ABSENCE OF MIXING FOR INTERVAL TRANSLATION MAPPINGS AND SOME GENERALIZATIONS

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ABSENCE OF MIXING FOR INTERVAL TRANSLATION MAPPINGS AND SOME

GENERALIZATIONS

Serge Troubetzkoy

To cite this version:

Serge Troubetzkoy. ABSENCE OF MIXING FOR INTERVAL TRANSLATION MAPPINGS AND SOME GENERALIZATIONS. In press. �hal-03138196v1�

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TRANSLATION MAPPINGS AND SOME GENERALIZATIONS

SERGE TROUBETZKOY

Abstract. We consider piecewise monotone maps, we show that an ergodic measure for which the map is invertible almost everywhere can not be mixing. It follows that every ergodic measure for an interval translation mapping is not mixing. We also show that double rotations without periodic points have an ergodic but not weakly mixing invariant measure.

This article is dedicated to the memory of Anatoly Mikhailovich Stepin.

1. Introduction

Interval translation mappings were introduced by Boshernitzan and Korneld [BoKo], they are a natural generalization of interval exchange transformations. A slightly more general class of maps, Interval translation mappings with flips, were introduced by Buzzi and Hubert, they form a rigid model for piecewise monotone maps of the interval without periodic points [BuHu].

Interval exchange transformations have been extensively studied, some aspects of their behavior have been extended to interval translation 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 transformation on three intervals is not mixing with respect to the Lebesgue measure ([KaSt][Remark 8.1]). In 1980 Katok extended this result to all interval exchange transformations, he showed that a non-atomic measures can never be mixing. The main results of our article is the extension of this result to interval translation mappings (with or without flips). This result, Corollary 4 follows from a more general result, Theorem 1, on non-mixing for piecewise monotone mappings with almost surely invertible invariant measures.

2020 Mathematics Subject Classification. 37A25, 28D05.

Key words and phrases. Interval translation mapping, mixing, double rotations.

Bibliographical references. 23

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2 Interval translation mappings

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, β_i a collection of m+ 1 (withm≥0) points satisfying a=β₀ < β₁ <· · ·<

β_m = b and I_j := (βj−1, β_j). A map f from I to itself is an piecewise monotone map (PMM for short) if for eachj = 1, . . . , mthe restriction of f to the intervalI_j 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 ageneralized 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 off to each I_i 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 prefixf 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 δ_x 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 metric entropy, then it is invertible a.s. [Wa][Cor 4.14.3], thus we have

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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 almost surely invertible. Then there exists an interval exchange transformation 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 ofg 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 thatgmay not be defined at a finite number of points.

In the same way as we concluded Corollary 3 Walters’ result yields Corollary 7. If µis 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. Ifµis 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,

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and Baron’s theorem that f-IETs are never mixing [Ba] (this is a generalization 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]) = 0andµ([a, b]) = 1. FurthermoreR 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.

If the setR⁻¹(x)does not contain any of the β_j forj = 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 pointsx 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 mostm−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 off, then the mapg =R◦fis continuous since it is the composition of a continuous function R withf 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.

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The map g preserves Lebesgue measure, to see this we calculate λ(g⁻¹([x₁, x₂])) = λ(R◦f⁻¹◦R⁻¹([x₁, x₂])) =µ(f⁻¹◦R⁻¹([x₁, x₂]))

= µ(R⁻¹([x₁, x₂])) =λ([x₁, x₂]).

The first equality holds by the definition of g, the second follows from equality (1), the third holds sinceµisf-invariant, while the final equality again follows from (1) .

Suppose thatx1 < 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)])

= µ(f⁻¹[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)])

= µ(f⁻¹[f(y₂), f(y₁)]) =µ([y₂, y₁])

= λ([x₂, x₁]) = x₁−x₂.

Thusg 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 x7→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 pointsx=R(y)for which the setR⁻¹(R(y))is a non- degenerate interval; there are only a countable set of pointsx⁰ =g(x) = R(f y)for which the set R⁻¹(R(f y))is a non-degenerate interval. If we choosexso thatx andx⁰ =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 µis non-atomic and thus r >1.

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In the case that f and g are CETs then the case r = 1can 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, ergodic invariant Borel probability measure which is not weakly mixing.

Bruin and Clack showed thatν-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 measure, 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 measureµon the space of interval 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 φ_f invariant measure ν is of the form ν=µ×λ withµ anf-invariant measure 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 almost surely invertible is not mixing.

Corollary 12. If φ_f is a flow (i.e., invertible), then no invariant measure is mixing.

Remark 13. - An analog of Katok’s theorem has not been studied for flows in the case that f has flips.

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- 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. Letν a φ_f-invariant measure which is almost surely invertible andµthe correspondingf-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).

LetS(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 ψ_g :X →X.

Define the map Rˆ : Y → X by R(y, t) = (R(y), t). As alreadyˆ 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

φ^g_s(x, t) = ˆR◦φ^f_s ◦Rˆ⁻¹(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 (φ_f Y,µ)ˆ with (φg, X,ˆλ).

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

φ_f is not mixing if f is an IET.

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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: serge.troubetzkoy@univ-amu.fr