aa r X i v : . [ m a t h . D S ] F e b OVER RECURRENCE FOR MIXING TRANSFORMATIONS
TERRENCE M. ADAMSA
BSTRACT . We show that every invertible strong mixing transforma-tion on a Lebesgue space has strictly over-recurrent sets. Also, we givean explicit procedure for constructing strong mixing transformations withno under-recurrent sets. This answers both parts of the first questionposed in [2].We define ǫ -over-recurrence and show that given ǫ > , any ergodicmeasure preserving invertible transformation (including discrete spec-trum) has ǫ -over-recurrent sets of arbitrarily small measure. Discretespectrum transformations and rotations do not have over-recurrent sets,but we construct a weak mixing rigid transformation with strictly over-recurrent sets.
1. I
NTRODUCTION
We answer a two-part question posed in [2]. It is the first question raisedin [3] on page 50.
Question 1.1. “Is it true that for any invertible mixing measurepreserving system ( X, B , µ, T ) there exists A ∈ B with µ ( A ) > such that for all n = 0 , µ ( A ∩ T n A ) < µ ( A ) ?How about the reverse inequality µ ( A ∩ T n A ) > µ ( A ) ” The answer is different for each part. Respectively, the answers are ”no”,and ”yes”. One of the key differences is a basic lemma on set intersec-tions which is presented in Lemma 4.2. However, this lemma alone is notsufficient, and in particular, this lemma is pointed out in [3]. In the nextsection, we prove that the answer to the second part is ”yes”. This is doneby constructing a set of positive measure such that the given mixing trans-formation mixes the set slowly. This answers the same question raised in[4].
Date : March 4, 2019.2010
Mathematics Subject Classification.
Primary 37A25; Secondary 28D05.
Key words and phrases. mixing, Khitchine recurrence, over-recurrence, under-recurrence, singular spectrum.
The notions of over-recurrent, under-recurrent, strictly over-recurrent andstrictly under-recurrent are defined in [4] as a means for addressing Ques-tion 1.1 and similar questions. We expand these definitions to include theweaker notion of ǫ -over-recurrent and ǫ -under-recurrent. See section 2 fordefinitions. It is straight-forward to show that any partially rigid transforma-tion has no under-recurrent set. The class of partially rigid transformationsis larger than the class of rigid transformations and was first introduced byN. Friedman in [6]. As a preliminary result, we give a short proof thatany discrete spectrum transformation (or rotation) does not have an over-recurrent set. However, we show that given an invertible ergodic measurepreserving transformation T and ǫ > , T has ǫ -over-recurrent sets witharbitrarily small measure. Also, we construct a rigid weak mixing transfor-mation that has a strictly over-recurrent set. The question of whether everyweak mixing transformation has an over-recurrent set remains open.To answer the first part of Question 1.1, we give a general procedure forconstructing a strong mixing transformation from an input mixing transfor-mation and an arbitrary rigid transformation. We gradually diminish theeffects of the rigid transformation, but in the process, build a strong mixingtransformation that acts like a rigid transformation on a shrinking part of themeasure space. We use a technique from [1] to produce this slow mixingtransformation.Finally, in the last section, we point out that the same construction forproducing a slow mixing transformation can be used to construct a strongmixing transformation with singular spectrum from any strong mixing trans-formation. Thus, any strong mixing transformation can be multiplexed withany rigid transformation to produce a transformation that is mixing of all or-ders. 2. P RELIMINARIES
All transformations are assumed to be invertible, ergodic and measurepreserving on a fixed Lebesgue probability space ( X, B , µ ) , and all sets areassumed to be measurable. Let IN = { , , . . . } be the natural numbers and Z the set of integers. The following definitions are expanded from [4]. Definition 2.1.
Let A be a measurable set such that < µ ( A ) < .(1) Set A is over-recurrent if µ ( T n A ∩ A ) ≥ µ ( A ) for n ∈ Z .(2) Set A is under-recurrent if µ ( T n A ∩ A ) ≤ µ ( A ) for n ∈ IN .(3) Set A is strictly over-recurrent if µ ( T n A ∩ A ) > µ ( A ) for n ∈ Z .(4) Set A is strictly under-recurrent if µ ( T n A ∩ A ) < µ ( A ) for n ∈ IN .(5) Set A is ǫ -over-recurrent if µ ( T n A ∩ A ) > (1 − ǫ ) µ ( A ) for n ∈ Z .(6) Set A is ǫ -under-recurrent if µ ( T n A ∩ A ) < (1+ ǫ ) µ ( A ) for n ∈ IN . VER RECURRENCE FOR MIXING TRANSFORMATIONS 3
These definitions are motivated by the Khintchine recurrence theorem[9]. It was shown in [4] that there exist mixing transformations with nounder-recurrent sets. Also, under-recurrent functions are defined, and it isshown that any transformation with singular maximal spectral type has nounder-recurrent function. While we give a general construction of mix-ing transformations with no under-recurrent sets, our main results concern(strictly) over-recurrent sets. All results were obtained independently of [4].3. O
VER - RECURRENT S ETS
This section focuses on results related to (strictly) over-recurrent sets.First, we prove that any strong mixing transformation has a strictly over-recurrent set.
Theorem 3.1.
Let T be an invertible mixing transformation on a Lebesgueprobability space. Then T has strictly over-recurrent sets A of arbitrarilysmall measure. In particular, µ ( T n A ∩ A ) > µ ( A ) for all n ∈ Z . We use the following lemma in the construction of the over-recurrentsets. For completeness, a proof is included.
Lemma 3.2.
Let a > be a real number. Define a i = ai ( i +1) for i ∈ IN .Then for k ∈ IN , ( k X i =1 a i ) + ∞ X i = k +2 a i > a (cid:16) k (1 − a ) + (1 − a ) a ( k + 1)( k + 2) (cid:17) . Proof:
First, note that ∞ X i =1 a i = a ∞ X i =1 ( 1 i − i + 1 ) = a. T. M. ADAMS
Thus, ( k X i =1 a i ) + ( ∞ X i = k +2 a i ) = ( a − ak + 1 ) + ak + 2= a − a k + 1 + a ( k + 1) + ak + 2 > a − a k + 1 + a ( k + 1)( k + 2) + ak + 2= a + − a ( k + 2) + a + a ( k + 1)( k + 1)( k + 2)= a + k ( a − a ) + ( a − a )( k + 1)( k + 2)= a (cid:16) k (1 − a ) + (1 − a ) a ( k + 1)( k + 2) (cid:17) . Proof of Theorem 3.1:
Let a ∈ IR be such that < a < and let a i = ai ( i +1) for i ∈ IN. For j ∈ IN, choose ǫ j > and decreasing such that (1 − ǫ j ) (cid:18) j (1 − a ) + (1 − a ) a ( j + 1)( j + 2) (cid:19) > . (1)We will define an infinite sequence A i of disjoint measurable sets such that µ ( A i ) = a i for i ∈ IN, and A = S ∞ i =1 A i . Let A be any set with measure a/ . Since T is mixing, there exists N ∈ IN such that for | n | ≥ N , | µ ( T n A ∩ A ) − µ ( A ) | < ǫ µ ( A ) . Choose m ∈ IN such that m > max { ǫ a , N } . Let B be the base of a Rohklin tower of height m such that µ ( m − [ i =0 T i B ) > − ǫ . Choose a subset I ⊂ B such that the set A = { T i x : x ∈ I , ≤ i < m , T i x / ∈ A } satisfies µ ( A ) = a . Note that for i ∈ IN such that | i | < N , µ ( T i A ∩ A ) > µ ( A ) − m > µ ( A ) − ǫ a = (1 − ǫ ) µ ( A ) . See the appendix for a visual representation of A and A . VER RECURRENCE FOR MIXING TRANSFORMATIONS 5
We repeat this inductively. Given A , A , . . . A k , choose A k +1 in thefollowing manner. Let C k = ∪ ki =1 A i . Since T is mixing, there exists N k > N k − such that for | n | ≥ N k , | µ ( T n C k ∩ C k ) − µ ( C k ) | < ǫ k µ ( C k ) . Choose m k ∈ IN such that m k > max { ǫ k a k +1 , N k } . Let B k be the base of a Rohklin tower of height m k such that µ ( m k − [ i =0 T i ( B k )) > − ǫ k . Choose a subset I k ⊂ B k such that the set A k +1 = { T i x : x ∈ I k , ≤ i < m k , T i x / ∈ C k } satisfies µ ( A k +1 ) = a k +1 . The set A k +1 has the property that for i ∈ INsuch that | i | < N k , µ ( T i A k +1 ∩ A ) > (1 − ǫ k ) µ ( A k +1 ) . Now, we show the set A = S ∞ i =1 A i is over-recurrent. If n is a naturalnumber such that | n | < N , then µ ( T n ( ∞ [ k =2 A k ) ∩ A ) > (1 − ǫ ) µ ( ∞ [ k =2 A k ) = 12 (1 − ǫ ) a>
12 (1 − − a a = ( 13 + a a > a . Let k ∈ IN, and n ∈ IN be such that N k ≤ n < N k +1 . The set A is a disjoint union of the following three sets: C k , A k +1 , S ∞ i = k +2 A i . Forconvenience, set C k, = C k , C k, = A k +1 and C k, = S ∞ i = k +2 A i . Thus, µ ( T n A ∩ A ) = X i =1 µ ( T n C k, ∩ C k,i ) + X i =2 µ ( T n C k,i ∩ A ) > (1 − ǫ k ) µ ( C k, ) + (1 − ǫ k +1 ) µ ( C k, ) ≥ (1 − ǫ k ) (cid:18) ( a − ak + 1 ) + ak + 2 (cid:19) > (1 − ǫ k ) a (cid:18) k (1 − a ) + 1 − aa ( k + 1)( k + 2) (cid:19) , by Lemma 3.2, > a , by ( 1 ) . T. M. ADAMS
Note, the method used for choosing A i can be used to show that mixingtransformations have no uniform rate over all measurable sets. Given anysequence δ i → , there exist parameters ǫ i , N i , m i and A i such that lim n →∞ δ n µ ( T n A ∩ A ) − µ ( A ) = 0 . This is already well known to be true, and follows from a general argumentof Krengel [8] on the lack of uniform rates for the ergodic theorem.3.1.
Over-recurrence for non-mixing transformations.
The previous re-sult can be used to construct a rigid weak mixing transformation that has astrictly over-recurrent set. First, we show that any discrete spectrum trans-formation does not have an over-recurrent set.
Proposition 3.3. If T has discrete spectrum, then T has no over-recurrentsets. Proof:
Let A be any measurable set such that < µ ( A ) < . Since T hasdiscrete spectrum, there exist a sequence of refining towers of heights h n and integers k n ≥ such that h n +1 = k n h n . Choose m such that the towerof height h m has a union J of levels that approximates A . In particular,choose δ and m such that δ < − µ ( A ) and µ ( A △ J ) < δµ ( A )4 . Note that µ ( T ih m J ∩ J ) = µ ( J ) for all i ∈ Z . Thus, µ ( T ih m A ∩ A ) > (1 − δ ) µ ( A ) > µ ( A ) for all i ∈ Z . By the L ergodic theorem, lim n →∞ h n h n − X i =0 µ ( T i A ∩ A ) = µ ( A ) . Since { ih m | i ∈ IN } forms a subsequence of positive density in IN, thenthere must exist i and j such that < j < h m and µ ( T ih m + j A ∩ A ) < µ ( A ) . A similar argument can be used to show that ergodic rotations do not haveover-recurrent sets.The following may seem a bit surprising intuitively, but it is not difficultto prove.
Proposition 3.4.
Given any invertible ergodic measure preserving trans-formation T and ǫ > , T has ǫ -over-recurrent sets of arbitrarily smallmeasure. VER RECURRENCE FOR MIXING TRANSFORMATIONS 7
Proof:
Let ǫ, δ > . Let S be a rank-one strong mixing transformation. ByTheorem 3.1, S has a strictly over-recurrent set A with measure less than δ . Choose a tower for S of height h and a union J of levels from the towerthat approximate A well, and such that the complement of the tower hasmeasure less than ǫµ ( A ) / . Also, assume µ ( A △ J ) < ǫ µ ( A ) . Choose a Rokhlin tower for T of height h such that the complement of thetower has measure less than ǫµ ( A ) / h . There is a one-to-one onto corre-spondence between the levels of the T tower and the levels of the S tower.Take the correspondence that preserves the order of the levels from top tobottom of the towers. The set J in the S tower matches a set K in the T tower. It is not difficult to prove that the set K is ǫ -over-recurrent for thetransformation T . Rigid weak mixing transformations with strictly over-recurrentsets.
We prove there exist rigid weak mixing transformations with strictlyover-recurrent sets.
Theorem 3.5.
There exist rigid weak mixing transformations T and sets A such that for all n ∈ Z , µ ( T n A ∩ A ) > µ ( A ) . Proof:
Let S be a rank-one mixing transformation such as Ornstein’s mix-ing rank-one, or the (Adams-Smorodinsky) staircase transformation. ByTheorem 3.1, there is an over-recurrent set A of arbitrarily small measure.By the technique used in Proposition 3.4, there exists a tower of height h such that the set A is ǫ/ -over-recurrent, even if we modify S to be discretespectrum from this point on. Similarly, we can cut this tower into r sub-columns of equal width, and stack to produce a rigid time (as r n → ∞ ).Resume the definition of the mixing transformation S similar to S . Thendefine a set A as in Theorem 3.1, such that iterates of A overlap itself fora long time (forward and backward in time). Since S is mixing, it will mix A ∪ A over time. Once this happens sufficiently well, then introduce an-other rigid time r . It will not disturb the near over-recurrence of A ∪ A .The error in the near over-recurrence can be forced to be much smaller thanthe size of set A . The set A is defined to be nearly fixed for a long periodof time compared to the last mixing times chosen for S , as it operates on A ∪ A .This is repeated inductively to produce a rigid weak mixing transforma-tion. The arguments used in Theorem 3.1 and Propostion 3.4 can be applied T. M. ADAMS here to show that the set A = S ∞ i =1 A i is strictly over-recurrent for the re-sulting transformation T . The transformation T may be defined as T = lim n →∞ S n . Although, each S n may be strong mixing, the limiting transformation T willbe rigid weak mixing, if r n → ∞ .
4. S
LOW S TRONG M IXING T RANSFORMATIONS
In this section, we prove the following theorem.
Theorem 4.1.
There exists a strong mixing transformation T such that forevery set A satisfying < µ ( A ) < , the following set is infinite: { n ∈ IN : µ ( T n A ∩ A ) − µ ( A ) > } . We use a technique from [1] to construct our example. In [1], a method isgiven for combining two transformations to produce a third ”multiplexed”transformation. In that paper, the two input transformations are a rigid er-godic transformation and a weak mixing transformation. The output trans-formation is a rigid weak mixing transformation. In this case, our inputtransformations are a strong mixing transformation and a rigid transforma-tion. The output is a strong mixing transformation.We use the following standard result from measure theory.
Lemma 4.2.
Let ( X, µ ) be a probability space. Given ǫ > and < α ≤ ,there exists N such that for any measurable sets A , A , . . . , A N satisfying µ ( A i ) = α for ≤ i ≤ N , there exist ≤ j < k ≤ N such that µ ( A j ∩ A k ) > α − ǫ. Proof: Z ( N X i =1 I A i ) dµ = X i = j µ ( A i ∩ A j ) + N X i =1 µ ( A i ) (2) ≥ ( N X i =1 µ ( A i )) = N α (3)Therefore, N X i = j µ ( A i ∩ A j ) ≥ α − αN (4)and we have our result. VER RECURRENCE FOR MIXING TRANSFORMATIONS 9
Mixing Counterexample.
The towerplex method was first defined insection 2 of [1]. The roles of the input transformations are different. In thiscase, we use S to represent the first input transformation which will be astrongly mixing transformation. The second input transformation will bea rigid transformation denoted by R . Thus, a sequence of transformations S n : Y n → Y n will be defined such that S n is isomorphic to S , and anothersequence R n : X n → X n such that R n is isomorphic to R . For each n ∈ IN, X n ∪ Y n = X and define T n ( x ) = (cid:26) R n ( x ) if x ∈ X n S n ( x ) if x ∈ Y n . Then the output transformation is defined by T ( x ) = lim n →∞ T n ( x ) for x ∈ X .Two main parameters are used to control the properties of T : s n = 12( n + 2) and r n = 12 . (5)The parameter s n represents the proportion of mass that transfers from Y n to X n at each stage. Similarly, r n represents the proportion of mass that trans-fers from X n to Y n at each stage. These settings cause lim n →∞ µ ( Y n ) = 1 and consequently lim n →∞ µ ( X n ) = 0 . Note, the fact that S n is mixing isnot sufficient to prove that T is mixing. On the other hand, the fact that T n is not ergodic does not prevent T from being ergodic. We are more carefulabout defining S n +1 based on S n and use a property called ”isomorphismchain consistency” to show that T is strongly mixing.This provides a general technique for constructing a slow strong mix-ing transformation from an arbitrary strong mixing transformation and anarbitrary rigid transformation. As in [1], we can have for each n ∈ IN, / ( n + 2) < µ ( X n ) < /n and ( n + 1) / ( n + 2) > µ ( Y n ) > ( n − /n .Also, since the measure of X n goes to zero slow enough, and the X n are ap-proximately independent, then X n will mix with any measurable set. Also, R n rigid on X n will cause T n to be approximately / ( n + 1) rigid on X . Inthis way, we can slow the rate of mixing, because T will resemble a rigidtransformation for arbitrarily long times on X n .4.2. Slow mixing from dissipating rigidity.
Lemma 4.2 will inform us onhow long S n should run before phasing in S n +1 to guarantee the intersection µ ( S in A ∩ A ) > (1 − δn ) µ ( A ) (6)for some i and δ . Let N be large enough to guarantee (6) holds for some i from a subset of at least N iterates. Let ρ k be a rigidity sequence for R n . Choose ρ , ρ , . . . , ρ N such that µ ( R ρ i n A ∩ A ) > (1 − δn ) µ ( A ) (7)for A ∈ P n . Using a similar approximation as in [1], then a rigidity condi-tion like (7) extends to all measurable sets A such that < µ ( A ) < . Thus,(6) and (7) together can be used to show that the following set is infinite: { n ∈ IN : µ ( T i A ∩ A ) − µ ( A ) > } .
5. M
IXING T OWERPLEX D ETAILS
Partition X into two equal sets X and Y (i.e. µ ( X ) = µ ( Y ) = 1 / ).Initialize R isomorphic to R and S isomorphic to S to operate on X and Y , respectively. Define T ( x ) = R ( x ) for x ∈ X and T ( x ) = S ( x ) for x ∈ Y . Produce Rohklin towers of height h with residual less than ǫ / for each of R and S . In particular, let I , J be the base of the R -towerand S -tower such that µ ( S h − i =0 R i I ) > / − ǫ ) and µ ( S h − i =0 S i J ) > / − ǫ ) . Let X ∗ = X \ S h − i =0 R i ( I ) and Y ∗ = Y \ S h − i =0 S i ( J ) bethe residuals for the R and S towers, respectively. Choose I ′ ⊂ I and J ′ ⊂ J such that µ ( I ′ ) = r µ ( I ) and µ ( J ′ ) = s µ ( J ) . Set X = X \ [ S h − i =0 R i ( I ′ )] ∪ [ S h − i =0 S i ( J ′ )] and Y = Y \ [ S h − i =0 S i ( J ′ )] ∪ [ S h − i =0 R i ( I ′ )] . We will define second stage transformations R : X → X and S : Y → Y . First, it may be necessary to add or subtract mea-sure from the residuals so that X is scaled properly to define R , and Y isscaled properly to define S .5.1. Tower Rescaling.
In the case where µ ( I ′ ) = µ ( J ′ ) , we give a pro-cedure for transferring measure between the towers and the residuals. Thisis done in order to consistently define R and S on the new inflated or de-flated towers. Let a = µ ( S h − i =0 R i I ) and b = h ( µ ( J ′ ) − µ ( I ′ )) . Let c be the scaling factor and d the amount of measure transferred between S h − i =0 S i ( J ′ ) and X ∗ . Thus, a + b − d = ca and / − a + d = c (1 / − a ) .The goal is to solve two unknowns d and c in terms of the other values.Hence, d = (1 − a ) b and c = 1 + 2 b .5.1.1. R Rescaling. If d > , define I ∗ ⊂ J ′ such that µ ( I ∗ ) = d/h . Let X ′ = X ∗ ∪ ( S h − i =0 R i ( I ∗ )) . If d = 0 , set X ′ = X ∗ . If d < , transfer mea-sure from X ∗ to the tower. Choose disjoint sets I ∗ (0) , I ∗ (1) , . . . , I ∗ ( h − contained in X ∗ such that µ ( I ∗ ( i )) = d/h . Denote I ∗ = I ∗ (0) . Begin bydefining µ measure preserving map α such that I ∗ ( i + 1) = α ( I ∗ ( i )) for i = 0 , , . . . , h − . In this case, let X ′ = X ∗ \ [ S h − i =0 I ∗ ( i )] . VER RECURRENCE FOR MIXING TRANSFORMATIONS 11 S Rescaling.
The direction mass is transferred depends on the signof b above. If d > , then µ ( J ′ ) > µ ( I ′ ) and mass is transferred from theresidual Y ∗ to the S -tower. Choose disjoint sets J ∗ (0) , J ∗ (1) , . . . , J ∗ ( h − contained in Y ∗ such that µ ( J ∗ ( i )) = d/h . Denote J ∗ = J ∗ (0) . Beginby defining µ measure preserving map β such that J ∗ ( i + 1) = β ( J ∗ ( i )) for i = 0 , , . . . , h − . In this case, let Y ′ = Y ∗ \ [ S h − i =0 J ∗ ( i )] . If d = 0 , set Y ′ = Y ∗ . If d < , transfer measure from the S -tower tothe residual Y ∗ . Define J ∗ ⊂ J \ J ′ such that µ ( J ∗ ) = d/h . Let Y ′ = Y ∗ ∪ ( S h − i =0 S i ( J ∗ )) .Note, if d = 0 , then both ǫ and µ ( X ∗ ) may be chosen small enough(relative to r ) to ensure the following solutions lead to well-defined setsand mappings. For subsequent stages, assume ǫ n is chosen small enough toforce well-defined rescaling parameters, transfer sets and mappings R n , S n .5.2. Stage 2 Construction.
We have specified three cases: d > , d = 0 and d < . The case d = 0 , can be handled along with the case d > . Thisgives two essential cases. Note the case d < is analogous to the case d > , with the roles of R and S reversed. However, due to a key distinctionin the handling of the R -rescaling and the S -rescaling, it is important toclearly define R and S in both cases. Case 5.1 ( d ≥ ) . Define τ : X ′ → X ∗ as a measure preserving mapbetween normalized spaces ( X ′ , B ∩ X ′ , µµ ( X ′ ) ) and ( X ∗ , B ∩ X ∗ , µµ ( X ∗ ) ) .Extend τ to the new tower base, τ : [ I \ I ′ ] ∪ [ J ′ \ I ∗ ] → I such that τ preserves normalized measure between µµ ([ I \ I ′ ] ∪ [ J ′ \ I ∗ ]) and µµ ( I ) . Define τ on the remainder of the tower consistently such that τ ( x ) = (cid:26) R i ◦ τ ◦ R − i ( x ) if x ∈ R i ( I \ I ′ ) for ≤ i < h R i ◦ τ ◦ S − i ( x ) if x ∈ S i ( J ′ \ I ∗ ) for ≤ i < h Define R : X → X as R = τ − ◦ R ◦ τ . Note R ( x ) = (cid:26) S ( x ) if x ∈ S i ( J ′ \ I ∗ ) for ≤ i < h − R ( x ) if x ∈ R i ( I \ I ′ ) for ≤ i < h − Clearly, R is isomorphic to R and R .Define ψ : Y ′ → Y ∗ as a measure preserving map between normalizedspaces ( Y ′ , B ∩ Y ′ , µµ ( Y ′ ) ) and ( Y ∗ , B ∩ Y ∗ , µµ ( Y ∗ ) ) . Extend ψ to the newtower base, ψ : [ J \ J ′ ] ∪ J ∗ ∪ I ′ → J such that ψ preserves normalized measure between µµ ([ J \ J ′ ] ∪ J ∗ ∪ I ′ ) and µµ ( J ) . Define ψ on the remainder of the tower consistently such that ψ ( x ) = S i ◦ ψ ◦ S − i ( x ) if x ∈ S i ( J \ J ′ ) for ≤ i < h S i ◦ ψ ◦ R − i ( x ) if x ∈ R i ( I ′ ) for ≤ i < h β i ◦ ψ ◦ β − i ( x ) if x ∈ J ∗ ( i ) for ≤ i < h In this case, define S : Y → Y such that S = ψ − ◦ S ◦ ψ . Note S ( x ) = R ( x ) if x ∈ R i I ′ for ≤ i < h − S ( x ) if x ∈ S i ( J \ J ′ ) for ≤ i < h − β ( x ) if x ∈ J ∗ ( i ) for ≤ i < h − ψ − ◦ S ◦ ψ ( x ) if x ∈ Y ′ ∪ S h − ( J \ J ′ ) ∪ R h − I ′ ∪ β h − J ∗ and S is isomorphic to S and S . Case 5.2 ( d < ) . Define τ : X ′ → X ∗ as a measure preserving mapbetween normalized spaces ( X ′ , B ∩ X ′ , µµ ( X ′ ) ) and ( X ∗ , B ∩ X ∗ , µµ ( X ∗ ) ) .Extend τ to the new tower base, τ : [ I \ I ′ ] ∪ I ∗ ∪ J ′ → I such that τ preserves normalized measure between µµ ([ I \ I ′ ] ∪ I ∗ ∪ J ′ ) and µµ ( I ) . Define τ on the remainder of the tower consistently such that τ ( x ) = R i ◦ τ ◦ R − i ( x ) if x ∈ R i ( I \ I ′ ) for ≤ i < h R i ◦ τ ◦ S − i ( x ) if x ∈ S i ( J ′ ) for ≤ i < h α i ◦ τ ◦ α − i ( x ) if x ∈ I ∗ ( i ) for ≤ i < h In this case, define R : X → X such that R ( x ) = S ( x ) if x ∈ S i J ′ for ≤ i < h − R ( x ) if x ∈ R i ( I \ I ′ ) for ≤ i < h − α ( x ) if x ∈ I ∗ ( i ) for ≤ i < h − τ − ◦ R ◦ τ ( x ) if x ∈ X ′ ∪ R h − ( I \ I ′ ) ∪ S h − J ′ ∪ α h − I ∗ Clearly, R is isomorphic to R and R .Define ψ : Y ′ → Y ∗ as a measure preserving map between normalizedspaces ( Y ′ , B ∩ Y ′ , µµ ( Y ′ ) ) and ( Y ∗ , B ∩ Y ∗ , µµ ( Y ∗ ) ) . Extend ψ to the newtower base, ψ : [ J \ ( J ′ ∪ J ∗ )] ∪ I ′ → J VER RECURRENCE FOR MIXING TRANSFORMATIONS 13 such that ψ preserves normalized measure between µµ ([ J \ ( J ′ ∪ J ∗ )] ∪ I ′ ) and µµ ( J ) . Define ψ on the remainder of the tower consistently such that ψ ( x ) = (cid:26) S i ◦ ψ ◦ S − i ( x ) if x ∈ S i ( J \ [ J ′ ∪ J ∗ ]) for ≤ i < h S i ◦ ψ ◦ R − i ( x ) if x ∈ R i ( I ′ ) for ≤ i < h Define S : Y → Y such that S = ψ − ◦ S ◦ ψ . Note S ( x ) = (cid:26) R ( x ) if x ∈ R i ( I ′ ) for ≤ i < h − S ( x ) if x ∈ S i ( J \ [ J ′ ∪ J ∗ ]) for ≤ i < h − Transformation S is isomorphic to S and S . Define T as T ( x ) = (cid:26) R ( x ) if x ∈ X S ( x ) if x ∈ Y Clearly, neither T nor T are ergodic. For T , X and Y are ergodic com-ponents, and X , Y are ergodic components for T .5.3. General Multiplexing Operation.
For n ≥ , suppose that R n and S n have been defined on X n and Y n respectively. Construct Rohklin towersof height h n for each R n and S n , and such that I n is the base of the R n tower, J n is the base of the S n tower, and µ ( S h n − i =0 R in I n ) + µ ( S h n − i =0 S in J n ) > − ǫ n . Let I ′ n ⊂ I n be such that µ ( I ′ n ) = r n µ ( I n ) . Similarly, suppose J ′ n ⊂ J n such that µ ( J ′ n ) = s n µ ( J n ) .We define R n +1 and S n +1 by switching the subcolumns { I ′ n , R n ( I ′ n ) , R n ( I ′ n ) , . . . , R h n − n ( I ′ n ) } and { J ′ n , S n ( J ′ n ) , S n ( J ′ n ) , . . . , S h n − n ( J ′ n ) } . Let X n +1 = [ h n − [ i =0 R in ( I n \ I ′ n )] ∪ [ h n − [ i =0 S in J ′ n ] ∪ [ X n \ h n − [ i =0 R in I n ] Y n +1 = [ h n − [ i =0 S in ( J n \ J ′ n )] ∪ [ h n − [ i =0 R in I ′ n ] ∪ [ Y n \ h n − [ i =0 S in J n ] . As in the initial case, it may be necessary to transfer measure between eachcolumn and its respective residual. We can follow the same algorithm asabove, and define maps τ n , α n , ψ n and β n . Thus, we get the following defi-nitions: Case 5.3 ( d ≥ ) . τ n ( x ) = (cid:26) R in ◦ τ n ◦ R − in ( x ) if x ∈ R in ( I n \ I ′ n ) for ≤ i < h n R in ◦ τ n ◦ S − in ( x ) if x ∈ S in ( J ′ n \ I ∗ ) for ≤ i < h n R n +1 ( x ) = S n ( x ) if x ∈ S in ( J ′ n \ I ∗ n ) for ≤ i < h n − R n ( x ) if x ∈ R in ( I n \ I ′ n ) for ≤ i < h n − τ − n ◦ R n ◦ τ n ( x ) if x ∈ X ′ n ∪ R h n − n ( I n \ I ′ n ) ∪ S h n − n ( J ′ n \ I ∗ n ) and R n +1 = τ − n ◦ R n ◦ τ n . ψ n ( x ) = S in ◦ ψ n ◦ S − in ( x ) if x ∈ S in ( J n \ J ′ n ) for ≤ i < h n S in ◦ ψ n ◦ R − in ( x ) if x ∈ R in ( I ′ n ) for ≤ i < h n β in ◦ ψ n ◦ β − in ( x ) if x ∈ J ∗ n ( i ) for ≤ i < h n S n +1 ( x ) = R n ( x ) if x ∈ R in I ′ n for ≤ i < h n − S n ( x ) if x ∈ S in ( J n \ J ′ n ) for ≤ i < h n − β n ( x ) if x ∈ J ∗ n ( i ) for ≤ i < h n − ψ − n ◦ S n ◦ ψ n ( x ) if x ∈ Y ′ n ∪ S h n − n ( J n \ J ′ n ) ∪ R h n − n I ′ n ∪ β h n − n J ∗ n and S n +1 = ψ − n ◦ S n ◦ ψ n . Case 5.4 ( d < ) . τ n ( x ) = R in ◦ τ n ◦ R − in ( x ) if x ∈ R in ( I n \ I ′ n ) for ≤ i < h n R in ◦ τ n ◦ S − in ( x ) if x ∈ S in ( J ′ n ) for ≤ i < h n α in ◦ τ n ◦ α − in ( x ) if x ∈ I ∗ n ( i ) for ≤ i < h n R n +1 ( x ) = S n ( x ) if x ∈ S in J ′ n for ≤ i < h n − R n ( x ) if x ∈ R in ( I n \ I ′ n ) for ≤ i < h n − α n ( x ) if x ∈ I ∗ n ( i ) for ≤ i < h n − τ − n ◦ R n ◦ τ n ( x ) if x ∈ X ′ n ∪ R h n − n ( I n \ I ′ n ) ∪ S h n − n J ′ n ∪ α h n − n I ∗ n and R n +1 = τ − n ◦ R n ◦ τ n . ψ n ( x ) = (cid:26) S in ◦ ψ n ◦ S − in ( x ) if x ∈ S in ( J n \ [ J ′ n ∪ J ∗ n ]) for ≤ i < h n S in ◦ ψ n ◦ R − in ( x ) if x ∈ R in ( I ′ n ) for ≤ i < h n S n +1 ( x ) = R n ( x ) if x ∈ R in ( I ′ n ) for ≤ i < h n − S n ( x ) if x ∈ S in ( J n \ [ J ′ n ∪ J ∗ n ]) for ≤ i < h n − ψ − n ◦ S n ◦ ψ n ( x ) if x ∈ Y ′ n ∪ S h n − n ( J n \ [ J ′ n ∪ J ∗ n ]) ∪ R h n − n ( I ′ n ) and S n +1 = ψ − n ◦ S n ◦ ψ n . VER RECURRENCE FOR MIXING TRANSFORMATIONS 15
The Limiting Transformation.
Define the transformation T n +1 : X n +1 ∪ Y n +1 → X n +1 ∪ Y n +1 such that T n +1 ( x ) = (cid:26) R n +1 ( x ) if x ∈ X n +1 S n +1 ( x ) if x ∈ Y n +1 The set where T n +1 = T n is determined by the top levels of the Rokhlintowers, the residuals and the transfer sets. Note the transfer set has measure d . Since this set is used to adjust the size of the residuals between stages,it can be bounded below a constant multiple of ǫ n . Thus, there is a fixedconstant κ , independent of n , such that T n +1 ( x ) = T n ( x ) except for x ina set of measure less than κ ( ǫ n + 1 /h n ) . Since P ∞ n =1 ( ǫ n + 1 /h n ) < ∞ , T ( x ) = lim n →∞ T n ( x ) exists almost everywhere, and preserves normalizedLebesgue measure. Without loss of generality, we may assume κ and h n are chosen such that if E n = { x ∈ X | T n +1 ( x ) = T n ( x ) } then µ ( E n ) < κǫ n for n ∈ IN. In the following section, additional structureand conditions are implemented to ensure that T inherits properties from R and S , and is also ergodic.For the remainder of this paper, assume the parameters are chosen suchthat(1) lim n →∞ s n = 0 ;(2) P ∞ n =1 r n = P ∞ n =1 s n = ∞ ;(3) lim n →∞ µ ( X n ) = 0 ;(4) P ∞ n =1 ǫ n < ∞ .5.5. Isomorphism Chain Consistency.
Suppose S is a strong mixing trans-formation on ( Y, B , µ ) . We will use the multiplexing procedure defined inthe previous section to produce a ”slow” mixing transformation T . Let µ n be normalized Lebesgue probability measure on Y n . i.e. µ n = µ/µ ( Y n ) .For n ∈ IN, let P n be a refining sequence of finite partitions whichgenerates the sigma algebra. By refining P n further if necessary, assume X n , Y n , X ∗ n , Y ∗ n ∈ P n . Also, assume R in ( I ′ n ) , R in ( I n \ I ′ n ) , S in ( J ′ n ) , S in ( J n \ J ′ n ) are elements of P n for ≤ i < h n . Finally, assume for ≤ i < h n − ,if p ∈ P n and p ⊂ R in ( I n ) then R n ( p ) ∈ P n . Likewise, assume for ≤ i < h n − , if p ∈ P n and p ⊂ S i ( J n ) then S n ( p ) ∈ P n . Previ-ously, we required that ψ n map certain finite orbits from the S n and R n towers to a corresponding orbit in the S n +1 tower. In this section, furtherregularity is imposed on ψ n relative to P n to ensure dynamical properties of S n are inherited by S n +1 .Let P ′ n = { p ∈ P n | p ⊂ S h n − i =0 S in ( J n \ J ′ n ) } . For each of the followingthree cases, impose the corresponding restriction on ψ n : (1) for d = 0 and p ∈ P ′ n , ψ n is the identity map (i.e. ψ n ( p ) = p );(2) for d > and p ∈ P ′ n , ψ n ( p ) ⊂ p ;(3) for d < and p ∈ P ′ n , p ⊂ ψ n ( p ) .This can be accomplished by uniformly distributing the appropriate massfrom the sets S in ( J ∗ n ) using ψ n . Note that ψ n either preserves Lebesguemeasure in the case d = 0 , or ψ n contracts sets relative to Lebesgue measurein the case d > , or it inflates measure in the case d < . In all three cases,for p ∈ P ′ n , µ ( p ) µ ( ψ n ( p )) = µ ( Y n +1 ) µ ( Y n ) . It is straightforward to verify for any set A measurable relative to P ′ n , µ ( A △ ψ n A ) < | µ ( Y n +1 ) µ ( Y n ) − | . The properties of ψ n allow approximation of S n +1 by S n indefinitely overtime. This is needed to establish mixing for the limiting transformation T .Since each S n is strongly mixing on Y n , then for all A, B ∈ P ′ n , lim i →∞ µ n ( A ∩ S in B ) = µ n ( A ) µ n ( B ) . Prior to establishing strong mixing, we prove a lemma which is part of asimilar lemma shown in [1]. For p ∈ P ′ n , µ ( p ) µ ( ψ n ( p )) = µ ( Y n +1 ) µ ( Y n ) . It is straightforward to verify for any set A measurable relative to P ′ n , µ ( A △ ψ n A ) = µ ( A ) − µ ( ψ n ( A )) ≤ µ ( ψ n A )[ µ ( Y n +1 ) µ ( Y n ) −
1] = µ ( ψ n A ) µ ( Y n ) [ µ ( Y n +1 ) − µ ( Y n )] . and for any measurable set C ⊂ Y n , | µ ( ψ − n C ) − µ ( C ) | < | µ ( Y n +1 ) µ ( Y n ) − | . These two properties are used in the following lemma to show S n +1 inheritsdynamical properties from S n indefinitely over time. Let Q n = { ψ n ( p ) : p ∈ P ′ n } . Lemma 5.5.
Suppose δ > and n ∈ IN is chosen such that ǫ n + µ ( X n ) < δ . Then for
A, B ∈ Q n and i ∈ IN , | µ ( S in +1 A ∩ B ) − µ ( A ) µ ( B ) | < | µ ( S in A ∩ B ) − µ ( A ) µ ( B ) | + δ. VER RECURRENCE FOR MIXING TRANSFORMATIONS 17
Proof.
For
A, B ∈ Q n , let A ′ = ψ − n A and B ′ = ψ − n B . Thus, µ ( A ′ △ A ) = µ ( ψ − n ( A \ ψ n A )) < δ/ and µ ( B ′ △ B ) < δ . By applying the triangleinequality several times, we get the following approximation: | µ ( S in +1 A ∩ B ) − µ ( S in A ∩ B ) |≤ | µ ( S in +1 A ′ ∩ B ′ ) − µ ( S in A ∩ B ) | + δ µ ( ψ − n S in ψ n A ′ ∩ B ′ ) − µ ( S in A ∩ B ) | + δ µ ( ψ − n ( S in ψ n A ′ ∩ ψ n B ′ )) − µ ( S in A ∩ B ) | + δ | µ ( ψ − n ( S in A ∩ B )) − µ ( S in A ∩ B ) | + δ < δ . Similarly, | µ ( S in +1 F ∩ F ) − µ ( S in F ∩ F ) | < δ . Therefore, | µ ( S in +1 A ∩ B ) − µ ( A ) µ ( B ) | < | µ ( S in A ∩ B ) − µ ( A ) µ ( B ) | + δ. (cid:3)
6. T
OWERPLEXES WITH SINGULAR SPECTRUM If Φ is the space of ergodic measure preserving transformations on a sep-arable probability space, then the tower multiplexing operation defines amapping M : Φ × Φ → Φ . The mapping also depends on a collection of parameters P . Thus, we maywrite T = M ( R, S, P ) to represent the multiplexed transformation T pro-duced from transformations R and S . In [1], the transformation R is ergodicand rigid, and S is weak mixing. In particular, S is set to the Chacon3 trans-formation, and the parameters are defined such that S is a ”dissipating”component. Given R ergodic with rigidity sequence ( ρ n ) ∞ n =1 , it is shownthere exists P such that T = M ( R, S, P ) is weak mixing with rigiditysequence ρ n .In this paper, we use the tower multiplexing technique to produce a trans-formation T = M ( S, R, P ) with continuous singular spectrum. Again, thesecond component transformation will be a dissipating component. How-ever, we flip the roles of R and S , so R (rigidity) is used in the secondcomponent. The parameter collection P includes sequences r n and s n . Asin [1], associate r n with R and s n with S . Associate X n with the second component transformation R , and Y n with S . Other parameters included in P are ǫ n and h n . We have the following main theorem. Theorem 6.1.
Let S be an invertible ergodic measure preserving transfor-mation with weak limit S m n → S as n → ∞ . There exist a rigid weakmixing transformation R , and parameter P such that T = M ( S, R, P ) is weak mixing with singular spectrum, and T m n → S . Prior to sketching a proof to the previous theorem, we will use the fol-lowing result from [5].
Proposition 6.2 (B. Fayad) . Let ( X, B , µ, T ) be an invertible ergodic mea-sure preserving system. If for any complex nonzero, mean zero function f ∈ L ( X, µ ) , there exists a measurable set E ⊂ X with µ ( E ) > , and astrictly increasing sequence ℓ n , such that for every x ∈ E , we have lim sup n →∞ n | n − X i =0 f ( T ℓ i x ) | > , then the maximal spectral type of the unitary operator associated to ( X, B , µ, T ) is singular. Proof of Theorem 6.1:
Let S be an invertible ergodic measure preservingtransformation with limit S = w ∗ − lim n →∞ S m n . We can define a rigidweak mixing transformation R such that the dissipating component R inthe multiplexed transformation T = M ( S, R, P ) will allow T to satisfy theprevious proposition. We still require that parameters r n and s n have thesame properties as in [1] except with roles reversed. In particular, r n for R satisfies r n = 1 / , and s n = 1 / n + 2) . This ensures that the base of R n , X n , satisfies lim n →∞ µ ( X n ) = 0 and P ∞ n =1 µ ( X n ) = ∞ with the X n approximately independent. The same technique to establish the rigiditysequence in [1] can be used to establish weak convergence to S along m n .Ergodicity and weak mixing may be established in a similar manner as in[1]. Singular spectrum is established using the previous proposition andthe fact that almost every point falls in a subset of X n infinitely often. Thetransformation R is defined such that ℓ i are ”strong” rigid times, and rigidmultiples of rigid times. Corollary 6.3.
Suppose S is an invertible strong mixing transformation.There exist an invertible rigid transformation R , and parameter P suchthat T = M ( S, R, P ) is strong mixing with singular spectrum. VER RECURRENCE FOR MIXING TRANSFORMATIONS 19
Corollary 6.4.
Suppose S is an invertible ergodic measure preserving trans-formation, and ( m n ) ∞ n =1 is a sequence such that the weak closure of { S m n : n ∈ IN } contains a countable set of limit points { S k : k ∈ IN } . Then thereexists a weak mixing transformation T with singular spectrum such that theweak closure of { T m n : n ∈ IN } contains the same countable set of limitpoints { S k : k ∈ IN } . Multiple mixing towerplexes.
In [7], it is shown that any mixingtransformation with singular spectrum is mixing of all orders. This impliesthe following corollary.
Corollary 6.5.
Given any strong mixing transformation S , there exist arigid transformation R , and parameter P such that T = M ( S, R, P ) is mixing of all orders. Question:
Given a mixing transformation S , is it possible to construct arigid transformation R and parameter P such that T = M ( S, R, P ) has thesame higher order mixing properties as S and has singular spectrum? Thiswould be sufficient to prove that strong mixing implies mixing of all orders. Acknowledgements.
The author wishes to thank Vitaly Bergelson for point-ing out the question in [3].A
PPENDIX
A. O
VER - RECURRENT SETS FOR MIXINGTRANSFORMATIONS
For the example in Figure 1, m = 6 . The height is m = 36 . The set A ⊃ A ∪ A . For i such that ≤ i < N < m , T i ( A ) will be containedin A except for part of the top m -block. Since m > / ( ǫ a ) , then the top m -block has measure less than ǫ a . Thus, the proportion of A that doesnot return to A under T i , ≤ i < N < m , is less than ǫ µ ( A ) . A similarargument holds for − N < i ≤ .R EFERENCES [1] A
DAMS , T.M. Tower multiplexing and slow weak mixing,
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