Rotation number of 2-interval piecewise affine maps

We study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps $f_{\p}$ are parametrized by a quintuple $\p$ of real numbers satisfying inequations. Viewing $f_{\p}$ as a circle map, we show that it has a rotation number $\rho(f_{\p})$ and we compute $\rho(f_{\p})$ as a function of $\p$ in terms of Hecke-Mahler series. As a corollary, we prove that $\rho(f_{\p})$ is a rational number when the components of $\p$ are algebraic numbers.


Introduction
Let R/Z denote the unit circle and f : R/Z → R/Z be an orientation-preserving circle homeomorphism.Any continuous lift F : R → R of f is strictly increasing and F − id is one-periodic.In order to study the dynamics of orientation preserving circle homeomorphisms, Poincaré introduced an invariant quantity ρ(f ) ∈ [0, 1), known as the rotation number of f that measures the average rotation along any orbit of f .Given any continuous lift F : R → R of f and x ∈ R, the rotation number of f is defined as ρ(f ) := lim n→∞ F n (x) n (mod 1).
This limit exists and is independent of x and the lift F .Moreover, ρ(f ) is rational if and only if f has a periodic point.
The theory of rotation number was extended to orientation-preserving circle maps which are not continuous, neither surjective, in particular by Rhodes and Thompson [27,28].Let M denote the set of circle maps f whose lifts F are strictly increasing and F − id is one-periodic.Rhodes and Thompson have shown that the rotation number is well defined for all maps f ∈ M and it varies continuously as a function of f on any continuous one parameter family contained in M.
By identifying the unit interval I := [0, 1) with R/Z through the canonical bijection I → R → R/Z, we may view any circle map in M as an orientation preserving injective map of I. Through this identification, we associate a rotation number to any orientation preserving injective map.
The study of the rotation number of injective piecewise affine increasing contractions with only one discontinuity point on R/Z and a unique slope was introduced in Canaiello's neuronic equations which are 2-interval piecewise contractions, see [16] for a discussion of the topic.The dynamics of piecewise contractions has been studied by many authors, in particular [8,13,15,24,25,26].A detailed study of contracted rotations (meaning that the two branches have equal slopes) can be found in [22, 11, . In [18], Laurent and Nogueira were the first to relate transcendental properties of the parameters of these maps to the irrationality of their rotation number.Later in [19], Laurent and Nogueira extended their work to allow injective maps with two different slopes and a single discontinuity point.However, they also studied the dynamics of maps which are not piecewise contractions, in particular see [19,Corollary,page 36], where the rotation number of a 2-interval piecewise affine circle homeomorphism is obtained.
This family of maps is also known in the literature as contractive piecewise linear Lorenz maps.It is sometimes claimed, see for instance [10, page 237], that the dynamics of such a contractive map is trivial, meaning that all its orbits converge to a periodic cycle.This is almost true, but not always.In fact, for an uncountable set of parameters, the corresponding maps have singularly continuous invariant probability measures and all the orbits converge to a Cantor set sharing fine arithmetical properties which are investigated in [5] and [6].
In the present work, we extend the framework of [19] by allowing an additional discontinuity to the circle map f at the break point (denoted η below) and even allowing f to be non-injective in some cases.Moreover, the family of maps we consider are not necessarily piecewise contractions, i.e., one of its branches may expand.
The article is organized as follows.In Section 2 we introduce our family of maps whose set of parameters is described in Section 3. In Section 4 we prove that the maps we consider have a well-defined rotation number.Our main results will be stated in Section 5. Theorem 5.2 describes the rotation number of f = f p as a function of the parameter p and Theorem 5.5 makes explicit the semi-conjugacy of f to a rotation following the approach given in [9].In order to prove our main results the dynamics of our map is reduced to that of a map on an invariant interval.Up to an isomorphism, this map belongs to the family of maps that have been studied in [19].In Corollary 5.3, using transcendence results on Hecke-Mahler series, it is proved that the rotation number takes a rational value when all the components of p are algebraic numbers.In Section 6, we explain our strategy for the proof of the results in Section 5.
The properties of two functions a and φ describing the dynamics of f are displayed in Section 7.
The proofs of the main results from Section 5 are given in Section 8.

The setting
Let I = [0, 1) be the unit interval identified with the circle R/Z as in Section 1.We are concerned with the dynamics of maps f : I → I as pictured in Figure 1 below.Namely, the graph of f is made up with two increasing segments such that f has no fixed point and is injective when restricted to a certain invariant subinterval of I.Such a map f has a rotation number ρ(f ) ∈ I and is semi-conjugated to the rotation R ρ : x → x + ρ (mod 1).Our goal is to make explicit these two assertions in terms of the parameters defining f .The precise statements are given in Theorem 5. We parametrize those maps f by a quintuple p = (λ, µ, a, b, c) as follows.
Definition 2.1.Let us denote by P the set of quintuple parameters p = (λ, µ, a, b, c) satisfying the inequalities Set η = b − a λ and define f p : I → I by the splitted formula The left branch of the graph has slope λ and endpoints (0, a) and (η, b), while the right branch has slope λµ and has for origin the point (η, c).Assumption (2) ensures that the break point η belongs to the open interval (c, b).It implies that f p has no fixed point.Assumption (3) ensures that f p is injective when restricted to the interval J := [c, b).Indeed, we have It remains to show that the image f p (I) is contained in I. Looking at Figure 1, it is sufficient to prove that f p (1 − ) < 1.We postpone the proof to Section 3 which makes use of the assumptions, There is no loss of generality assuming 0 < λ < 1 in Assumption (1), since the homeomorphism (x, y) ∈ (0, 1) 2 → (1 − x, 1 − y) ∈ (0, 1) 2 exchanges the two branches and at least one slope must be less than 1 by (5).

Description of the parameter set P
In order to study the dynamics of f p , we shall basically view the four parameters λ, µ, b, c as fixed and regard the fifth parameter a as a variable.It turns out that a ranges over an interval once (λ, µ, b, c) has been fixed.The description of this interval is the following.
and define By (2), we have the inequalities b − bλ < a < b − cλ.Now, we solve inequality (3) with respect to a.We have three cases: We conclude in this case that ( 2) as required, since λµ < 1.

Rotation number of f p
Let p ∈ P be given.Define a real function F = F p : R → R by Notice that F is not always strictly increasing, so we cannot apply immediately the theory of Rhodes and Thompson developed in [27] that generalizes the classical theory of the rotation number of Poincaré to circle maps having some strictly increasing lift.Nevertheless, we show in Lemma 4.2 below, that f has a well-defined rotation number.
Lemma 4.1.F satisfies the following properties: ( Proof.Using F , we define a new piecewise-affine function F : R → R which equals F inside X, but outside X has a graph which is obtained by connecting with a straight segment the end points (b+p, F (b − )+p) and (c+p+1, F (c)+p+1) for every p ∈ Z. See Figure 3.It follows from Lemma 4.1, that the function F is strictly increasing and F − id is 1-periodic.Therefore, by [27, Corollary 1], the limit ρ := lim n→+∞ (mod 1) exists and is independent of x.Now, again by Lemma 4.1, F (X) ⊂ X and for every x ∈ R there is n 0 = n 0 (x) ≥ 0 such that F n 0 (x) ∈ X.Since F | X = F | X , we conclude that F n+n 0 (x) = F n (y) for every n ≥ 0 where y = F n 0 (x).Therefore,
It will be proved in Proposition 7.1 that the map ρ → a(λ, µ, b, c, ρ) is increasing in the interval 0 < ρ < r λ,µ and that it has a left discontinuity at any rational value and is right continuous everywhere.Our main result is the following.(1) For every irrational number ρ with 0 < ρ < r λ,µ , the rotation number ρ(f λ,µ,a,b,c ) equals ρ if and only if a = a(λ, µ, b, c, ρ); (2) Let p/q be a rational number with co-prime positive integers p < q and 0 < p/q < r λ,µ .
Then, ρ(f λ,µ,a,b,c ) = p/q if and only if a(λ, µ, b, c, (p/q) − ) ≤ a ≤ a(λ, µ, b, c, p/q), where As a consequence of Theorem 5.2 and a classical result due to Loxton and Van der Poorten [20,21], we obtain the following result: f k (x) be the ω-limit set of x under f .Definition 5.4.Let p = (λ, µ, a, b, c) ∈ P and 0 < ρ < 1 be such that λµ ρ < 1.Let φ = φ p,ρ : R → R be defined by where with the convention that a sum indexed by an empty set equals zero.
The following result describes the dynamics of f on the limit set C. (2) If ρ = p/q is rational with p and q co-prime positive integers and a(λ, µ, b, c, (p/q) − ) ≤ a < a(λ, µ, b, c, p/q), then, for every x ∈ I, is a cycle of order q and the following diagram commutes, where R p/q denotes the rotation of angle p/q.(3) When a = a(λ, µ, b, c, p/q), the limit set C is empty, φ(I) is a finite set containing b and ω(x) = φ(I) for every x ∈ I.

Reduction of parameters
The overall idea of the proof of the results in Section 5 is that, for any p ∈ P, the dynamics of the map f p is determined by its restriction f p | J to the invariant interval J := [c, b).It turns out that, up to an isomorphism, the dynamics of f p | J has already been studied in [19].See Figure 4 below.
Define the map Θ : A simple computation shows that for every (λ, µ, b, c) ∈ Q.Notice that the second equality follows from the identity (10).
Let h : J → I be the affine map x → (x − c)/(b − c).Recall that P 0 is the set of parameters defined in (9) which coincides with the set of parameters defined in [19].

1 Figure 1 .
Figure 1.A plot of f p

Figure 4 .Lemma 6 . 2 .
Figure 4. (A) Plot of f p and the square J 2 in red.(B) Zoom of the square J 2 using the affine map h and plot of f Θ(p) .