is the following.

Theorem 2.12. Suppose is a filter. Then a minimal -mixing system does not have -rigid factor where.

Proof. Let be a minimal -mixing. Then by Theorem 2.9 and Theorem 2.10, every factor of is -mixing. Thus it is sufficient to show that if is -rigid, then it must be trivial. Assume that is rigid with respect to an sequence.

Let, where and. Note that is and so

. Therefore, there exists a subsequence of such that

and. Which implies that is a proximal pair. Since was arbitrary the system is proximal. But in a minimal system, is distal for any x and this in turn implies that must be trivial.

Corollary 2.13.

1) A minimal -mixing system does not have any rigid factor.

2) A minimal -transitive system is not rigid.

3) Any -mixing, IP^*-mixing or -transitive system does not have a non-trivial uniformly rigid factor.

Proof.

1) By Theorem 2.10, it suffices to show that a minimal -mixing is not rigid along any. Assume the contrary and let be a sequence decreasing to zero. Let, where are -balls containing and respectively. Then there exists such that for any

,. Since is

,. Now if, then there exists a subsequence and a sequence such that and. Now an argument as in the proof of Theorem 2.12 gives the proof.

2) The proof is similar to (1).

3) Recall that any -mixing, IP^*-mixing and -transitive is trivially an -transitive. Now the conclusion follows from the fact that mild mixing systems do not have non-trivial uniform rigid factors.

REFERENCES