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.