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A circle diffeomorphism with breaks that is smoothly linearizable

In this paper we answer positively a question of whether it is possible for a circle diffeomorphism with breaks to be smoothly conjugate to a rigid rotation in the case when its breaks are lying on pairwise distinct trajectories. An example constructed is a piecewise linear circle homeomorphism that has four break points lying on distinct trajectories, and whose invariant measure is absolutely continuous w.r.t.\ the Lebesgue measure. The irrational rotation number for our example can be chosen a Roth number, but not of bounded type.


Introduction
Any orientation-preserving homeomorphism T of the unit circle T 1 = R/Z with an irrational rotation number ρ = ρ(T ) is uniquely ergodic [1], i. e. it possesses a unique invariant probability measure µ = µ(T ). If T is piecewise C 1 , and the total variation of log T ′ is finite, then by the renown Denjoy's theorem it is topologically conjugate to the rigid (linear) rotation R ρ : ξ → ξ + ρ, i. e. there exists a homeomorphism φ (which is unique up to an additive constant) of T 1 such that The invariant measure µ and the conjugacy (i. e., the linearizing change of variables) φ are related by the equality φ(ξ 1 ) − φ(ξ 0 ) = ξ 1 ξ 0 dµ(ξ) for any arc [ξ 0 , ξ 1 ] ⊂ T 1 . The question of how smooth this conjugacy φ is, depending on smoothness of T , has been studied extensively since 1960's.
The case when T is a diffeomorphism, is covered by Herman's theory and its later developments [2,3,4].
Quite different is the case when T is a diffeomorphism with breaks, i. e. it is piecewise C 1 , but not C 1 (and we still assume that the total variation of log T ′ is finite). Let us call the 'size of the break' the ratio of derivatives at the break point from the left and from the right. Since the first paper [5], in which it was shown that the invariant measure of a circle diffeomorphism with a single break of non-unit size is singular (with respect to the Lebesgue measure ℓ), there appeared a number of publications proving the singularity of the invariant measure in different cases. Recently, two groups of scientists have independently proved [6,8] the following most general result: if the product of sizes of all breaks is non-unit, then the invariant measure is singular.
The case of unit product of sizes of breaks appeared to be difficult to investigate. Let us notice at this point that several breaks lying on the same trajectory are in a sense equivalent to one single break of a size that is the product of their sizes, therefore without lost of generality one may consider only diffeomorphisms with breaks that lie on pairwise distinct trajectories. It is shown in [2] (although not formulated as a statement) that a piecewise linear diffeomorphism with two breaks (the product of their sizes is unit automatically) has singular invariant measure unless the break points lie on the same trajectory. In [7,8], it was proved that for a diffeomorphism with multiple breaks lying on distinct trajectories, the invariant measure is singular provided that rotation number is of bounded type (i. e., its continued fraction expansion is a bounded sequence). Generally, there was a colloquial belief among specialists on the subject that the invariant measure is singular w. r. t. ℓ if at least for one trajectory the product of sizes of all breaks it contains is non-unit.
In this paper, we disprove that hypothesis presenting an example of piecewise linear homeomorphism of the unit circle that has four breaks lying on distinct trajectories, but nevertheless is absolutely continuously linearizable. Its rotation number can be chosen a Roth number (those are the numbers Diophantine with any positive Diophantine exponent), but not of bounded type. The linearizing conjugacy is essentially absolutely continuous in the sense that it is not piecewise C 1 , possessing an everywhere dense set of points where it is not differentiable.
The example we construct is also interesting in a framework of rigidity theory for circle diffeomorphisms with breaks [9], which asks a more general question of whether given two such maps are smoothly conjugate. It was proved in [10] that in the case of break equivalence of the two maps (a property assuming that a conjugacy sends each break point of the first map to a break point of the second one with the same size of break), unit product of sizes of breaks and bounded combinatorics (this notion generalizes bounded type of rotation number), that conjugacy is C 1 . On the other hand, very recently it was shown [11,12] that if two circle diffeomorphisms with breaks with the same irrational rotation number have different products of sizes of breaks, then every conjugacy between them is a singular function. Our example suggests that in the case of the equal products, the condition of generalized break equivalence (i. e., a break equivalence adjusted by a piecewise smooth change of coordinates) is not necessary for absolute continuity of the conjugacy.
In Section 2, we define our example for given sizes of breaks and rotation number and formulate the theorem about absolute continuity of its invariant measure w. r. t. ℓ. In Section 3, we prove that theorem and make some additional comments.
The author is grateful to A. Dzhalilov for useful discussions.
It is easy to see from the definition that (as soon as d s = ∆ s , and (k n ) n is strictly increasing, the two latter displayed inequalities give us an estimate It follows that In particular, all d By scaling them all down by putting d n = d * n /d * −1 , n ≥ −1, we obtain the sequence that satisfies the conditions of the lemma with α = d * −1 . We assume δ n = 0 for n odd. Notice that (2) and (3) imply the inequalities: Now let us produce our main example of a piecewise linear circle homeomorphism T through its lift function f 0 : R → R, f 0 (x + 1) ≡ f 0 (x) + 1, given by (It easily follows from (6) that d 0 +aδ 0 +aδ 2 < 1.) As one can see, f 0 (1) = 1+d 0 = f 0 (0)+1 indeed, and thus defined T has four breaks: two of size a and two of size a −1 . In the next section we will show that the rotation number of T is ρ, its break points lie on four distinct trajectories, and the invariant measure of T is absolutely continuous w. r. t. the Lebesgue measure ℓ. Theorem 1. The circle diffeomorphism with breaks T defined above has rotation number ρ, its four break points lie on pairwise distinct trajectories, and its invariant measure µ is absolutely continuous w. r. t. the Lebesgue measure ℓ.
In what follows, we will represent a continuous piecewise linear function defined on a segment by indicating the values of its derivative over those subsegments where it differs from 1, rather than expressions for the function itself, as it makes formulas much more intuitive. (Such information together with a value at a single point lets one restore the whole function.) For example: and this information together with the value 3 Proof Of The Claim

Renormalization Basics
For analyzing metrical properties of the dynamical system induced on T 1 by the homeomorphism T we will apply the renormalization approach, described in detail, for ex., in [9] as follows. Given a circle homeomorphism T with irrational rotation number ρ, one may 'mark' a point ξ 0 ∈ T 1 , consider its trajectory ξ i = ξ i (ξ 0 ) = T i ξ 0 ∈ T 1 , i ∈ Z, and pick out of it the sequence of the 'dynamical convergents' {ξ qn } n indexed by the denominators of the consecutive rational convergents to ρ. The dynamical convergents approach the marked point ξ 0 , alternating their order in the following way: ξ q −1 < ξ q 1 < ξ q 3 < · · · < ξ q 2m+1 < · · · < ξ 0 < · · · < ξ q 2m < · · · < ξ q 2 < ξ q 0 (the point ξ q −1 on the circle coincides with ξ 0 ; here it may be seen as ξ q −1 = ξ 0 −1). Define the nth 'fundamental segment' ∆ cover the whole circle without overlapping beyond their endpoints, thus forming the nth 'dynamical partition' For given n ≥ 0, one may consider the pair of functions (f n,ξ 0 , g n,ξ 0 ) called sometimes the n-th 'pre-renormalization' of T , which are the mappings T qn and T q n−1 restricted to ∆ (n−1) 0 and ∆ (n) 0 respectively, in the coordinate system with the origin at ξ 0 : where π ξ 0 is a linear mapping in a neighbourhood of ξ 0 that sends ξ 0 to 0 and changes neither length no orientation. We will omit the index ξ 0 , when it is clear which point is marked. Both f n and g n are strictly increasing continuous functions defined on the segments [−ℓ(∆ ) and f n (g n (0)) = g n (f n (0)), the functions f n and g n are said to form a 'commuting pair'. (Notice that our pairs (f n , g n ) differ from those in [9] by having not been rescaled.) This definition is slightly ambiguous for n = 0, in which case we define f 0 on the segment [−1, 0] representing the unit circle T 1 cut by the marked point ξ 0 (one can write ∆ , so that f 0 is a lift of T onto R, shifted to the marked point, while Notice that g n+1 is f n restricted to a smaller segment, while f n+1 = f k n+1 n • g n , and the 'renormalization height' k n+1 is the largest integer k such that f k n (g n (0)) has the same sign as g n (0) (it is also a partial quotient in the continued fraction ρ = [k 1 , k 2 , . . . , k n , . . . ]), n ≥ 0.
In this paper, we shall also consider the 'backward dynamical partition segments' ∇ if n is odd and [ξ −qn , ξ 0 ] if n is even, and define the 'backward pre-renormalization' as the pair of functionš respectively. These functions have properties similar to f n and g n , in particularf n+1 =f k n+1 n •ǧ n .

Measurements
Now, let us turn to the specific homeomorphism T that was defined at the end of Section 2 (notice that the meaning of f 0 there is consistent with Subsection 3.1, though the expressions were written for the segment [0, 1] rather than [−1, 0]). Proposition 1. For every n ≥ 0, we have ℓ(∆ (n) 0 ) = d n , and for every n ≥ 1 The rotation number of T is ρ = [k 1 , k 2 , . . . , k n , . . . ].
Remark 1. When we state that a formula like those above holds, it is a part of our statement that the segments in it are positioned correctly (for example, formula for f 2n−1 includes the claim that the segments (0, δ 2n ) and (δ 2n−2 , δ 2n−2 + aδ 2n ) do not overlap and are both contained within the segment [0, d 2n−2 ]). Such correctness is usually easy to check due to (6) and the assumed lower bounds on k n .
Proof. On each induction step m ≥ 1, one must follow the sequence of points g m−1 (0), f m−1 (g m−1 (0)), . . . , f k m−1 (g m−1 (0)), . . . until the moment right before it jumps over zero, and each moment k calculate the function f k m−1 • g m−1 by composing a previously calculated f k−1 m−1 • g m−1 with f m−1 on an appropriate segment. The last moment will be the renormalization height k m (a partial quotient in the continued fraction expansion for the rotation number of T ), while the function f km m−1 • g m−1 will be the new f m .
Proof. First we calculatef 0 as f −1 0 restricted to [0, 1], and then analyze consecutive iterates similarly to the proof of Proposition 1.
Proof. The definition of N n implies that for every i ∈ N n there exist integers 0 ≤ j m+1 ≤ k m+1 − 3, 0 ≤ m ≤ n, such that i = n m=0 (q m−1 + j m+1 q m ). Accordingly, ∇ 0 . In view of Proposition 2, it is easy to check that consecutive images of ∇ (n) 0 in the latter sequence never fall onto the teeth of non-unit derivative off m andǧ m . Proposition 3. The limit set Θ = n≥0 Θ n has positive Lebesgue measure.
Proof. It follows from Lemma 2 that ℓ(Θ n ) = d n · #(N n ), and by definition of N n we have #(N n ) = (k n+1 −2)·#(N n−1 ). Hence, ℓ(Θ n ) = (k n+1 −2)dn d n−1 ℓ(Θ n−1 ). The relations (2), (6) and increasing of k n imply that d n−1 < (k n+1 +1)d n and therefore ℓ( for large enough n. The statement follows from assumed convergence of the series n k −1 n . Proposition 4. Let η 0 ∈ Θ. Then for every n ≥ 0, we have ℓ(∆ (n) 0 (η 0 )) = d n , Proof. It is easy to follow the sequence of iterates like we did in the proof of Proposition 1, this time using η 0 as the marked point. What is important here are the bounds on values (b n ) n≥0 that follow directly from our construction of the set Θ. Indeed, b n is nothing else but a coordinate of an appropriate preimage of the break point ξ 0 in vicinity of η 0 . Since passing from Θ n−1 to Θ n we have cut away from every ∇ (d 2n )).

Differentiability
In what follows, all the renormalization structures are counted from η 0 as the marked point, and we omit the letter η 0 by f n , g n , P n and ∆ (n) i . Proposition 5. The conjugacy φ from (1) is differentiable at every point η 0 ∈ Θ.
First we need to prove the following lemma about the lengths of segments from dynamical partitions of several deeper levels lying within the fundamental segments. Denote by Γ n,s the set of all segments ∆ (n+s−1) i ∈ P n+s , which lie within the segment ∆ and for all those ∆ It is easy to see that Γ n,s = Γ n+2,s−2 ∪ k n+1 −1 j=0 f j n • g n (Γ n+1,s−1 ). We will use this recurrent relation in order to estimate the lengths of ∆ (n+s−1) i ∈ Γ n,s for s ≥ 1. As seen from Proposition 4, g n does not distort length of any segment, while f n may once stretch and once shrink it, with the maximal possible change in length equal to (a − 1)δ n for n even and (a − 1)δ n+1 for n odd.
For s = 1, the segments in Γ n,1 are the images of ∆ (n) 0 under application of g n (once) and then f n (from 0 to k n+1 − 1 times). Using Proposition 4, one can easily check that all of them have length d n for n odd, while for n even all but one have length d n with that exceptional one of length d n + (a − 1)δ n .
For s = 2, the segments in Γ n,2 \Γ n+2,0 are the images of the segments in Γ n+1,1 under f j n • g n , 0 ≤ j < k n+1 . Proposition 4 allows one to calculate that their lengths are either d n+1 or d n+1 + (a − 1)δ n for n even, and either d n+1 or d n+1 + (a − 1)δ n+1 for n odd. For s = 3, the segments in Γ n,3 \Γ n+2,1 are the images of the segments in Γ n+1,2 under f j n • g n , 0 ≤ j < k n+1 . If n is odd, then their lengths differ from d n+2 by not more than 2(a − 1)δ n+1 . If n is even, then this difference formally can reach (a − 1)δ n+2 + (a − 1)δ n , but let us look closer at this case. Notice that for n even the teeth of non-unit derivative in f n are positioned in such a way that the maximal possible change in length of a segment under their cumulative action (i.e. the action of f 2 n on ∆ i ) is (a − 1)δ n+2 rather than (a − 1)δ n . In fact, there are exactly two segments in Γ n,3 that are possibly stretched by more than 2(a − 1)δ n+2 : those whose preimages under f n intersect the upper tooth (b 2n , b 2n + δ 2n ) (while they themselves intersect the lower tooth and are being shrank back on the next application of f n ). It also follows from Proposition 4 that these two exceptional segments do not intersect [f k n+1 −1 n (−d n−1 ), 0] = ∆ (n) q n+1 −q n−1 . Since δ n = o(d n+1 ) (in fact δ 2n /d 2n+2 → 1 as n → +∞), the estimates we proved imply the statement of the lemma.
Finally, we obtain which tends to zero as well.

Conclusions
Proof of Theorem 1. The theorem contains three statements, which we will prove now. 1. The rotation number of T is ρ by Proposition 1. 2. Four break points lie on pairwise distinct trajectories by Proposition 4. Indeed, by the construction, the break points of f n are nothing else but the projections of first preimages of break points of T within the segment ∆ (n−1) 0 . If two of the latter points lied on the same trajectory, then for large enough n their first preimages within ∆ (n−1) 0 would coincide, which contradicts to the fact that f n has four distinct break point for any n.
3. For a circle diffeomorphism with breaks, its invariant measure µ can be either absolutely continuous, or singular w. r. t. ℓ (see, for ex., [1], where this result is proved for pure diffeomorphisms, and is easily extended to those with breaks). If µ is singular, then the conjugacy φ is a singular function, and φ ′ = 0 almost everywhere ℓ, which contradicts to Propositions 3 and 5. Therefore, µ is absolutely continuous.
Remark 2. The linearizing change of variables φ in (1) is absolutely continuous and therefore possesses a density h ∈ L 1 (T 1 ). This density is positive a. e. ℓ due to the fact that T is ergodic w. r. t. ℓ (see [1]).
Remark 3. The conjugacy φ is essentially absolutely continuous, i. e. it is not piecewise C 1 , moreover, there is an everywhere dense subset of T 1 , over which φ is not differentiable.
Remark 4. An irrational number ρ is said to belong to Diophantine class D δ if there exists a constant C > 0 such that |ρ − p/q| ≥ Cq −2−δ for any rational number p/q. The set of Roth numbers is R = ∩ δ>0 D δ . The numbers from D 0 are those whose continued fraction expansion is formed by a bounded sequence (k n ) n , therefore they are called 'numbers of bounded type'. The restrictions we have imposed on (k n ) n allow one to choose ρ ∈ R (for ex., k n = (n + 5) 2 will work), but not ρ ∈ D 0 .