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2. Definitions and statement of the main results In this article we will consider various classes of maps, we begin by defining these classes. Let I be an interval with end points a < b, βi a collection of m +1 (with m ≥ 0) points satisfying a = β0 < β1 < ··· < βm = b and Ij := (βj−1, βj). A map f from I to itself is an piecewise monotone map (PMM for short) if for each j =1,...,m the restriction of f to the interval Ij is continuous and strictly monotone. We always assume 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 transformation with flips (gf-IET) if it is invertible. A PMM is called an interval translation mapping with flips (f-ITM) if the restriction of f to each Ii is an isometry and is called an interval exchange transformation with flips (f-IET) is f is both an gf-IET and an f-ITM. If additionally we identify the points a and b then we will refer to such a map as a circle translation 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. A 2-CET without flips is called a double rotation. Note that by definition a nontrivial circle rotation is an 2-IET, but is a 1-CET; in fact by our maximality assumption 2-CETs can not exist since they would be continuous everywhere. In all of these notations we remove the prefix f if no interval is flipped, for example IET will stand for interval exchange transformation in the classical sense. An f-invariant Borel probability measure µ is called almost surely invertible if the set {x ∈ I : #{f −1(x) ∩ supp(µ)} > 1} has µ-measure 0. We are interested in mixing measures, the identity map on I is a trivial example of a 1-PMM which is invertible and for any x ∈ I the atomic invariant measure δx is mixing. The main result of this article states 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 Borel probability measure which is almost surely invertible If µ is not the Dirac measure on a fixed point, then f :(I,µ) → (I,µ) is not mixing. The almost sure invertibility assumption is clearly necessary since for example the Lebesgue measure is a mixing measure for the PMM x → 2x mod 1. Generalized f-IETs are everywhere invertible thus we have Corollary 2. Except for Dirac measures on a fixed points, invariant Borel probability measures are never mixing for gf-IETs. If an invariant measure has zero entropy, then it is invertible a.s. [Wa][Cor 4.14.3], thus we have SERGE TROUBETZKOY 3

Corollary 3. Except for Dirac measures on a fixed points, zero entropy measures for PMMs are never mixing. Buzzi showed that the topological entropy of piecewise isometries is always zero [Bu], and thus all invariant measures have zero entropy. Thus we obtain Corollary 4. Except for the Dirac measure on a fixed point invariant measures for f-ITMs are not mixing. Kryzhevich showed that every ITM has a non-atomic invariant Borel probability [Kr], but any invariant Borel probability measure is the con- vex combination of ergodic ones, and ITMs have only a finite number of 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 version of Kryzhevich’s theorem in this case. It is quite likely that Kryzhevich’s generalizes to this case without difficulty.

3. Reduction to an IET To prove Theorem 1 we show that any non-atomic, ergodic, almost surely invertible PMM is metrically isomorphic to an f-IET with respect to Lebesgue measure. Versions of this result were shown for IETs by Katok [Ka], gIETs by Yoccoz [Yo], for some ITMs in [Kr] and [Pi], and for f-IETs in [Ba]. Lemma 6. Let f : I → I be an PMM on m intervals and µ a nonatomic, f-invariant, ergodic, Borel probability measure which is al- most surely invertible. Then there exists an interval exchange trans- formation with flips g : [0, 1) → [0, 1) on r intervals, with 2 ≤ r ≤ m, such that f :(I,µ) → (I,µ) is metrically isomorphic to g : ([0, 1),λ) → ([0, 1),λ) where λ is the Lebesgue measure. The flip set of g is a subset of 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 1 ≤ r ≤ m. Moreover, the isomorphism is effected by a monotone continuous surjective function R : I → [0, 1), 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 for a 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 > 1 then it also can not be mixing. In all the other cases we apply the Lemma, 4 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 isomorphism invariant we conclude that f :(I,µ) → (I,µ) is not mixing. 

Proof of Lemma 6. We define R : I → [0, 1) as follows R(y) := µ[a, y]. 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, but not necessarily strictly increasing and thus not necessarily bijective. By definition we have

(1) R∗µ = λ, i.e., R :(I,µ) → ([0, 1),λ) is an isomorphism of measure spaces. We define g : [0, 1) → [0, 1) by g(x)= R(f(y)) where y ∈ I is any point satisfying Ry = x. In particular, once we have 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 of points. The map g is clearly well defined for those x such that the set R−1(x) consists of a single point. If this set contains more than a single point then by monotonicity it is an interval. In this case f(R−1(x)) is a union of a finite number of intervals. −1 If the set R (x) does not contain any of the βj for j =1,...,m − 1 then f(R−1(x)) is a single interval. In this case R(f(R−1(x))) can be either 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 −1(f(J))) = µ(J) Using successively equalities (1), (2), then (1) yields λ(R(f(R−1(x)))) = µ(f(R−1(x))) = µ(R−1(x)) = λ({x})=0. Thus R(f(R−1(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−1(x) contain a point of discontinuity of f. There are at most m − 1 such points. This finishes the proof of the claim. If x is a point such that R−1(x) does not contain one of the m − 1 discontinuity points of f, then the map g = R◦f is continuous since it is the composition of a R with f which is continuous on the set R−1(x). Thus the map g is continuous except at at most m − 1 points. A priori it is possible that g is continuous at some of these points. We (re)define g at these points by continuity from the right. SERGE TROUBETZKOY 5

The map g preserves Lebesgue measure, to see this we calculate −1 −1 −1 −1 −1 λ(g ([x1, x2])) = λ(R ◦ f ◦ R ([x1, x2])) = µ(f ◦ R ([x1, x2])) −1 = µ(R ([x1, x2])) = λ([x1, x2]). The first equality holds by the definition of g, the second follows from equality (1), the third holds since µ is f-invariant, while the final equal- ity again follows from (1) . Suppose that x1 < x2 belong to same segment of continuity of g and that yi are such that R(yi)= xi. Then if f(y1) < f(y2) we obtain

g(x2) − g(x1) = R(f(y2)) − R(f(y1)) = µ([f(y1), f(y2)]) −1 = µ(f [f(y1), f(y2)]) = µ([y1,y2]) = λ([x1, x2]) = x2 − x1. Here the first equality holds by the definition of g, the second by the definition of R and the assumption that f(y1) < f(y2), 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 definition of Lebesgue measure. On the other hand if f(y1) > f(y2), i.e. we are in a flipped interval, by the same reasoning we obtain

g(x2) − g(x1) = R(f(y2)) − R(f(y1)) = µ([f(y2), f(y1)]) −1 = µ(f [f(y2), f(y1)]) = µ([y2,y1]) = λ([x2, x1]) = x1 − x2. Thus g is a fITM. The metric isomorphism statement in the theorem follows since if g is not a fIET then there are two distinct intervals whose images coincide, which contradicts the invariance of Lebesgue measure. Finally we claim that r ≥ 2. If r = 1 then since it is a f-IET preserving the Lebesgue measure g is the identity map or the map x 7→ 1 − x. Thus, for any y ∈ I we have f 2j(y) ∈ R−1(g2(R(y))) = R−1(R(y)) for all j ≥ 0. But since R is monotonically increasing, there are only a countable set of points x = R(y) for which the set R−1(R(y)) is a non- degenerate interval; there are only a countable set of points x′ = g(x)= R(fy) for which the set R−1(R(fy)) is a non-degenerate interval. If we choose x so that x and x′ = g(x) are not in this countable set, then the forward orbit of y = R−1(x) consists of two points, R−1(R(y)) ∪ R−1(R(f(y))) = y ∪ f(y). Since µ is ergodic and non-atomic, we can apply this observation to a point y whose forward orbit is dense in supp(µ). The conclusion contradicts the assumption that µ is non-atomic and thus r > 1. 6 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. 

4. Weak mixing There are no 2-CETs, thus applying the lemma to a double rotation produces a 1-CET, i.e., a rotation. Thus in the same way Lemma 6 and Corollary 5 yield Corollary 8. - An almost surely invertible, non-atomic, invariant Borel 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 uniquely ergodic 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 weakly mixing. 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 ergodic and not weakly mixing.

5. Suspension flows Let f be a PMM without flips. Let h : I → R be a strictly positive function with bounded variation. Consider the space Y := {(y, t): y ∈ I, 0, t ≤ h(y)} and the suspension semi-flow φf : Y → Y defined as follows, we flow (y, t) with unit speed until we reach the top of Y and then continue after identifying the points (y, h(y)) with (f(y), 0). A φf invariant measure ν is of the form ν = µ × λ with µ an f-invariant measure on I and λ the Lebesgue measure in the vertical direction of Y .

Theorem 11. A invariant measure of the semi-flow φf which is almost surely invertible is not mixing.

Corollary 12. If φf 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 for flows in the case that f has flips. SERGE TROUBETZKOY 7

- In the much more delicate case when the the roof function has logarith- mic singularities Ulcigrai has shown that the typical φf is not mixing [Ul], while examples of mixing flow exist [ChWr], [Ko].

Proof. Let ν a φf -invariant measure which is almost surely invertible and µ the corresponding f-invariant measure. We proceed as in Lemma 6, 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, 1),λ). Let S(x) be the left most point of the interval R−1(x). Consider the space X := {(x, t): x ∈ [0, 1), 0 ≤ t ≤ h(S(x))} and the special flow ψg : X → X. Define the map Rˆ : Y → X by Rˆ(y, t)=(R(y), t). As already mentioned in the proof of Lemma 6, the interval R−1(x) is a singleton except for an at a countable collection of points. It follows that g ˆ f ˆ−1 φs(x, t)= R ◦ φs ◦ R (x, t) for all point {(x, t): } outside the countable set discussed above. Since this countable set has Lebesgue measure 0 The map Rˆ is a measure theoretic isomorphism of (φf Y, µˆ) with (φg,X, λˆ). Since h is assumed to have bounded variation and S is increasing we have h ◦ S is of bounded variation, and thus by Katok’s theorem dλ × dt is not mixing, and thus µˆ is not mixing. Katok also showed that every invariant measure for the special flow φf is not mixing if f is an IET. 

References

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Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France Address: I2M, Luminy, Case 907, F-13288 Marseille CEDEX 9, France Email address: [email protected]