Absence of mixing for interval translation mappings and some generalizations
aa r X i v : . [ m a t h . D S ] F e b ABSENCE OF MIXING FOR INTERVALTRANSLATION MAPPINGS AND SOMEGENERALIZATIONS
SERGE TROUBETZKOY
Abstract.
We consider piecewise monotone maps, we show thatan ergodic measure for which the map is invertible almost every-where can not be mixing. It follows that every ergodic measure foran interval translation mapping is not mixing. We also show thatdouble rotations without periodic points have an ergodic but notweakly mixing invariant measure.This article is dedicated to the memory of Anatoly Mikhailovich Stepin. Introduction
Interval translation mappings were introduced by Boshernitzan andKorneld [BoKo], they are a natural generalization of interval exchangetransformations. A slightly more general class of maps, Interval trans-lation mappings with flips, were introduced by Buzzi and Hubert, theyform a rigid model for piecewise monotone maps of the interval withoutperiodic points [BuHu].Interval exchange transformations have been extensively studied,some aspects of their behavior have been extended to interval transla-tion mapping, the main results can be found in the following references[ArFoHuSk, BoKo, Br, BrCl, BrTr, BuHu, Ka, Kr, Pi, ScTr, SuItAi,Vo, Yo, Zh].In 1967 Katok and Stepin showed that an interval exchange trans-formation on three intervals is not mixing with respect to the Lebesguemeasure ([KaSt][Remark 8.1]). In 1980 Katok extended this result toall interval exchange transformations, he showed that a non-atomicmeasures can never be mixing. The main results of our article is theextension of this result to interval translation mappings (with or with-out flips). This result, Corollary 4 follows from a more general result,Theorem 1, on non-mixing for piecewise monotone mappings with al-most surely invertible invariant measures.
Mathematics Subject Classification.
Key words and phrases.
Interval translation mapping, mixing, double rotations.
Bibliographical references. Definitions and statement of the main results
In this article we will consider various classes of maps, we begin bydefining these classes. Let I be an interval with end points a < b , β i acollection of m + 1 (with m ≥ ) points satisfying a = β < β < · · · <β m = b and I j := ( β j − , β j ) . A map f from I to itself is an piecewisemonotone map (PMM for short) if for each j = 1 , . . . , m the restrictionof f to the interval I j is continuous and strictly monotone. We alwaysassume that these intervals are maximal domains of continuity of f .We will refer to those intervals on which f is strictly decreasing as flipped . A PMM is called a generalized interval exchange transformationwith flips (gf-IET) if it is invertible. A PMM is called an intervaltranslation mapping with flips (f-ITM) if the restriction of f to each I i is an isometry and is called an interval exchange transformation withflips (f-IET) is f is both an gf-IET and an f-ITM. If additionally weidentify the points a and b then we will refer to such a map as a circletranslation mapping with flips resp. circle exchange transformation flips (f-CTM, resp. f-CET). We will sometimes write m -f-IET, m -f-ITM,etc. to emphasize the number of intervals in the definition of f . A2-CET without flips is called a double rotation . Note that by definitiona nontrivial circle rotation is an 2-IET, but is a 1-CET; in fact byour maximality assumption 2-CETs can not exist since they would becontinuous everywhere. In all of these notations we remove the prefix f if no interval is flipped, for example IET will stand for interval exchangetransformation in the classical sense.An f -invariant Borel probability measure µ is called almost surelyinvertible if the set { x ∈ I : { f − ( x ) ∩ supp ( µ ) } > } has µ -measure0. We are interested in mixing measures, the identity map on I is atrivial example of a -PMM which is invertible and for any x ∈ I theatomic invariant measure δ x is mixing. The main result of this articlestates that this is the only way to obtain a mixing measure. Theorem 1.
Let f : I → I be a PMM and µ an f -invariant Borelprobability measure which is almost surely invertible If µ is not theDirac measure on a fixed point, then f : ( I, µ ) → ( I, µ ) is not mixing. The almost sure invertibility assumption is clearly necessary sincefor example the Lebesgue measure is a mixing measure for the PMM x → x mod 1 .Generalized f-IETs are everywhere invertible thus we have Corollary 2.
Except for Dirac measures on a fixed points, invariantBorel probability measures are never mixing for gf-IETs.
If an invariant measure has zero metric entropy, then it is invertiblea.s. [Wa][Cor 4.14.3], thus we have
ERGE TROUBETZKOY 3
Corollary 3.
Except for Dirac measures on a fixed points, zero entropymeasures for PMMs are never mixing.
Buzzi showed that the topological entropy of piecewise isometries isalways zero [Bu], and thus all invariant measures have zero entropy.Thus we obtain
Corollary 4.
Except for the Dirac measure on a fixed point invariantmeasures for f-ITMs are not mixing.
Kryzhevich showed that every ITM has a non-atomic invariant Borelprobability [Kr], but any invariant Borel probability measure is the con-vex combination of ergodic ones, and ITMs have only a finite numberof ergodic measures [BuHu], thus we have
Corollary 5.
Every ITM without periodic points has a non-atomic,ergodic invariant Borel probability measure which is not mixing.
Corollary 5 should also hold for f-ITMs, but one must prove a versionof Kryzhevich’s theorem in this case. It is quite likely that Kryzhevich’sgeneralizes to this case without difficulty.3.
Reduction to an IET
To prove Theorem 1 we show that any non-atomic, ergodic, almostsurely invertible PMM is metrically isomorphic to an f-IET with respectto Lebesgue measure. Versions of this result were shown for IETs byKatok [Ka], gIETs by Yoccoz [Yo], for some ITMs in [Kr] and [Pi], andfor f-IETs in [Ba].
Lemma 6.
Let f : I → I be an PMM on m intervals and µ anonatomic, f -invariant, ergodic, Borel probability measure which is al-most surely invertible. Then there exists an interval exchange trans-formation with flips g : [0 , → [0 , on r intervals, with ≤ r ≤ m ,such that f : ( I, µ ) → ( I, µ ) is metrically isomorphic to g : ([0 , , λ ) → ([0 , , λ ) where λ is the Lebesgue measure. The flip set of g is a subsetof the flip set of f , in particular if f has no flips, then g has not flips.If f is an m -CTM then g is an r -CET with ≤ r ≤ m . Moreover, theisomorphism is effected by a monotone continuous surjective function R : I → [0 , , thus g is a topological factor of the restriction f . We emphasize that g may not be defined at a finite number of points.In the same way as we concluded Corollary 3 Walters’ result yields Corollary 7. If µ is a non-atomic, zero entropy invariant measure fora PMM f , then ( f, I, µ ) is metrically isomorphic to a f-IET. Proof of Theorem 1. If µ is not ergodic then it is can not be mixing.If µ is atomic supported on a periodic orbit with period > then italso can not be mixing. In all the other cases we apply the Lemma, Interval translation mappings and Baron’s theorem that f-IETs are never mixing [Ba] (this is a gen-eralization of Katok’s theorem [Ka]), since mixing is an isomorphisminvariant we conclude that f : ( I, µ ) → ( I, µ ) is not mixing. (cid:3) Proof of Lemma 6.
We define R : I → [0 , as follows R ( y ) := µ (cid:0) [ a, y ] (cid:1) . The map R is continuous since µ is nonatomic and surjective since µ ([ a, a ]) = 0 and µ ([ a, b ]) = 1 . Furthermore R is clearly increasing, butnot necessarily strictly increasing and thus not necessarily bijective.By definition we have(1) R ∗ µ = λ, i.e., R : ( I, µ ) → ([0 , , λ ) is an isomorphism of measure spaces.We define g : [0 , → [0 , by g ( x ) = R ( f ( y )) where y ∈ I is any point satisfying Ry = x . In particular, once wehave shown that g is well defined we will have g ◦ R ( y ) = R ◦ f ( y ) . We claim that g is well defined except possible for a finite set ofpoints. The map g is clearly well defined for those x such that the set R − ( x ) consists of a single point. If this set contains more than a singlepoint then by monotonicity it is an interval. In this case f ( R − ( x )) isa union of a finite number of intervals.If the set R − ( x ) does not contain any of the β j for j = 1 , . . . , m − then f ( R − ( x )) is a single interval. In this case R ( f ( R − ( x ))) can beeither a point or an interval. Notice that for any interval J ⊂ I since µ is f -invariant and f is almost surely invertible we have(2) µ ( f ( J )) = µ ( f − ( f ( J ))) = µ ( J ) Using successively equalities (1), (2), then (1) yields λ ( R ( f ( R − ( x )))) = µ ( f ( R − ( x ))) = µ ( R − ( x )) = λ ( { x } ) = 0 . Thus R ( f ( R − ( x ))) is a point. We have shown that the only points x at which the map g is possibly not well defined are those x such that R − ( x ) contain a point of discontinuity of f . There are at most m − such points. This finishes the proof of the claim.If x is a point such that R − ( x ) does not contain one of the m − discontinuity points of f , then the map g = R ◦ f is continuous since it isthe composition of a continuous function R with f which is continuouson the set R − ( x ) . Thus the map g is continuous except at at most m − points. A priori it is possible that g is continuous at some ofthese points. We (re)define g at these points by continuity from theright. ERGE TROUBETZKOY 5
The map g preserves Lebesgue measure, to see this we calculate λ ( g − ([ x , x ])) = λ ( R ◦ f − ◦ R − ([ x , x ])) = µ ( f − ◦ R − ([ x , x ]))= µ ( R − ([ x , x ])) = λ ([ x , x ]) . The first equality holds by the definition of g , the second follows fromequality (1), the third holds since µ is f -invariant, while the final equal-ity again follows from (1) .Suppose that x < x belong to same segment of continuity of g andthat y i are such that R ( y i ) = x i . Then if f ( y ) < f ( y ) we obtain g ( x ) − g ( x ) = R ( f ( y )) − R ( f ( y )) = µ ([ f ( y ) , f ( y )])= µ ( f − [ f ( y ) , f ( y )]) = µ ([ y , y ])= λ ([ x , x ]) = x − x . Here the first equality holds by the definition of g , the second by thedefinition of R and the assumption that f ( y ) < f ( y ) , the third be-cause µ is f invariant, the fourth since µ is almost surely invertible,the fifth follows from (1), and the last equality holds by the definitionof Lebesgue measure.On the other hand if f ( y ) > f ( y ) , i.e. we are in a flipped interval,by the same reasoning we obtain g ( x ) − g ( x ) = R ( f ( y )) − R ( f ( y )) = µ ([ f ( y ) , f ( y )])= µ ( f − [ f ( y ) , f ( y )]) = µ ([ y , y ])= λ ([ x , x ]) = x − x . Thus g is a fITM. The metric isomorphism statement in the theoremfollows since if g is not a fIET then there are two distinct intervalswhose images coincide, which contradicts the invariance of Lebesguemeasure.Finally we claim that r ≥ . If r = 1 then since it is a f-IETpreserving the Lebesgue measure g is the identity map or the map x − x . Thus, for any y ∈ I we have f j ( y ) ∈ R − ( g ( R ( y ))) = R − ( R ( y )) for all j ≥ . But since R is monotonically increasing, there are only acountable set of points x = R ( y ) for which the set R − ( R ( y )) is a non-degenerate interval; there are only a countable set of points x ′ = g ( x ) = R ( f y ) for which the set R − ( R ( f y )) is a non-degenerate interval. If wechoose x so that x and x ′ = g ( x ) are not in this countable set, then theforward orbit of y = R − ( x ) consists of two points, R − ( R ( y )) ∪ R − ( R ( f ( y ))) = y ∪ f ( y ) . Since µ is ergodic and non-atomic, we can apply this observationto a point y whose forward orbit is dense in supp ( µ ) . The conclusioncontradicts the assumption that µ is non-atomic and thus r > . Interval translation mappings
In the case that f and g are CETs then the case r = 1 can happen,it is simply a circle roation. (cid:3) Weak mixing
There are no 2-CETs, thus applying the lemma to a double rotationproduces a 1-CET, i.e., a rotation. Thus in the same way Lemma 6and Corollary 5 yield
Corollary 8. - An almost surely invertible, non-atomic, invariantBorel probability measure for a double rotation is not weakly mixing.- Every double rotation without periodic points has a non-atomic, er-godic invariant Borel probability measure which is not weakly mixing.
Bruin and Clack showed that ν -almost every double rotation is uniquelyergodic where ν is any invariant measure for the Suzuki induction[BrCl][Theorem 5]. They did not show the existence of such a mea-sure, none the less if such a measure exists it follows that Corollary 9. ν -a.e. double rotation is uniquely ergodic and not weaklymixing. Artigiani et. al. prove the existence of a measure µ on the space of in-terval translation mappings which invariant under Artigiani–Fougeron-Hubert-Skripchenko induction. For this measure they showed µ -a.e.double rotation is uniquely ergodic, thus we have Corollary 10. µ -a.e. double rotation of infinite type is uniquely ergodicand not weakly mixing. Suspension flows
Let f be a PMM without flips. Let h : I → R be a strictly positivefunction with bounded variation. Consider the space Y := { ( y, t ) : y ∈ I, , t ≤ h ( y ) } and the suspension semi-flow φ f : Y → Y defined asfollows, we flow ( y, t ) with unit speed until we reach the top of Y andthen continue after identifying the points ( y, h ( y )) with ( f ( y ) , . A φ f invariant measure ν is of the form ν = µ × λ with µ an f -invariantmeasure on I and λ the Lebesgue measure in the vertical direction of Y . Theorem 11.
A invariant measure of the semi-flow φ f which is almostsurely invertible is not mixing. Corollary 12. If φ f is a flow (i.e., invertible), then no invariant mea-sure is mixing. Remark 13. - An analog of Katok’s theorem has not been studied forflows in the case that f has flips. ERGE TROUBETZKOY 7 - In the much more delicate case when the the roof function has logarith-mic singularities Ulcigrai has shown that the typical φ f is not mixing [Ul] , while examples of mixing flow exist [ChWr] , [Ko] . Proof.
Let ν a φ f -invariant measure which is almost surely invertibleand µ the corresponding f -invariant measure. We proceed as in Lemma6, but with some extra care for the vertical direction in Y . The map f of ( I, µ ) is isomorphic to an IET g of the space ([0 , , λ ) .Let S ( x ) be the left most point of the interval R − ( x ) . Consider thespace X := { ( x, t ) : x ∈ [0 , , ≤ t ≤ h ( S ( x )) } and the special flow ψ g : X → X .Define the map ˆ R : Y → X by ˆ R ( y, t ) = ( R ( y ) , t ) . As alreadymentioned in the proof of Lemma 6, the interval R − ( x ) is a singletonexcept for an at a countable collection of points. It follows that φ gs ( x, t ) = ˆ R ◦ φ fs ◦ ˆ R − ( x, t ) for all point { ( x, t ) : } outside the countable set discussed above. Sincethis countable set has Lebesgue measure 0The map ˆ R is a measure theoretic isomorphism of ( φ f Y, ˆ µ ) with ( φ g , X, ˆ λ ) .Since h is assumed to have bounded variation and S is increasingwe have h ◦ S is of bounded variation, and thus by Katok’s theorem dλ × dt is not mixing, and thus ˆ µ is not mixing.Katok also showed that every invariant measure for the special flow φ f is not mixing if f is an IET. (cid:3) References [ArFoHuSk] M. Artigiani, Ch. Fougeron, P. Hubert and A. Skripchenko,
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