Infinite symmetric ergodic index

# Infinite symmetric ergodic index and related examples in infinite measure

Isaac Loh Cesar E. Silva Department of Mathematics and Statistics, Williams College,
Williamstown, MA 01267
and  Ben Athiwaratkun
###### Abstract.

We define an infinite measure-preserving transformation to have infinite symmetric ergodic index if all finite Cartesian products of the transformation and its inverse are ergodic, and show that infinite symmetric ergodic index does not imply that all products of powers are conservative, so does not imply power weak mixing. We provide a sufficient condition for -fold and infinite symmetric ergodic index and use it to answer a question on the relationship between product conservativity and product ergodicity. We also show that a class of rank-one transformations that have infinite symmetric ergodic index are not power weakly mixing, and precisely characterize a class of power weak transformations that generalizes existing examples.

###### Key words and phrases:
Ergodic, infinite measure, power weak mixing, infinite ergodic index
###### 2010 Mathematics Subject Classification:
Primary 37A40, 37A25; Secondary 28D05

## 1. Introduction

In [11], Kakutani and Parry constructed infinite measure-preserving Markov shifts such that all their finite Cartesian products are ergodic and called this property infinite ergodic index. They proved that ergodic Cartesian square does not imply infinite ergodic index as it does in the finite measure-preserving case. In [2], Adams, Friedman and the second-named author constructed a rank-one transformation that has infinite ergodic index but such that is not conservative, hence not ergodic. A transformation such that all finite Cartesian products of all its nonzero powers are ergodic is called power weakly mixing [8], and if all finite Cartesian products of all its powers are conservative it is said to be power conservative. As we only consider invertible transformations on nonatomic spaces, under these assumptions, if a transformation is ergodic, then it is conservative (see e.g. [14, 3.9.1]). It follows from [2] that infinite ergodic index does not imply power conservative, so it does not imply power weak mixing. In [1], Adams, Friedman and the second-named author modified the classic Chacon transformation by adding at each stage of the rank-one construction enough new intervals so that the measure of a column at each stage is twice the measure of the previous column. This results in an infinite measure-preserving transformation which is shown in [1] to have infinite ergodic index, and that has been called the infinite Chacon transformation. It was later shown that the infinite Chacon is not power weakly mixing [9] (in fact, not power conservative). Soon after [2], Bergelson asked if there existed a transformation that has infinite ergodic index but such that is not ergodic. In [4], Clancy et al. construct rank-one transformations such that is ergodic but is not ergodic; it is also shown in [4] that for a rank-one , the product must always be conservative. The example with not ergodic was later generalized to more general group actions in Danilenko [7], but the full Bergelson question remains open.

We define a transformation to have infinite symmetric ergodic index if has infinite ergodic index. We show that infinite symmetric ergodic index does not imply power conservativity, so does not imply power weak mixing, by demonstrating in Section 3 that a condition, shared by the infinite Chacon transformation and several other existing examples with infinite ergodic index, implies infinite symmetric ergodic index. A sufficient condition for finite symmetric ergodic index is also given in Proposition 3.2, and is used in Theorem 3.5 to answer Question 1 of Danilenko [7]. In Section 4, we give a condition guaranteeing that a rank-one is not power conservative and explicitly bound the for which is conservative. We show that a large class of infinite Chacon transformations meet this condition and consequentially are not power weakly mixing, although they have infinite symmetric ergodic index. In Section 5 we show that power weak mixing holds in a class of transformations containing the primary example of [8] (and also the examples in [13]), and the bounded-recurrence example of [9]. For terms not defined here the reader may refer for [14] or [4].

### 1.1. Acknowledgements

We would like to thank partial support provided by the National Science Foundation REU Grant DMS-0850577 and DMS - 1347804 and the Bronfman Science Center of Williams College.

## 2. Preliminaries

### 2.1. Rank-One Cutting and Stacking

Our main constructions will be obtained through rank-one cutting and stacking. We define a Rokhlin column to be a ordered and finite collection of levels, which are intervals in of the same measure. A column is associated with a map taking every point, except for those in the topmost level, to the point directly above it in the above level. The levels of the column are at heights through , where is called the height of the column (note that the column contains levels). The rank-one method involves iteratively constructing the columns and building an associated transformation as follows:

1. Take the first column to be the unit interval.

2. To build from , cut into subcolumns of equal width and add new levels (spacers) above the th subcolumn for . The spacers are levels taken from that are disjoint from the levels of . Then stack every subcolumn under the subcolumn to its right.

3. Build such that .

There are many means of characterizing rank-one transformations but the one we use is the notation of descendants. We say that a level in splits into descendants in , which have heights indexed by . If was at height in , then it is easy to see that , where is the height set defined to be

 Hn={0}∪{k∑ℓ=0(hn+sn,k+1):0≤k

It is then clear that . By an inductive argument, , where by abuse of notation we use to refer to the sum of sets (i.e. ) and shifts of sets by integer addition.

### 2.2. Product Ergodicity and Conservativity

Lemma 2.1 provides distinct necessary and sufficient conditions for ergodicity of product transformations. Its proof is standard (see e.g. Lemma 2.1 in [4]).

###### Lemma 2.1.

Let be rank-one measure preserving transformations on measure spaces and let be nonzero integers. Let , and . Let be the sufficient semiring of rectangles of the form , where is a level of some column of . If is conservative ergodic, then for all , we have

 A⊂⋃nTnB\rm\ (mod~{}μ).

Furthermore, is ergodic if for every of the form for the base of column , , and , where for all , , there exists a one-to-one map satisfying for all and:

 and μ(D(τ))≥β(b0,…,bk−1)μ(A), and dμ∘τdμ(v)≥δ(b0,…,bk−1) for all v∈D(γ).

where are positive functions of the heights of sides of , and and denote the domain and range of a map, respectively.

Lemma 2.2 provides necessary conditions for the ergodicity of products of rank-ones, and follows from Lemma 2.1 by argument of [4], Lemma 2.2. We denote the descendant set of the indexed transformation by .

###### Lemma 2.2.

Let be a product of (nonzero) integer powers of rank-one transformations in the product space . Fix . Let be the base of column and be the product of such base levels, and , where for . Then is conservative ergodic only if for every and every choice of and , there is a natural number such that for at least tuples of descendants we have for for some tuple and .

We now state these results as conditions on elements of products of the descendant sets:

###### Proposition 2.3.

For rank-one transformations and nonzero integers , is conservative ergodic only if for every , , and -tuple , there is a natural number such that for at least -tuples of descendants of the form , we have corresponding -tuples such that

 (1) a0−d0−b0α0=aℓ−dℓ−bℓαℓ∈Z∖{0}

for each .

is ergodic if there are constants for all -tuples such that the following product ergodicity condition holds: There exists some such that to at least some fraction of the -stage descendant tuples with a complementary descendant -tuple meeting (1), we can associate a unique such -tuple.

###### Proof.

Necessity of (1) follows from Lemma 2.2. For sufficiency of the second, write and . Let denote all of the descendant tuples in which are matched with unique complementary tuples meeting the stated conditions. By supposition, we can pick a such that . Set . Then for each tuple ,

 Tn(Ta00J0×⋯×Tak−1k−1Jk−1)=Td0+b00J0×⋯×Tdk−1+bk−1k−1Jk−1⊂B,

for some , where . Because the correspondence of rectangles in indexed by to covered rectangles in is one-to-one and measure preserving, we can infer by Lemma 2.1 (with ) that is ergodic. ∎

The following result on the conservativity of products of rank-one transformations is [4, Proposition 4.2]. For , let and define similarly.

###### Proposition 2.4.

Let be a rank-one transformation on a measure space , let and let be a -tuple of nonzero integers, and the product of -fold copies of . Then the product transformation on is conservative if and only if for every , where is the base of column , for every there is such that at for at least of the -tuples , there exist complementary -tuples satisfying for .

## 3. Infinite Symmetric Ergodic Index on a General Class of Rank-One

In this section, we consider a fairly broad class of transformations having infinite ergodic index. These incorporate subsets of a nice additive form into height sets infinitely often. In these subsets, elements will be spaced and apart in such a way that, under fairly light conditions, is ergodic for .

Let be a level of column . As a general technique for rank-one transformations for elements and in , we write the standard descendant sum decomposition as and , where and are elements of the height set . For any , we also write the expansion of as , where is the tuple of height set elements corresponding to the subrectangle containing . Throughout, we will let .

###### Lemma 3.1.

Let be a rank-one transformation, a level of , and positive integers. Furthermore fix a sequence such that . For each , suppose that there are sets such that , and for all and , where is a positive constant. Let . Then there exists a positive number such that:

 (2) μ({x∈I(k): the first γ indices m for which xnm∈Fnm have: xnm∈E1,nmfor b1 times, then xnm∈E2,nmfor b2 times, then ⋮ xnm∈Ek−1,nmfor bk−1 times, }) > K(b1,…,bk−1)μ(I(k)).
###### Proof.

Enumerate the set of all possible sets of distinct, nonnegative integers by . Let . Then clearly forms a partition for the subset of points of having for at least indices in . To see that this subset is just , recall that by assumption. By the construction of rank-one transformations, we can treat the elements as elements of a probability space over infinite sequences of events (with the events being subsets of height sets). For , set as events with

 P(Fnm)≡μ(x:xnm∈Fnm)μ(I(k))=|Fnm||Hnm|k≥δknm;

then the are independent insofar as the probability of their finite or countable intersections is just the product of the probabilities. Because , the second Borel-Cantelli Lemma implies that the probability that occurs infinitely often is , which is to say that , which shows that , i.e. almost every point of lies in for some .

Now consider any fixed . Call the set in (2) , and let be the subset of ranging from its highest element to (including) its . Then we have

 μ(W∩Rq)μ(I(k)) =∏p∈Rq|b1−10|E1,nM+p|⋯∏p∈Rq|γ−1bk−2|Ek−1,nM+p|∏p∈Rq|b1−10|HnM+p|k⋯∏p∈Rq|γ−1bk−2|HnM+p|k μ(Rq)μ(I(k)) =∏p∈Rq|b1−10|FnM+p|⋯∏p∈Rq|γ−1bk−2|FnM+p|∏p∈Rq|b1−10|HnM+p|k⋯∏p∈Rq|γ−1bk−2|HnM+p|k μ(W∩Rq)μ(Rq) =∏p∈Rq|b1−10|E1,nM+p|⋯∏p∈Rq|γ−1bk−2|Ek−1,nM+p|∏p∈Rq|b1−10|FnM+p|⋯∏p∈Rq|γ−1bk−2|FnM+p| ≥D−γ.

Because the are disjoint, we have

 μ(W)=μ(⨆q(W∩Rq))≥D−γμ(⨆qRq)=D−γμ(I(k)).

We finish by taking .

###### Proposition 3.2.

Let be a rank-one transformation such that there is a sequence indexing positive integers such that

 |Snm(znm)|=δnm|Hnm| and |Snm(znm+1)|=δnm|Hnm|

for a sequence satisfying . Then all -fold products of of the form:

 (3) T×⋯×Tt times×T−1×⋯×T−1k−t times ,

with , are ergodic.

###### Proof.

Assume that (this is without loss of generality, because the -fold product of is ergodic if and only if it is for ). By Proposition 2.3, the result follows if we show that for any , , and (with the base of ), there exists a such that to at least of the -tuples , we can associate unique -tuples satisfying for and for . Set a vector . For ease of notation, reorder the indices so that all satisfying fall in the range , for . Then we may assume by adding to all equations that , for all satisfying or , and for all other .

Let and be high enough such that there exist tuples satisfying the following properties through ; note that the indices satisfying these conditions are necessarily distinct.

 (4) For all ℓ∈{1,…,s−1}: ℓ: aℓ′,nm∈Snm(znm) for all ℓ′≠ℓ and aℓ,nm∈Snm(znm+1) for at least bℓ indices m∈{M,…,M′} For all ℓ∈{s,…,t−1}: ℓ: aℓ′,nm∈Snm(znm+1) for all % ℓ′≠ℓ and aℓ,nm∈Snm(znm) for at least −bℓ indices m∈{M,…,M′} For all ℓ∈{t,…,k−1}: ℓ: aℓ′,nm∈Snm(znm+1) for all % ℓ′≠ℓ and aℓ,nm∈Snm(znm)−znm for at least bℓ indices m∈{M,…,M′}.

For such a tuple, we can uniquely assign a complementary tuple meeting the product ergodicity condition of Proposition 2.3. For all , do the following: for the first indices satisfying , set and . Repeat similarly for and but replace the differences according to index (e.g. when is chosen to be in , set ). Then, everywhere has been assigned but has not, choose such that (for ) or (for ). Elsewhere, set .

Now we use Lemma 3.1 to show that there is a set of size such that the assignment of rectangles to rectangles on is unique. In the statement of Lemma 3.1, take to be the -tuple in meeting the condition relevant for in (4). For instance, is the subset of of tuples with . Also, take

 Fnm=(Snm(znm)∪Snm(znm+1)∪Snm(znm)−znm∪Snm(znm+1)−znm−1)k.

Then , and . Lemma 3.1 implies we can use .

Now suppose that and are assigned to the same rectangle. It must be the case that every time is assigned a , we have . If in this stage, then we have , implying by definition of that meets conditions through in indices strictly below . But everywhere that we must have , and in indices , does not meet conditions through (else, we would not have ). This contradicts that meets conditions in the first indices in which it is in . So , i.e.  and belong to the same partition block in Lemma 3.1. Thus, it is straightforward to check by the definition of and the algorithm assigning rectangles to rectangles that . Proposition 2.3 finishes, with given by .

###### Corollary 3.3.

If for all , then has infinite symmetric ergodic index.

In the case of infinite rank-one integer actions, Proposition 3.2 implies claim 1 of Theorem 0.1 in [7] (consider ). Other well known examples, such as the main construction in [3] and the infinite measure Chacon transformation with or more cuts, satisfy the conditions of Corollary 3.3 to obtain infinite ergodic index. To address the notion of mixing for rank-one transformations, we require the following lemma. Its proof is similar to that of Claim 6, [7]. We say that is Koopman mixing if for all sets of finite measure, ; in the finite measure preserving case this is equivalent to the spectral characterization of (strong) mixing, and in infinite measure it is also known as zero-type.

###### Lemma 3.4.

Let be a rank-one transformation with height sets chosen such that with and , and such that if with and , then . If adds at least spacers on the right subcolumn for each , then it is Koopman mixing.

###### Proof.

Let be a collection of levels in column . For any , consists of copies of indexed by . We claim that for satisfying: we have

 (5) μ(Tm(F)∩F)≤max{|Hn|−1,δn}μ(F)

Note that such an ensures that a copy of in cannot intersect with itself. Observe that for such (by addition of the absorbing spacers on the right subcolumn)

 (6) μ(Tm(F)∩F)≤μ(F)∣∣(x,y)∈H2n:(D(F,n)+x+m)∩D(F,n)+y≠∅∣∣|Hn|.

If one of is in , then the condition on implies that for all , so the right side of (6) can be bounded by . On the other hand, the maximal intersection between -indexed copies of has size . It follows that for such an , , so (5) holds. As , is mixing for collections of levels, whence mixing for arbitrary sets (see e.g. Theorem 9.6 in [12]). ∎

With the use of Proposition 3.2, we are able to answer Question 1 of Danilenko in [7]:

###### Theorem 3.5.

Fix any and (including ). Then there exists a Koopman mixing rank-one transformation such that all -fold products of the form (3) are ergodic, has ergodic index , and has conservative index .

###### Proof.

Let to start. In the previous construction, take to be the odd numbers. For every even , take to be a set of elements satisfying (i.e. if the difference of differences is nonzero, then it is large).

For ease of notation, let . For every odd , choose and such that . This is a stronger version of condition (2-1) in [7], and similar to the restriction discussed in Remark 1, [4]. Add spacers on the rightmost subcolumn for every , and choose such that but . Lastly, pick such that but . Throughout, we let denote the set of -tuples .

First, we argue that is not conservative. By selection of and , there exists an so high and a level of such that

 ∣∣( Hp+1i∖Sp+1i+Hp+1i+1+Hp+1i+2∖Sp+1i+2+⋯) ∩(Hp+1i∖diagp+1(Hi)+⋯+Hp+1j−1∖diagp+1(Hj−1)∣∣ >ε|D(I,j)|p+1

for all . Assume that is an element of the above set in . If there is a complementary descendant tuple such that , there exists a highest such that are not all . Fix an such that this is true; by assumption, there is another such that . Since , the inequality

 |aℓ,q−dℓ,q−aℓ′,q+d