Mild mixing of certain interval exchange transformations
aa r X i v : . [ m a t h . D S ] S e p MILD MIXING OF CERTAIN INTERVAL EXCHANGETRANSFORMATIONS
DONALD ROBERTSON
Abstract.
We prove that irreducible, linearly recurrent, type W interval ex-change transformations are always mild mixing. For every irreducible permu-tation the set of linearly recurrent interval exchange transformations has fullHausdorff dimension. Introduction
Fix a permutation π of { , . . . , d } and positive lengths λ , . . . , λ d that sum to 1.Put λ = 0. Write I i = [ λ + · · · + λ i − , λ + · · · + λ i )for each 1 ≤ i ≤ d . The interval exchange transformation on [0 ,
1) determinedby the data ( λ, π ) is the map T : [0 , → [0 ,
1) given by
T x = x − X j
For every irreducible, type W permutation π on { , . . . , d } the set oflengths ( λ , . . . , λ d ) for which ( λ, π ) is mild mixing has full Hausdorff dimension. In fact, we will prove in Section 5 that whenever π is type W and λ is such that( λ, π ) is linearly recurrent (see Section 3) then ( λ, π ) is mild mixing. It followsfrom work of Kleinbock and Weiss [KW04] that, for a fixed irreducible permutation π , the set of such λ in the simplex has full Hausdorff dimension. (See Theorem 3below.) Chaika, Cheung and Masur [CCM13] have extended Klienbock and Weiss’sresult by showing that the set is winning for Schmidt’s game, which implies fullHausdorff dimension.For interval exchanges on three intervals Theorem 1 is a consequence of work byFerenczi, Holton and Zamboni [FHZ05, Theorem 4.1] who showed that, for suchinterval exchanges, linear recurrence implies minimal self-joinings and hence mildmixing. Boshernitzan and Nogueira [BN04, Theorem 5.3] showed that all intervalexchange transformations that are type W and linearly recurrent are weakly mixing.We remark that examples due to Himli [Hmi06] prove this is not the case in general:indeed taking m = 4 in the first example of Section 3 therein gives a permutationthat is not type W and an interval exchange transformation that has a non-constanteigenfunction; one can verify that the interval exchange transformation is linearlyrecurrent provided α is badly approximable.For flows on surfaces Frączek and Lemańczyk [FL06] showed that special flowsover irrational rotations with bounded partial quotients whose roof function is piece-wise absolutely continuous and has non-zero sum of jumps is always mild mixing.Subsequently, Frączek, Lemańczyk and Lesigne [FLL07] gave a criterion for a piece-wise constant roof function over an irrational rotation with bounded partial quo-tients to be mild mixing. Using this criterion Frączek [Fra09] has shown that, whenthe genus is at least two, the set of Abelian differentials for which the vertical flow ismild mixing is dense in every stratum of moduli space. More recently, Kanigowskiand Kułaga-Przymus [KKP15] showed that roof functions over interval exchangetransformation having symmetric logarithmic singularities at some of the discon-tinuities of the interval exchange transformation give rise to special flows that aremild mixing.Sections 2 and 3 contain the facts we need about type W and lienarly recurrentinterval exchange transformations respectively. In Section 4 we discuss rigidity andmild mixing.We would like to thank Jon Chaika for suggesting this project and for his helpand patience during conversations related to it. The author gratefully acknowledgesthe support of the NSF via grant DMS-1246989.2. Type W permutations
Fix a permutation π on { , . . . , d } . Define a permutation σ of { , . . . , d } by σ ( j ) = π − (1) − j = 0 d π ( j ) = dπ − ( π ( j ) + 1) − π ) for the set of orbits of σ . This auxiliary permutation was introducedby Veech [Vee82]. It describes which intervals are adjacent after an application ofany interval exchange transformation defined by the permutation π .We can represent Σ( π ) as a directed graph with { , ω , . . . , ω d − , } as its set ofvertices: writing ω = 0 and ω d = 1 there is an edge from ω i to ω j if and only if σ ( i ) = j . The edge with source 0 and the edge with target 1 correspond, respec-tively, to the first two cases in the definition of σ . Call this graph the endpointidentification graph . Say that π is type W if the vertices 0 and 1 are in distinct ILD MIXING OF CERTAIN INTERVAL EXCHANGE TRANSFORMATIONS 3 T − p JT − JJT JT q JTT
Figure 1.
The tower determined by the interval J . Each intervalis a translate of the one below under T .components of the graph. Some examples of type W permutations are given in[CDK09, Section 5].Define T + ( a ) = lim x → a + T ( x ) T − ( a ) = lim x → a − T ( x )for all a in [0 ,
1) and (0 ,
1] respectively. The endpoint identification graph containsthe edge 0 → ω k if and only if 0 = T + ( ω k ) and the edge ω j → T − ( ω j ) = 1. All other edges ω j → ω k correspond to equalities T − ( ω j ) = T + ( ω k )where ω j = 0 and ω k = 1. 3. Linear recurrence
Fix an irreducible permutation π on { , . . . , d } . Given lengths λ , . . . , λ d put β i = λ + · · · + λ i for all 1 ≤ i ≤ d −
1. Let D = { β , . . . , β d − } . One says that( λ, π ) satisfies the infinite distinct orbits condition if D ∩ T − n D = ∅ for all n in N . Keane [Kea75] showed that the infinite distinct orbits condition impliesminimality.Assuming the infinite distinct orbits condition, the set [ { T − i D : 0 ≤ i ≤ n } partitions [0 ,
1) into sub-intervals of positive length. Write ǫ n for the length of theshortest interval in this partition. It was observed in [Bos88] that if T satisfies theinfinite distinct orbits condition and J ⊂ [0 ,
1) is an interval of length at most ǫ n then there are times p, q ≥ p + q = n − T − p J, T − p +1 J, . . . , J, . . . , T q − J, T q J are disjoint intervals. Moreover T k J = T k − J + α k for all − p < k ≤ q , which isto say that each interval is a translate of the previous one under T . We call (2)the tower defined by J . Call T − p J the bottom floor of the tower and T q J the top floor of the tower – see Figure 1 for a schematic. Write τ for the union ofthe sets in (2). If J contains a discontinuity of T then q = 0, and if J containsa discontinuity of T − then p = 0. Note that disjointness of the intervals implies nǫ n ≤ ǫ n → n → ∞ .An interval exchange transformation is linearly recurrent ifinf { nǫ n : n ∈ N } ≥ c DONALD ROBERTSON for some positive constant c . Thus for linearly recurrent interval exchange trans-formations the tower (2) determined by any interval J of length ǫ n has measure atleast c .Fix an irreducible permutation π . We conclude this section with a proof of thefollowing result. Theorem 3.
For every irreducible permutation π the set of λ for which the intervalexchange transformation ( λ, π ) is linearly recurrent is strong winning. An interval exchange transformation T is badly approximable ifinf { n | q − T n ( p ) | : n ∈ N } > p, q of T . Note that badly approximable implies the infi-nite distinct orbits condition. It follows from the proof of Theorem 1.4 in [CCM13]that those λ for which ( λ, π ) is badly approximable is strong winning and in par-ticular has full Hausdorff dimension. To prove Theorem 3 it therefore suffices toprove that every badly approximable interval exchange transformation is linearlyrecurrent. We give a proof of this folklore result for completion. Proposition 4.
Fix an irreducible permutation π . If for some λ the interval ex-change transformation defined by ( λ, π ) is badly approximable then it is linearlyrecurrent.Proof. Fix an interval exchange transformation T that is badly approximable. Let c be the minimum of the quantitiesinf { n | q − T n ( p ) | : n ∈ N } over all discontinuities p, q of T . By hypothesis c is positive. Let η be the minimalspacing between discontinuities of T .Suppose that T is not linearly recurrent. Then nǫ n ≤ c/ ǫ n < η for some n ≥
2. Fix 0 ≤ l ≤ m ≤ n and discontinuities p, q of T such that ǫ n = | T l ( q ) − T m ( p ) | . We claim that if 1 ≤ l there is no discontinuity of T − between T m ( p ) and T l ( q ). Indeed, all discontinuities of T − are of the form T i ( r ) for some i ∈ { , } and some discontinuity r of T , so the existence of a discontinuity of T − between T m ( p ) and T l ( q ) contradicts ǫ n = | T l ( q ) − T m ( p ) | . Thus | T l − ( q ) − T m − ( p ) | = ǫ n .Iterating gives | q − T m − l ( p ) | = ǫ n . Since ǫ n < η we must have 0 < m − l . But then c ≤ inf { n | q − T n ( p ) | : n ∈ N } ≤ ( m − l ) | q − T m − l ( p ) | = ( m − l ) ǫ n ≤ nǫ n ≤ c/ (cid:3) Rigidity
A measure-preserving transformation T on a probability space ( X, B , µ ) is rigid if there is a sequence i n i in N with n i → ∞ such that for every f in L ( X, B , µ )one has T n i f → f in L ( X, B , µ ). We remark (see, for instance [BJLR14, Sec-tion 2]) that if a measure-preserving transformation T on a probability space( X, B , µ ) has the property that for every f in L ( X, B , µ ) there is a a sequence i n i such that T n i f → f in L ( X, B , µ ) then it is rigid. Lemma 5.
Let T be a rigid, measure-preserving transformation on a probabilityspace ([0 , , B , µ ) where X = [0 , and µ is Lebesgue measure. Then T is rigid ifand only if there is a sequence n i → ∞ such that (6) µ ( { x ∈ [0 ,
1) : | T n i x − x | > ǫ } ) → ILD MIXING OF CERTAIN INTERVAL EXCHANGE TRANSFORMATIONS 5 for every ǫ > .Proof. If T is rigid then || T n i ι − ι || → ( X, B , µ ) and (6) holds by Chebychev’sinequality. Conversely, one can use (6) to prove that || T n i f − f || → f on [0 , (cid:3) A measure-preserving transformation T is mildly mixing if it has no non-trivialrigid factors. One can show that this is equivalent to the absence of non-constantfunctions f in L ( X, B , µ ) such that T n i f → f in L ( X, B , µ ). Indeed, given aparticular sequence i n i the subspace { f ∈ L ( X, B , µ ) : T n i f → f } can be shown to be of the form L ( X, C , µ ) for some T invariant sub- σ -algebra C , and the corresponding factor is non-trivial if the above subspace contains non-constant functions. Conversely, any rigid function on a factor lifts to a rigid functionon the original system. 5. Proof of main theorem
In this section we will prove Theorem 1. We begin with the following lemma.
Lemma 7.
Let T be an ergodic, measure-preserving transformation on a probabilityspace ( X, B , µ ) . If, given f in L ( X, B , µ ) , one can find a constant ρ > such that µ ( { x ∈ X : | f ( T i +1 x ) − f ( T i x ) | < δ for all − b ≤ i ≤ b } ) ≥ ρ for all δ > and all b ∈ N then f is constant.Proof. For each δ > { x ∈ X : | f ( T i +1 x ) − f ( T i x ) | < δ for all − b ≤ i ≤ b } are decreasing as b → ∞ so their intersection has measure at least ρ by hypothesis.This intersection is T invariant so has full measure by ergodicity. In particular { x ∈ X : | f ( T x ) − f ( x ) | < δ } has full measure. Since δ > f is T invariant almost surely and therefore constant by ergodicity. (cid:3) By Theorem 3 the following result implies Theorem 1.
Theorem 8.
A linearly recurrent, type W interval exchange transformation mustbe mildly mixing.Proof.
Fix a type W interval exchange transformation T on [0 ,
1) that is linearlyrecurrent. Let 0 → ζ → · · · → ζ s → c = inf { nǫ n : n ∈ N } . Assume that for some f in L ( X, B , µ ) there is a sequence n i → ∞ suchthat T n i f → f in L ( X, B , µ ). Write f l for f ◦ T l . We show that for every δ > b ∈ N the set { x ∈ [0 ,
1) : | f i ( T x ) − f i ( x ) | < δ for all − b ≤ i ≤ b } has measure at least c . It will then follows from Lemma 7 and ergodicity of alllinearly recurrent, type W interval exchange transformations ([BN04, Theorem 5.2])that f is constant. Thus the only rigid functions in L ( X, B , µ ) are the constantfunctions and T is mildly mixing. DONALD ROBERTSON
Fix δ > b ∈ N . Using Lusin’s theorem one can find a compact set K ⊂ [0 ,
1) with µ ( K ) ≥ − c s on which each of T − b f, . . . , f, . . . , T b f is uniformlycontinuous. Fix η > x, y ∈ K with | x − y | < η one has | f ( T i x ) − f ( T i y ) | < δ s for all − b ≤ i ≤ b . Fix using Lemma 5 and the fact that ǫ n → n ∈ N such that(1) ǫ n < min { c , η, ǫ } ;(2) the set G = { x ∈ [0 ,
1) : | f i ( T n x ) − f i ( x ) | < δ s for all − b ≤ i ≤ b } has measure at least 1 − c s .Put H = G ∩ K ∩ T − n K . We have µ ( H ) ≥ − c s .For each 1 ≤ i ≤ s let I i be the interval [ ζ i − ǫ n , ζ i + ǫ n ). As described inSection 3 each I i determines a tower T − ( n − I i , T − ( n − I i , . . . , T − I i , I i of disjoint intervals with total measure at least c . Write τ i for this tower. Put I = [0 , ǫ n ). Similarly, it is the roof of a tower τ with total measure at least c .We claim that 90% of the points x in τ have all the following properties:(1) x ∈ T − ℓ I with ℓ = n − x ∈ H and T − x ∈ H ;(3) for every 1 ≤ i ≤ s one can find points y i in H ∩ T − ℓ [ ζ i − ǫ n , ζ i ) and z i in H ∩ T − ℓ [ ζ i , ζ i + ǫ n ).This follows from the following arguments.(1) 99% of the points in τ are not in the bottom level since 100 ǫ n < c/ µ ( τ ∩ H ) ≥ c s and µ ( τ ∩ T − H ) ≥ c s so µ ( τ ∩ H ∩ T − H ) ≥ c s .Thus 98% of the points in τ satisfy Property 2.(3) Suppose one can find r distinct levels T − l I , . . . , T − l r I in τ and for eachsuch level a “defective” tower τ m i with either H ∩ T − l i [ ζ m i − ǫ n , ζ m i ) = ∅ or H ∩ T − l i [ ζ m i , ζ m i + ǫ n ) = ∅ . By pigeonhole at least r/s of these levelshave the same defective tower, say τ m . But then µ ( τ m \ H ) ≥ rs ǫ n ≥ rs c n .But µ ( τ m \ H ) ≤ c s so rn ≤ τ enjoyProperty 3.All told, at least 90% of the points in τ satisfy all three properties.Fix a point x in τ such that all three properties hold. In particular, let y , . . . , y s , z , . . . , z s and ℓ be as in Property 3. For every 1 ≤ j ≤ s − ζ j → ζ j +1 we have the estimates(9) | f i ( y j ) − f i ( T n y j ) | < δ s | f i ( T n z j +1 ) − f i ( z j +1 ) | < δ s for all − b ≤ i ≤ b since both y j and z j +1 belong to G . We also have | z j +1 − y j +1 | < η since { y j +1 , z j +1 } ⊂ T − ℓ [ ζ j +1 − ǫ n , ζ j +1 + ǫ n ) and | T n y j − T n z j +1 | < η since [ T ζ j +1 − ǫ n , T ζ j +1 + ǫ n ) contains { T ℓ +1 y j , T ℓ +1 z j +1 } and is the bottom floorof a tower of width ǫ n . (See Figure 2 for a schematic.) These two inequalities imply | f i ( z j +1 ) − f i ( y j +1 ) | < δ s | f i ( T n y j ) − f i ( T n z j +1 ) | < δ s ILD MIXING OF CERTAIN INTERVAL EXCHANGE TRANSFORMATIONS 7
T Ty j z j y j +1 z j +1 T n y j T n z j +1 ζ j ζ j +1 T ζ j +1 Figure 2.
The towers and their relationships for an edge ζ j → ζ j +1 in the endpoint identification graph with 0 / ∈ { ζ j , ζ j +1 } .for all − b ≤ i ≤ b since { y j , y j +1 , z j +1 } ⊂ K ∩ T − n K . Combined with (9) thesegive(10) | f i ( y j ) − f i ( y j +1 ) | < δ s for all − b ≤ i ≤ b .Considering next the edge ζ s →
0, we have the estimates(11) | f i ( x ) − f i ( T n x ) | < δ s | f i ( T n y s ) − f i ( y s ) | < δ s for all − b ≤ i ≤ b since { x, y s } ∈ G . Also | T n x − T n y s | < η since [ T (0) − ǫ n , T (0) + ǫ n ) contains { T ℓ +1 x, T ℓ +1 y s } and is the bottom floor of atower of width ǫ n . (See Figure 3 for a schematic.) Thus | f i ( T n x ) − f i ( T n y s ) | ≤ δ s for all − b ≤ i ≤ b since { x, y s } ⊂ T − n K . Combined with (11) we get(12) | f i ( x ) − f i ( y s ) | ≤ δ s for all − b ≤ i ≤ b .Using (10) for all 1 ≤ j ≤ s − | f i ( x ) − f i ( y ) | < δ for all − b ≤ i ≤ b . Now we use the edge 0 → ζ to pick up some invariance. First,note that(14) | f i ( z ) − f i ( T n z ) | < δ s | f i ( T n ( T − x )) − f i ( T − x ) | < δ s for all − b ≤ i ≤ b since { z , T − x } ⊂ H . We also have | y − z | < η because { y , z } ⊂ T − ℓ [ ζ − ǫ n , ζ + ǫ n ) and | T n z − T n ( T − x ) | < η DONALD ROBERTSON
T T Ty s z s ζ s y z ζ x T ζ T n y s T n xT n z T Figure 3.
The towers and their relationships for the edges 0 → ζ and ζ s → T n z and T n − x are in the same level since T ζ = 0.because T n z and T n ( T − x ) are both in the interval T n − ℓ − [0 , ǫ n ). (See Figure 3for a schematic.) These two estimates imply | f i ( y ) − f i ( z ) | < δ s | f i ( T n z ) − f i ( T n ( T − x )) | < δ s for all − b ≤ i ≤ b because { y , z , T n ( T − x ) , T n z } ⊂ K . Together with (14) theseimply | f i ( y ) − f i ( T − x ) | < δ s for all − b ≤ i ≤ b . Finally, combined with (13) we get | f i ( x ) − f i ( T − x ) | < δ for all − b ≤ i ≤ b .Since x was an arbitrary point in τ satisfying Properties 1, 2 and 3, the set { x ∈ [0 ,
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