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The Period adding and incrementing bifurcations: from rotation theory to applications

This survey article is concerned with the study of bifurcations of piecewise-smooth maps. We review the literature in circle maps and quasi-contractions and provide paths through this literature to prove sufficient conditions for the occurrence of two types of bifurcation scenarios involving rich dynamics. The first scenario consists of the appearance of periodic orbits whose symbolic sequences and"rotation"numbers follow a Farey tree structure; the periods of the periodic orbits are given by consecutive addition. This is called the {\em period adding} bifurcation, and its proof relies on results for maps on the circle. In the second scenario, symbolic sequences are obtained by consecutive attachment of a given symbolic block and the periods of periodic orbits are incremented by a constant term. It is called the {\em period incrementing} bifurcation, in its proof relies on results for maps on the interval. We also discuss the expanding cases, as some of the partial results found in the literature also hold when these maps lose contractiveness. The higher dimensional case is also discussed by means of {\em quasi-contractions}. We also provide applied examples in control theory, power electronics and neuroscience where these results can be applied to obtain precise descriptions of their dynamics.


Introduction
Piecewise-smooth (piecewise-defined or non-smooth) systems are non-regular or discontinuous systems induced by dynamics associated with sharp changes in position, velocity, or other magnitudes undergoing a jump in their value. While exhibiting richer dynamics than they smooth versions, this type of systems provide more natural and simpler models in many applications from different disciplines, such as switching systems in power electronics, sliding-mode techniques in control theory, hybrid systems with resets in neuroscience or impact systems in mechanics. As a consequence, in the last decade, the interest in such type of systems has considerably grown (see [ML12] for a recent general survey). In particular, piecewise-smooth maps have focused the attention of many researchers, who have studied them from very different perspectives. One of the most reported dynamical aspects are the different bifurcations scenarios that they may exhibit, which turn out to be extraordinarily rich. Seduced by their graphical beauty and mainly supported by computations, many authors have recurrently observed and reported these bifurcation phenomena. However, although they assemble many well known results on circle maps, they have been considered as new and exclusive of piecewise-smooth maps.
In this review article we show that, after some extensions and modifications, many of the well known results for circle maps developed in the 80's and early 90's can be used to obtain rigorous proofs of general results that can be systematically applied to piecewise-smooth maps. More precisely, we consider a piecewise-smooth map undergoing a discontinuity at x = 0, and consider as parameters the two lateral images at this point. To our knowledge, such a map was first studied by Leonov [Leo59], and later on obtained as an approximation of a Poincaré map of smooth flow near a homoclinic bifurcation of the figure eight and butterfly types [Spa82,Hom96a] (see §2.2 for more references). Depending on the signs and magnitude of the slopes of the map at both sides of the discontinuity, the bifurcation scenario in this two-dimensional parameter space may be very different. These signs are determined by the number of twists exhibited by the invariant manifolds involved in the homoclinic bifurcation. When the map is contracting in both sides of the discontinuity and both slopes have different sign, the so-called period incrementing scenario occurs. This bifurcation was reported in [Hom96a], and the details of the proof for this case were given in [AGS11]. However, if both slopes are positive, then the so-called period adding scenario occurs. Although this has been widely reported in the literature (see §2.2 for references) a complete proof has not been reported until now. We first note, in the increasing-increasing case, the piecewise-smooth map can be reduced to a discontinuous circle map. We then assemble and extend many well known results for circle maps to the discontinuous case in order to provide a straight path to prove this bifurcation scenario.
This work is organized as follows. In Section 2 we provide basic definitions and a detailed statement of the results. In Section 3 we review and extend results for circle maps and quasi-contractions to provide a proof of the period adding bifurcation scenario (increasing-increasing or orientation preserving case). In Section 4 we revisit the proof provided in [AGS11] for the period incrementing bifurcation scenario (increasing-decreasing or non-orientable case).

System definition and properties
Let us consider a piecewise-smooth map (2.1) with x ∈ R and f L , f R smooth functions satisfying h.1 f L (0) = f R (0) = 0.
h.2 0 < (f L (x)) < 1, x ∈ (−∞, 0) h.3 0 < |(f R (x)) | < 1, x ∈ (0, ∞) For convenience, we do not define at this point the map f at x = 0. Roughly speaking, the only difference between choosing the image of f at x = 0 the one given by one side, the other one, none (or even both) will imply the persistence or not of invariant objects (fixed points or periodic orbits) at their bifurcations. We will focus on this question whenever it becomes relevant. Due to condition h.1, the map (2.1) is discontinuous at x = 0 if µ L = µ R . As we will show, this discontinuity introduces exclusive dynamical phenomena which are not possible in smooth (C 1 ) one-dimensional systems. As discussed in the introduction, one observes similar phenomena (the bifurcation scenarios described below) in smooth flows of dimension three near homoclinic bifurcations. They are also observed in smooth maps, when restricted to the circle instead of R. This discontinuity represents a boundary in the state space abruptly separating two different dynamics: the ones given by the maps f L and f R . These dynamics will strongly depend on the sign of f R (x) on (0, ∞) leading to completely different families of periodic orbits. Note that the cases when f L (x) and f R (x) have different slopes in their respective domains are conjugate through the symmetry x ←→ −x. Moreover, as it will be shown below, when both f L (x) and f R (x) are decreasing functions in their respective domains, the possible dynamics will be easy. Therefore we can restrict to the case that f L (x) is an increasing function in (−∞, 0), as stated in h.2.
One of the differences between the families of periodic orbits that one can find depending on the sign of f R (x) in (0, ∞) will be given by the sequence of steps that periodic orbits perform at each side of the boundary x = 0. Therefore, we will introduce the symbolic dynamics given by the following symbolic encoding. Given a point x ∈ R, we associate to its trajectory by f , (x, f (x), f 2 (x), . . . ), a symbolic sequence given by As a(x) provides a symbol of length one (L or R), one can omit the comas separating the symbols in Eq. (2.2) without introducing imprecisions. We call this the itinerary of x by f or the symbolic sequence associated with the trajectory of x by f . Let us now consider the shift operator acting on symbolic sequences Clearly, the shift operator satisfies Of special interest for us will be the symbolic sequences associated with periodic orbits. In this case, the symbolic sequences will be also periodic and we will represent them by the repetition of the generating symbolic block. For example, let (x 1 , x 2 ) be a periodic orbit, and assume x 1 < 0 and x 2 > 0. Then, the symbolic sequences associated with x 1 and x 2 are where ∞ indicates infinite repetition. Due to property (2.5), the shift operator acts on the generating blocks as a cyclic permutation of offset 1, as it moves the first symbol to the last position. More precisely, if (x 1 , . . . , x n ), x i ∈ R, is a periodic orbit of f and (x 1 . . . x n ) ∞ , x i ∈ {L, R}, is the symbolic sequence associated with the periodic trajectory of Hence, a periodic orbit of length n can be represented by n different symbolic sequences obtained by cyclic permutations one from each other.
Definition 2.1. Symbolic sequences can be ordered by lexicographical order induced by L < R. That is, if and only if there exists some j ≥ 1 such that Definition 2.2. Given a periodic symbolic sequence x = (x 1 . . . x q ) ∞ , we will say that it is minimal if and similarly for a maximal symbolic sequence.
Definition 2.3. We will say that a periodic orbit of length n is a x 1 . . . x nperiodic orbit, x i ∈ {L, R}, if there exists some point of this periodic orbit, Usually, in order to represent the symbolic sequence of a periodic orbit we will choose its minimal representative. For example, assume (x 1 , . . . , x 5 ) is a 5-periodic orbit such that I f (x 1 ) = (LRLLR) ∞ . Then, we will say that (x 1 , . . . , x 5 ) is a L 2 RLR-periodic orbit or a periodic orbit of type L 2 RLR, where the superindex 2 means that there two symbols L.
Given a periodic orbit, besides its period, one important characteristic associated with its symbolic sequence is the number of symbols R and L and how are they distributed along the sequence. The latter will be explained in detail below. For the former we define the η-number: q be a symbolic sequence, and let p be the number of symbols R contained in γ. We then define the η-number as η = p q . (2.6) As it will detailed below ( §3.1, see Remark 3.4), under certain conditions, the piecewise-smooth map (2.1) becomes a circle map with rotation number the η-number. Hence, the η-number as defined above is frequently referred to as rotation number in the context of piecewise-smooth maps, even when these conditions are satisfied.
We now focus on the question of, for a map f of type (2.1) satisfying h.1-h3, what are the possible periodic orbits, their symbolic sequences and their bifurcations in the parameter space given by the offsets µ L and µ R , µ L × µ R . To this end, we first note that, if µ L , µ R < 0, as the maps f L and f R are contracting, then f possesses two attracting coexisting L and R-fixed points x L < 0 and x R > 0: The domains of attraction are separated by the boundary x = 0. Indeed, if both f L and f R are increasing maps, then these domains become (−∞, 0) and (0, ∞), respectively. Note that, although x = 0 is not an invariant point (an equilibrium), it acts as a separatrix between these domains of attraction. If one of these two parameters vanishes and becomes positive, the corresponding fixed point collides with the boundary x = 0 and undergoes a border collision bifurcation. Depending on how the map f (2.1) is defined at x = 0, at the moment of the bifurcation the fixed point may still exist or not. Just after this bifurcation, the fixed point and no longer exists, while the other one remains and becomes the unique global attractor. Hence, the origin of this parameter space consists of a codimension-two bifurcation point. But then the question arises: what does exist when both parameters are positive and both fixed points disappear in border collision bifurcations? The answer to this question (summarized in §2.2 and §2.3) depends on the signs of the slopes of the maps f L and f R for x < 0 and x > 0, respectively. Recalling that the increasing-decreasing and decreasing-cases are conjugate we will only distinguish between two cases: increasing-increasing and increasing-decreasing. These are also typically referred as orientation preserving and non-orientable cases. As will be argued in 2.4, the decreasing-decreasing case is straightforward under the assumption of contractiveness.
2.2 Overview of the orientation preserving case: the period adding The bifurcation scenario when both f L and f R are both increasing is shown in Figs. 2.1 and 2.2. As shown in Fig. 2.1, there exist an infinite number of bifurcation curves emerging from the origin the parameter space µ L × µ R . These separate regions of existence of periodic orbits whose periods are given by "successive addition" of the ones of "neighbouring regions" 1 . We will make this As explained in §1, to our knowledge, this bifurcation scenario was first described by Leonov in the late 1950s ([Leo59, Leo60a, Leo60b, Leo62]), when studying by means of direct computations a piecewise-linear map similar to (2.1). Later on, this was studied in more detail in different contexts ([Spa82, CGT84, GPTT86, TS86, Mir87, PTT87, GGT88, GT88, LPZ89, GH94, Hom96b]), rediscovered in [AS06] and named period adding. This is precisely defined in Definition 3.5.

Overview of the non-orientable case: the period incrementing
For the increasing-decreasing case one finds the bifurcation scenario shown in As mentioned in §1, this bifurcation scenario was first described by Leonov [Leo59,Leo60a,Leo60b,Leo62]. Later on, it was studied due to its relevance in homoclinic bifurcations involving non-orientable homoclinic manifolds [Hom96b,GH94]. It was rediscovered in [AS06] when studying a linear piecewise-smooth map and named period incrementing. Full details of this proof were given in [AGS11].

Summarizing theorem
The results presented in the previous sections are summarized in the following Theorem 2.5. Let f be a map as in Eq. (2.1) satisfying conditions h.1-h3. Let γ be a C 1 curve in the parameter space satisfying H.1 µ L (λ) > 0 and µ R (λ) > 0 for λ ∈ (0, 1)   Then, the bifurcation diagram exhibited by the map f λ obtained from Eq. (2.1) after performing the reparametrization given by γ, follows a For a description of the bifurcation scenarios announced in i) and ii), the period adding and period incrementing, see Sections 2.2 and 2.3 for an overview, and Sections 3 and 4 for more details and proofs.
We now explain briefly the case when condition h.2 is not satisfied (f L is decreasing): , only a LR-periodic orbit can exist for all λ ∈ (0, 1)., iv) if f R (x) > 0, for x ∈ (−∞, 0) is equivalent to ii) and interchanging L and R in the symbolic dynamics.
Clearly, iii) comes from the fact that, under these conditions, f L ((−∞, 0)) ⊂ (µ L , ∞) and f R ((0, ∞)) ⊂ (−∞, −µ R ), and hence, due to the contractiveness of these maps, f must possess a LR-periodic orbit. The fact that the cases iv) and ii) are conjugate comes from applying the symmetries given by the change of variables x ←→ −x.
3 Orientation preserving case

Detailed description
We first provide a detailed description of the period adding bifurcation structure by stating some results which will be proved in the rest of this section.
The bifurcation structure given by the so-called period adding is strongly linked with the ordering of the rational numbers given by the Farey tree. In order to explain how this tree is generated, we first define the Farey neighbours. Recall that a rational number p/q is irreducible if (p, q) = 1, where (·, ·) denotes the greatest common divisor.
Definition 3.1. Given two irreducible rational numbers p/q and p /q , we say that they are Farey neighbours if |pq − p q| = 1.  Remark 3.2. The definition of the Farey neighbours does not provide uniqueness. That is, given a rational number it does not have a unique Farey neighbour /1/2 and 1/3 are Farey neighbours, but also 1/2 and 2/5 are). Uniqueness is obtained when one also fixes the order, n, of the Farey tree shown in Fig. 3.5(a); that is, if one considers rational numbers with denominator smaller or equal to n.
Definition 3.3. Given and irreducible rational number P/Q we define its Farey parents as the two unique irreducible rational numbers p/q and p /q such that and p/q and p /q are Farey neighbours. Equivalently, two Farey neighbours p/q and p /q produce the child P/Q given in Eq. (3.1). The rational number P/Q is also called the mediant of p/q and p /q .
As shown in Fig. 3.5(a), starting with the Farey neighbours 0/1 and 1/1, the Farey tree is generated by obtaining rational numbers by adding their numerators and denominators. That is, given two Farey neighbours p/q and p /q , they generate the child (p + p )/(q + q ), which is an irreducible rational number. Note that this provides all rational numbers. This is because, for fixed n, the Farey tree contains in all levels up to n all combinations of rational number with denominator smaller or equal to n (also called Farey sequence of order n, see [HW60]). For example, the Farey sequence of order 6 is Hence, the Farey parents of an irreducible rational numbers are unique. Under conditions i) in Theorem 2.5, the map f (2.1) possesses periodic orbits for a set of values of λ with full measure. When this parameter is varied from 0 to 1, the periods of these periodic are given by the denominators of the rational numbers numbers given in the Farey tree (see Fig. 3.5(a)). As noted by some authors [FG11], other type of trees, as the Stern-Brocot, can also generate this sequence of periods. However, the most interesting relation between the Farey tree and the sequence of periodic orbits given by the period adding shown in Fig. 2.2 regards their associated symbolic sequences. To explain this, we recall that the η-number (Def. 2.4) is given by the ratio between the number of R s contained in a symbolic sequence and its length. When λ in Theorem 2.5 is varied from 0 to 1, η is continuous and monotonically increasing taking all values between 0 and 1. However, the set of values of λ for which η is irrational forms a Cantor set of zero measure. Such a function is called a devil's staircase, and it is constant (taking rational values) almost everywhere except for a Cantor set of zero measure. This function is shown in Fig. 2.2(b), and is the well known one formed by the rotation numbers of the periodic orbits of the so-called Arnold circle map when Ω is varied from 0 to 1. Note that η = 0 and η = 1 correspond to the fixed points L and R which undergo border collision bifurcations for λ = 0 and λ = 1, respectively.
Although each periodic orbit for λ ∈ (0, 1) is in one-to-one correspondence with a rational number in the Farey tree through the η-number, their symbolic sequences are, in principle, not uniquely identified. For example, assume that for a certain value λ = λ 2/5 there exists a periodic orbit with η = 2/5. However, its symbolic sequence could be given by any of the generating minimal blocks L 2 RLR or L 3 R 2 , as they have length 5 and contain two R symbols. However, the symbolic sequence that corresponds to this periodic orbit is L 2 RLR. To see this, we identify each periodic orbit with a symbolic sequence on the Farey tree of symbolic sequences shown in Fig. 3.5(b), which is constructed as follows. Starting with the sequences L and R, one concatenates those sequences whose η-numbers are Farey numbers. By construction, this provides a unique correspondence between rational numbers and symbolic sequences through their η-number and the Farey tree. That is, to each rational number P/Q in the Farey tree one associates a symbolic sequence ∆ given by the concatenation where α and γ are the (minimal) symbolic sequences of the Farey neighbours rational numbers p/q and p /q , respectively: As we will be discussed in §3.5, this concatenation provides the so-called maximin sequences (see definition 3.36).
We are now ready to provide a formal description of the period adding bifurcation structure. iii) if f λ∆ possesses a ∆-periodic orbit with η = P/Q, (P, Q) = 1, then there exist values λ α < λ ∆ < λ γ such that f λα and f λγ possess α and γ-periodic orbits, respectively, whose η-numbers are Farey neighbours and their mediant is P/Q. Moreover, ∆ is the concatenation of α and γ: ∆ = αγ

Summary of the proof
In order to facilitate the lecture of the proof of the result described in detail in §3.1, we provide a schematic summary of the necessary steps.
1) By performing the change of variables given in Eq. (3.2) we first show that a piecewise-defined map f as in (2.1) satisfying i) in Theorem 2.5 is an orientation preserving circle map (see Definition 3.6). Although this circle map will be discontinuous, in Section 3.3 we will show how classical results for continuous circle maps also hold. In particular, we will get (a) existence of the rotation number (Prop. 3.10) (b) Existence of a unique and stable periodic orbit when the rotation number is rational (Prop. 3.13 + contractivness).
2) In §3.4 we provide symbolic properties of the itineraries of periodic orbits of orientation preserving circle maps. More precisely: (a) In Prop. 3.31 we show that the symbolic itinerary of a twist periodic orbit is a p, q-ordered symbolic sequence. This identifies each periodic orbit of an orientation preserving map with a unique symbolic itinerary through its rotation number.
(b) In Proposition 3.31 we show that the itineraries of p, q-ordered periodic orbits are given by concatenation of the itineraries of the periodic orbits with rotation numbers the Farey parents of p/q. Hence, they are in the Farey tree of symbolic sequences shown in Fig. 3.5(b).
3) In §3.4 we also study one-parameter families of orientation preserving (discontinuous) circle maps. This is equivalent to varying the parameter λ under the conditions of Theorem 2.5 i). Using the continuity of the rotation number (Proposition 3.20) and a result of Boyd (Theorem 3.34), we show that the rotation number (and hence the η-number) follows a devil's staircase leading to the adding scenario when λ is varied from 0 to 1.
We emphasize that the previous steps provide (to our knowledge) the shortest path to prove i) of Theorem 2.5. However, as it will be explained along the rest of this Section, this is not the only one. In Section 3.5 we provide two partial alternatives to cover some of the steps mentioned above. These involve the concept of maximin sequences and quasi-contractions.

Reduction to an orientation preserving circle map and some properties
In this section we first show that, under condition i) of Theorem 2.5, the piecewise-smooth map (2.1) can be reduced to class of orientation preserving (increasing) discontinuous circle maps. Then, we present results on circle maps, which, are well known for continuous circle maps. However, by making little modifications of the classical proofs we adapt them to the discontinuous case. We will make this clear for each particular case. Let us observe that, under condition i) of Theorem 2.5, the piecewise-smooth map (2.1) is increasing on both sides of the discontinuity x = 0. Hence, all the dynamics are attracted into the interval Fig. 3.6). By identifying these two values, the map becomes a circle map which is continuous at x = 0, but not necessarily at x = −µ R ∼ µ L (see Fig. 3.6). As we are interested on varying the parameters µ R and µ L , we perform the change of variables which is homeomorphic, strictly increasing and maps −µ R to 0 and µ L to 1. Of special interest will be the value which separates the behaviour given by φ Hence, we have reduced the piecewise-smooth map to a class of circle maps, which we make precise in the following definition.
Definition 3.6. We say that Due to the existence of c fulfilling C.3, such a circle map is of degree one, as the image of S 1 by f twists at most once around S 1 . When also considering condition C.4 we ensure that such an orientation preserving map is invertible. However, condition C.4 allows this class of maps to be not necessary continuous at x = 0. Hence, at x = 0 and x = 1 one can choose between the image from the left or right of x = 0 (or indeed any other value). When convenient, we will choose both values at x = 0 and x = 1 and deal with a bi-valued function.
Notice that, at this point, we are not requiring contractiveness, and all results in this section hold also for expansive maps as long as conditions C.1-C4 are satisfied.
Definition 3.7. Let f be a map satisfying conditions C.1-C4. We will say that F is a lift of f of degree N ≥ 0 if to the map If N = 1, we will refer to F just as the lift of f .
From now on, we will restrict to lifts of degree one.
Remark 3.8. Due to condition C.4, the lift of an orientation preserving map is an increasing map, probably discontinuous at the integer numbers, where it undergoes a positive gap.
Remark 3.9. In definition 3.7 the value of the lift F at integer numbers, n, is not uniquely defined. When convenient, we will take the one given by f (1 − ) + n, f (0 + ) + n or both. The latter will lead to a bi-valued lift.
The following result is well known and provides the definition of the rotation number of an orientation preserving circle map satisfying C.1-C4. This was introduced by Poincaré [Poi81] for homeomorphisms of the circle of degree 1 and later studied and extended to rotation intervals by many authors (see [ALM00] and references therein). For discontinuous orientation preserving circle maps, this was proven in [RT86,Gam87]. However, if one considers a bi-valued lift at integer numbers (see Remark 3.9), then the standard proof holds. exists and is independent of x.
Proof. We give the standard proof a slight modification to overcome the discontinuities at integer numbers. As F is increasing, we get that, for 0 ≤ x ≤ 1 Noting that we take F be-valued at integer values, we can write this as where F (0) and F (1) can be each of the lateral values.
Using that F (x + 1) = F (x) + 1 and applying it recursively to F n (1) we have that which we can write as Hence, taking limits we get and the limit does not depend on x. We next show that, indeed, this limit exists. We apply Proposition 1 of [RT86], which states that, if a sequence a n satisfies |a m+n − a m − a n | ≤ A, (3.5) for all n, m ≥ 1 and some constant A, then there exists some ρ such that The sequence F n (0) satisfies (3.5) with A = 1. To see this, we use that F (x + n) = F (x) + n to obtain, for any x ≥ 0, where [·] denotes the integer part. Then, taking x = F m (0), we get Note that this proof differs from the one given in [ALM00]. There, it is first proved that, if f possesses a periodic orbit, then this limit exists and is rational; then it is shown that it does not depend on x as above. In this approach we show the existence of this limit and then we discuss the dynamics of the map depending on its value.
The previous result permits one to define the rotation number of a circle map f fulfilling C.1-C4.
Definition 3.11 (Rotation number). Given a map f satisfying C.1-C4 and F its lift, we define the rotation number of f as for any x ∈ R.
Remark 3.12. The fact that the limit given in Eq. (3.4) exists implies that F n (x) grows linearly with n: Recalling the properties shown in (3.3), the rotation number ρ(f ) can be seen as the average number of times that the lift F crosses an integer number per iteration.
The next result is also standard result for continuous circle maps (Prop. 3.7.11 of [ALM00]), it provides the existence of a periodic obit if the rotation number is rational. Below we prove that it also holds for discontinuous circle maps.
Proposition 3.13. Let F be a lift of an orientation preserving circle map satisfying C.1-C4. Then, ρ(F ) = p/q, (p, q) = 1, if and only if there exists Proposition 3.13, which we prove below, is stated assuming that the map F is bi-valued at integer values (see Remark 3.9). This ensures that one always finds a periodic orbit if the rotation number is rational. However, if one considers only one image at integer numbers, given by f (0 + ) or f (1 − ), one can lose the existence of a periodic orbit and get a ω-limit consisting of q points mimicking a periodic orbit. That is, one recovers a result given in [RT86], which we repeat below for completeness.
Proposition 3.14 ([RT86] Th. 2). Let F be the lift of an orientation preserving circle map satisfying C.1-C4 and p, q ∈ Z with q > 0. If ρ(F ) = p/q then exactly one of the following holds.
i) There exists x 0 ∈ R such that F q (x 0 ) = x 0 + p.
ii) For all x ∈ R, F q (x) > x + p, and there exists x 0 ∈ R such that iii) For all x ∈ R, F q (x) < x + p, and there exists x 0 ∈ R such that Conversely, if either i), ii) or iii) holds then ρ(F ) = p/q. For completeness, we will provide a proof of Proposition 3.13 based on [ALM00] but adapting it to the discontinuous case. It relies on the following two lemmas. The first lemma is equivalent to Lem. 2 of [RT86]. We announced it as in Lem. 3.7.10 of [ALM00] but we adapt its proof to hold also for discontinuous circle maps.
Lemma 3.16. Let F be the lift of a circle map satisfying C.1-C4, and let p ∈ Z. Then, Proof. One can proceed as in [RT86]. However, by considering that F is bivalued at integer numbers, we can proceed as in the proof for the continuous case [ALM00], which we repeat for completeness. We show i), ii) is analogous. Due to the strict inequality, there exists some Then, for all k we get Hence, as by Prop. 3.10 the rotation number exists and is unique, we have that which proves i) with ε = δ/q.
Note that the proof given in [ALM00] is slightly different, as it does not use the uniqueness of the rotation number. The following lemma is trivial for the continuous case.

Lemma 3.17 ([RT86] Lem. 3). Let
F : R −→ R be a (not necessary continuous) non-decreasing map fulfilling F (x + 1) = x + 1 for all x ∈ R. Assume that F (x 1 ) > x 1 and F (x 2 ) < x 2 for some x 1 , x 2 ∈ R. Then there exists some x 0 such that F (x 0 ) = x 0 , and F is continuous on the left at x 0 .
We provide more intuitive proof of this lemma.
Proof. Assume F (x) = x for all x and recall that, if F undergoes a discontinuity, the jump must be positive. If F (0) > 0, F lies above the diagonal, otherwise it crosses it or undergoes a negative gap, which contradicts the existence of x 2 . If F (0) < 0, we argue that F lies below the diagonal, which contradicts the existence of x 1 . If it happens that F skips the diagonal by a positive jump, as F (1) = F (0) + 1 < 1, it must cross it afterwards or undergo a negative jump, which is not possible.
Remark 3.18. The class of maps considered in Lemma 3.17 is not necessary a lift of a circle map satisfying C.1-C4, it may undergo discontinuities with positive jumps between 0 and 1.
We now prove Proposition 3.13. by adapting the proof given in [ALM00] to the discontinuous case.
Proof. (Of Proposition 3.13) Assume that F q − p has a fixed point. Then, we get that F nq (x 0 ) − np = x 0 for all n > 0, which implies that ρ(F ) = p/q. Assume that ρ(F ) = p/q and that F q (x) − p does not have a fixed point. Then, by Lemma 3.17, we get that either F q (x) − p < x or F q (x) − p > x. But then, by Lemma 3.16, there exists ε > 0 such that either ρ(F ) > p/q + ε or ρ(F ) < p/q − ε, which is a contradiction.
Note that, besides the fact that we deal with a discontinuous lift, the previous proof differs from the one given in [ALM00] by the fact that we can use the existence and uniqueness of the rotation number provided by Prop. 3.10.
Remark 3.19. Proposition 3.13 does not provide the uniqueness nor stability of periodic orbits. However, if, in addition to C.1-C4, one adds contractiveness, f (x) < 1 for x ∈ [0, 1], then one gets that the such periodic orbit is unique and attracting.
The next result provides the continuity of the rotation number; i.e., if two lifts of orientation preserving maps are "close" (using the uniform norm), so are their rotation numbers. Note that, assuming that these maps are bi-valued at integer values, one can always choose the proper images to properly compare them. Hence, its proof for the discontinuous case becomes the standard one but taking into account this fact and Lemmas 3.16 and 3.17. Proof. We proceed as in [ALM00] by adapting the proof to the fact that F is bi-valued in order to to overcome the discontinuities at integer numbers.
Assume ρ(F ) = p/q. Then, the function G(x) = F q (x) − p − x is away from zero. By Lemma 3.17 (applied to G(x) + x), we get that either G(x) < 0 or G(x) > 0 for all x ∈ R. Then, by Lemma 3.16, we have that either ρ(F ) < p/q or ρ(F ) > p/q, respectively. We now note that G(x) has degree 0; that is, . Hence, we can ensure that, ifG is in small enough neighbourhood of G, then eitherG < 0 orG > 0, respectively. This implies that, ifF is in a small enough neighbourhood of F , then either ρ(F ) < p/q or ρ(F ) > p/q, respectively. Hence, we have shown that, if there exist p 1 /q 1 and p 2 /q 2 such that andF is sufficiently close to F , then and hence F −→ ρ(F ) is continuous.
The next Lemma shows that the rotation number is increasing as a function of F . Lemma 3.21. Let f and g be two orientation preserving maps satisfying C.1-C4. If f ≥ g then ρ(f ) ≥ ρ(g).
Proof. Let x < c and F and G be two lifts of f and g such that F (x + 1) = F (x) + 1 and G(x + 1) = G(x) + 1. As F and G are increasing functions undergoing a positive gap at x = k, F n (x) > G n (x), even if, for some n, F n (x) or G n (x) reach any of the discontinuities at x = k. Hence, As the rotation number does not depend on x, we get ρ(f ) ≥ ρ(g).
We next study properties of the periodic orbits of orientation preserving circle maps satisfying C.1-C4. These will be crucial to show symbolic properties of periodic orbits in §3.4. To this end, we provide the following definitions.
Definition 3.22 (Well ordered sequence of points, twist periodic orbit). Let F : R −→ R be an continuous monotonically increasing map, and let be a sequence of q points. Let Π be the projection map and consider the points given by Π −1 (Π(x i )), which consist of adding integers to the original sequence (3.6). We write the points Π −1 (Π(x i )) as the sequence given by We call this a lifted sequence. Moreover, we say that the sequence (x i ) i∈N is p, q-ordered by F if it satisfies Notice that, if F is the lift an orientation preserving circle map f and a sequence is p, q-ordered by F , then, by definition, the sequence x 0 , . . . , x q−1 must be periodic by f . In this case, such periodic orbit satisfying (3.8)-(3.9) is called a twist cycle or twist periodic orbit of f , and the corresponding lifted sequence is a lifted cycle. The following result tells us that this is reciprocal; that is, any periodic orbit must be a twist cycle.
be the points of a periodic orbit of an orientation preserving map f with rotation number p/q. Let F be the (degree one) lift of f . Then, the sequence of points given by Π −1 (Π(x i )) is p, q-ordered by F .

The proof of this result is as in the continuous case ([ALM00] Lem. 3.7.4) but taking into account that the lift F is bi-valued at integer numbers.
Proof. By properly choosing the image of F at integer values, the lift F restricted to the lifted cycle of the periodic orbit is an order preserving bijection. That is, by iterating the points x 0 < · · · < x q−1 one visits all the points of the lifted cycle exactly once, and the order is preserved (F (x j ) < F (x i ) iff x j < x i ). Hence, F (x i ) = x i+r for all i and some integer r. If not, then one gets that F (x j ) = x j+r1 and F (x i ) = x i+r2 . If r 1 = r 2 , then the number of points of the lifted sequence in [x j+r1 , x i+r2 ] and in [x j , x i ] does not coincide, and hence the order cannot be preserved. As the rotation number is ρ(F ) = p/q, we have that necessarily x i + p = F q (x i ) = x i+qr . As x i belongs to q-periodic cycle, we get x i+qr = x i + r and hence r = p and the cycle is p, q-ordered.
The next result is also very well known for continuous circle maps. As in previous results, we provide a standard proof adapted for the discontinuous case.
Proposition 3.24. Let f be an orientation preserving circle map satisfying C.1-C4 and assume that ρ(f ) ∈ R\Q. Then, if condition C.4 is a strict inequality, the ω-limit set is a Cantor set. f −1 , which is a function with a "hole", as it is not defined in U . After filling this hole with the value 1, we obtain the continuous function (3.11) (see Fig. 3.7) which has the same rotation number as f . Note that Hence, if for some n, c ∈ f n (U ), then g n (c) ∈ U , g n+1 (c) = 1, g n+2 (c) = c, and hence g has a periodic orbit of period n + 2, which is not compatible with having irrational rotation number. Next we show that lim is a Cantor set. As long as c / ∈ f n (U ), at each iteration, f n (S 1 ) = S 1 \f n−1 (U ) consists of subtracting a nonempty interval to the interior of f n−1 (U ) (see Fig. 3.8). By Corollary 3.3 of [Vee89], the total removed amount, ∪ i≥0 f i (U ), is dense. This comes from the fact that, although the map g is not differentiable, Denjoy theorem holds and, if the rotation number is irrational, g has no "homtervals"; in particular, U is not a homterval and the sequence g n (U ) is pairwise disjoint. Therefore, by construction, S 1 \ ∪ i≥0 f i (U ) is a Cantor set. Moreover, also by construction, the images of x − and x + are dense in this set. Thus, every point in S 1 \∪ i≥0 f i (U ) has a dense orbit and hence this set becomes the ω-limit of f .
Remark 3.25. If condition C.4 is satisfied by an equality (the map becomes continuous) and the rotation number is irrational, then the ω-limit of f may also be a Cantor set or the whole circle. If f is C 2 , then Denjoy theorem holds and the latter occurs. However, if it is C 1 or C 0 , f may become a Denjoy counterexample (see [Nit71]), and it's ω-limit may be a Cantor set.
To conclude this section, we recover a piecewise-smooth map as defined in Eq. (2.1). As mentioned above, after applying the change of variable given in Eq. (3.2), the mapf ( After applying the reparameterization γ given in Eq. (2.7), the value becomes an strictly decreasing function of λ such that lim λ→0 + c λ = 1 lim λ→1 − c λ = 0. Moreover, for λ = 0 and λ = 1, the mapf possesses fixed points at x = 1 and x = 0, respectively.
Given a piecewise-smooth map f satisfying h.1-h3, we will define its rotation number as the rotation number of the mapf = φ • f • φ −1 obtained after a reduction to a circle map: For simplicity, we will also refer to the lift of f as the lift off .
Remark 3.26. Let (x 0 , . . . , x q−1 ) be a periodic orbit of a piecewise-smooth map f satisfying conditions C.1-C4, and let x = (x 0 . . . x q−1 ) ∈ {L, R} q be its associated symbolic sequence regarding the symbolic encoding given in (2.3): Then, recalling Remark 3.12 and the fact that the image of x by the lift of f , F (x), crosses an integer number m when x < c+m < F (x), the rotation number of f , ρ(f ), becomes the η number defined in (2.6). That is, it becomes the ration of number of symbols R contained in x to the length of the sequence x, q.

Symbolic dynamics and families of orientation preserving maps
In this section we will show some dynamical properties of maps satisfying conditions C.1-C4, focusing specially on periodic orbits, their symbolic itineraries and their relation with the rotation number. In some of the results, we will additionally require the map to be contractive, however, we emphasize that, when not specified, the results that we present here do not require contractiveness. The main result in this section is the following Theorem, which is, recalling that periodic orbits of orientation preserving circle maps satisfying C.1-C4 are well ordered, a straightforward consequence of Proposition 3.33.
Theorem 3.27. The symbolic sequence of the itinerary of a periodic orbit of a circle map satisfying C.1-C4 with rotation number P/Q is the one in the Farey tree of symbolic sequences associated with rotation number P/Q.
At the end of this section (Lemma 3.35) we show that, for a piecewise-smooth map (2.1) satisfying i) of Theorem 2.5, the η-number defined in Def. 2.4 follows a devil's staircase. This is a consequence of Theorem 3.34 proved in [Boy85].
To show this, we start with the following Definition 3.28. We call W p,q the set of periodic symbolic sequences generated by a symbolic block of length q containing p symbols R: Of special interest will be the well ordered symbolic sequences contained in these sets: Definition 3.29. Let x ∈ W p,q be a periodic symbolic sequence. Consider the (lexicographically) ordered sequence given by iterates of x by σ (3.12) We say that the sequence x is a p, q-ordered (symbolic) sequence if In other words, σ acts on the sequence (3.12) as a cyclic permutation: there exists some k ∈ N, 0 < k < q, such that The next example illustrates the previous definition.
The following result identifies the symbolic sequences of twist periodic orbits.  1-2). Under the conditions of Proposition 3.23, if the sequence of points (Π −1 (Π(x i ))) 0≤i≤q−1 is p, q-ordered by F (see Def. 3.22) then the itinerary I f (x i ) ∈ W p,q is a p, q-ordered symbolic sequence (see Def. 3.29).
Proof. Let k be such that We first note that, for 0 ≤ i, j ≤ q − 1 we have We then write kp = q + r, r ≥ 0.
Then, the results comes from the fact that which occurs iff 0 < r ≤ p. Assume that r > p. Then q + r − p > q and hence (k − 1)p > q, which contradicts (3.13). Letting x = I f (x 0 ), this implies where N is the smallest such that N k = 0 (mod q).
The next result recovers what was stated in Remark 3.26: the η-number (Def. 2.4) for a piecewise-smooth map satisfying i) of Theorem 2.5 becomes the rotation number (Def. 3.11) of the orientation preserving circle map obtained after the change (3.2).
Corollary 3.32. Let f be an orientation preserving map, and let x belong to a q-periodic orbit with symbolic sequence I f (x) = x ∞ ∈ W p,q . Then, the rotation number becomes ρ(f ) = p q .
That is, it is given by the ratio between the number of R symbols contained in x and the period, q, of the sequence (η-number).
Our next step consists of showing that the symbolic sequence associated with a periodic orbit of an orientation preserving circle map belongs to the Farey tree of symbolic sequences shown in Fig. 3.5(b). More precisely, we show that such a symbolic sequence is obtained by the concatenation of the symbolic sequences associated with the periodic orbits of the Farey parents of its rotation number. As a consequence of that, ones obtains Theorem 3.27, announced above.
In Section 3.5 we will present an alternative approach using the maximin properties of these sequences.
Proposition 3.33. Let f be an orientation preserving circle map (satisfying C.1-C4), and assume it has a periodic orbit with rotation number P/Q, (P, Q) = 1. Let ∆ be its symbolic sequence, and assume that ∆ is minimal, Let p, q, p and q natural numbers such that and let α and ω be the minimal symbolic sequences corresponding to the periodic orbits with rotation numbers p/q and p /q , respectively. Then ∆ is the concatenation of α and ω, ∆ = αω.
Proof. Let 0 < z 0 < z 1 < · · · < z Q−1 < 1 be the periodic orbit with rotation number P/Q. Let us consider the sequence of points given by Π −1 (Π(z i )), where Π is the projection map given in (3.7). Let F be the lift of f . Due to Proposition 3.23, the points z i are P, Q-ordered by F : Let us now split the first Q points of the sequence z iP , 0 ≤ i < Q, in two subsequences defined as follows As before, we extend these sequences by means of Π −1 . On one hand, by construction, the sequence x n fulfills On the other hand, using that p q − pq = 1, we get that Hence, by translating the subindexes of y i → y i+pq we get a sequence of points 0 < y 0 < y 1 < · · · <y q −1 < 1 < y q < y q +1 < · · · < y 2q −1 < 2 < y 2q < . . .
By Proposition 3.23, these sequences follow the symbolic dynamics given by the periodic orbits with rotation numbers p/q and p /q , as they are p, q and p , q -ordered: By construction, the symbolic sequence ∆ is the concatenation of α and ω, as we wanted to show.
In the rest of this section we study the rotation number for families of orientation preserving circle maps; that is, under the variation of the parameter c in condition C.3, which, by means of the change of variables given in Eq. (3.2), is equivalent to the parameter λ of parametrizing the curve in parameter space mentioned in Theorem 2.5.
Recalling Proposition 3.20 and Lemma 3.21, we already have that, when varying λ from 0 to 1, the rotation number (and hence the η-number) is continuous and monotonically increases from 0 to 1. In order to show that, moreover, it is a devil's staircase, we need to show that, in addition, it is constant for all values of λ except for a Cantor set of zero measure. This is will come from the following Then m(E) = 0, where m denotes Lebesgue measure, and furthermore E has zero Hausdorff dimension.
Generalizations of the previous theorem can be found in [Vee89,Swi89]. Finally, we show that Theorem 3.34 extends to a piecewise-smooth system of the form (2.1) satisfying h.1-h3 and i) of Theorem 2.5 (or an orientation preserving circle map satisfying C.1-C4). This will prove that the η-number follows a devil's staircase. Proof. We note that the mapf λ is invertible. Let φ as in (3.2) and define Then, the inversef −1 λ (y) is an increasing expanding map with a "hole" for y ∈ [a, b] (see fig. 3.9(b)). As the trajectories of any point x ∈ [0, 1] byf do not reach the interval [a, b], we can proceed as the proof of Proposition 3.24 and complete the mapf −1 λ with a horizontal part equal to 1 for y ∈ [a, b]. This allows us to consider a map be a parameterization satisfying H.1-H3 of Theorem 2.5. Then, the interval [a, b] smoothly varies from [0, φ(f L (−µ R (0)))] to [φ(f R (µ L (1))), 1] when λ is varied from λ = 0 to λ = 1. Let g λ (y) by g(y) after applying the reparameterization γ. Then, we have Maybe we need the length of [a, b] to be kept constant.
g λ (y) = g 0 (y) + λ, and hence we can apply Theorem 3.34 and get the result.

The maximin approach
In this section we first present an alternative approach to identify sequences in the Farey tree of symbolic sequences with the itineraries of orientation preserving circle maps. This approach is based on results presented in the 80's on maximin symbolic sequences (see Def. 3.41). In some sense, this approach provides stronger results, as they give more information about the symbolic sequences associated with periodic orbits of orientation preserving circle maps.
In particular, given the rotation number, p/q, the maximin property permits to obtain the proper symbolic itinerary without constructing the whole Farey tree of symbolic sequences. Instead, one needs to find in W p,q which is the symbolic sequence that verifies a certain condition (maximin/minimax). The path of this alternative proof is as follows. By Propositions 3.13 and 3.23, we know that, if its rotation number is rational, p/q, then any periodic orbit has a p, q-ordered symbolic itinerary. Then, by Theorem 3.40, such an itinerary is maximin. Finally, Proposition 3.39 tells us that a symbolic sequence is maximin if and only if it belongs to the Farey tree of symbolic sequences. The most difficult step of this alternative approach relies on proving Theorem 3.40. Once we get that the symbolic itineraries of orientation preserving circle maps belong to the Farey tree of symbolic sequences, one proceeds as in §3.4 to study the bifurcation structure when the parameters c or λ are varied in order to prove i) in Theorem 2.5.
In addition, in this section we also recover a result on quasi-contractions (see Def. 3.41) from the late 80's. This shows show that, in the contractive case, any periodic orbit must have a maximin itinerary, which permits to avoid going trough the concept of well ordered sequences. This is given by Theorem 3.42, which states that, under the assumption of contraction, any periodic orbit of an orientation preserving map has a maximin symbolic itinerary. However, not only the proof of this result is significantly more difficult than the ones shown in Sections 3.3 and 3.4, but also we emphasize that it only holds for the contractive case, whereas most of the results presented in those sections do not.
We start by defining the maximin/minimax properties.
Definition 3.36. We say that a symbolic sequence x ∈ W p,q is As it was proven in [Ber82,GIT84] using different techniques, the maximin and minimax properties are equivalent. That is, one has the following Theorem 3.37 ( [Ber82,GIT84]). Let x ∈ W p,q . Then, x is maximin if and only if it is minimax.
Example 3.38. Let η = 2/5. Up to cyclic permutations, there exist only two periodic sequences in W 2,5 , which are represented by means of the minimal and maximal blocks Then, as the sequence L 2 RLR is minimax and maximin.
The following results tell us that all the sequences shown in the Farey tree of symbolic sequences (Fig. 3.5(b)) (given by consecutive concatenation), are maximin (minimax).
Proposition 3.39 ([Gam87] Proposition II.2.4-3). Let p/q < p /q be the irreducible form of two Farey neighbours, and let x ∞ ∈ W p,q and y ∞ ∈ W p ,q two maximin sequences, with x ∈ {L, R} q and y ∈ {L, R} q minimal blocks.
Then, the sequences given by the concatenation of these two blocks (xy) ∞ ∈ W (p+p )/(q+q ) is maximin.
Proof. One first sees that x and y belong to the same domain of some deflation; i.e, a map which collapses blocks contained in sequences as follows: Then, one proceeds by induction using that π(x) and π(y) are maximin, and also is π −1 (xy). See [Gam87] for more details.
Finally, together with Proposition 3.31, the following theorem gives us and alternative way to the one shown in §3.4 to show that the symbolic sequences of periodic orbits of orientation preserving circle maps are given by the Farey tree of symbolic sequences.
For completeness, we finally present an alternative way to obtain that, in the contractive case, any periodic orbit of an orientation preservinc circle map must have a maximin itinerary. This result was proven in [Gam87, GGT88,GT88] and presented in a quite abstract and general form. However, it can be applied, under the hypothesis of contraction, to orientation preserving circle maps (or equivalently to piecewise-smooth maps as in (2.1)) to show that the symbolic itineraries of periodic orbits of such maps are maximin and, hence, they are found in the Farey tree of symbolic sequences.
We first difine a quasi contraction in an abstract way.
Definition 3.41. Let (E 0 , d 0 ) and (E 1 , d 1 ) be two metric spaces and F 0 ∈ E 0 and F 1 ∈ E 1 two points. Then, a map is a quasi contraction if there exists 0 ≤ k ≤ 1 such that for any Note that the sets E 0 and E 1 can be of arbitrary dimension. In addition, notice that an orientation preserving circle map satisfying C.1-C4 under the assumption of contraction, is a quasi-contraction with E 0 = [0, c], E 1 = [c, 1], and F 0 = F 1 = c. Equivalently, E 0 = (−∞, 0], E 1 = [0, ∞) and F 0 = F 1 = 0 for the piecewise-smooth map (2.1).
We now note that the definition of an itinerary of a trajectory defined in Eq. (2.2) extends naturally to the general context of a quasi-contraction. Given x ∈ E 0 ∪ E 1 and f as in Eq. (3.14), its itinerary by f becomes Note that also the η-number defined in Eq. (2.6) also extends to the type of maps as in Eq. (3.14). We then have the following result: Theorem 3.42.
[GT88] Let f be a quasi-contraction. Then, i) f admits 0, 1 or 2 periodic orbits ii) any periodic orbit of f has an itinerary which is maximin iii) if f has two periodic orbits, then their itineraries belong to W p,q and W p ,q and p/q and p /q are Farey neighbours.
The previous theorem was proved in [GGT88] (Theorem A) for the case that quasi-contraction constant k satisfies 0 ≤ k ≤ 1/2. The version given in [GT88] not only extends to the case of the quasi-contractions with 0 ≤ k ≤ 1, but it also provides more information; we give here a mutilated version which adapts to our purposes.
Note that, by the uniqueness of the rotation number for the orientation preserving case, when applied to quasi-contracting maps, one only gets 0 or 1 periodic orbit. The situation in Theorem 3.42 for 2 periodic orbits is only possible in the non-orientably case, which will be treated in §4. The case for which one gets 0 periodic orbits correponds to an irrational rotation number.

Non-orientable case
The non-orientable (increasing-decreasing) case occurs under conditions ii) of Theorem 2.5. In this case, the map (2.1) becomes increasing for x < 0 and decreasing for x > 0. As mentioned in §1 and §2.3, this situation was discussed in [Hom96b] §3.3 and proven in full detail in [AGS11]. Independently some of these results were proven afterwards by other authors [JMB + 13], who seem not aware of these previous works. For completeness, we provide in this section an overview of this proof.
We first observe (see Fig. 4.10) that a map f as in Eq. (2.1) satisfying conditions h.1-h3 and ii) of Theorem 2.5 is a map on the interval (4.1) We now wonder about the symbolic itineraries of periodic orbits for a map (4.1) under the mentioned conditions. Clearly, when the parameters µ L and µ R are varied along the parametrization (2.7) satisfying H.1-H3, the map (4.1) becomes negative for x ∈ [0, µ L ]. Hence, periodic orbits cannot have symbolic itineraries with consecutive R symbols. In order to show that only possible itineraries for periodic orbits are of the form L n R, we define the sequences of preimages of x = 0 by f L and f R : a 0 = 0, a n = f −1 L (a n−1 ) if n > 0, (4.2) b n = f −1 R (a n ) if n ≥ n 0 . (4.3) where m indicates the length of the interval. Then we have the following Lemma 4.1. If f is a map of type (2.1) fulfilling conditions h.1-h3 and ii) of Theorem 2.5. Then there exists at most one a j (equiv. b j ) such that a j ∈ f r ((0, µ L ]) (equiv. b j ∈ (0, µ L ]).
Using the property shown in Eq. and, since f R is a contracting one obtains m(f r ((0, µ L ]) < m([a n+1 , a n ]) ∀n.
Therefore, at most one a n can be located in f ((0, µ L ]).
The next Lemma tells us what symbolic itineraries are possible.
Lemma 4.2 ([AGS11] Lem. 7). Let f be a piecewise-smooth map as considered above, and let x 0 < · · · < x n , x i ∈ [f R (µ L ), µ L ] be a periodic orbit. Then, I f (x 0 ) = (L n R) ∞ . The proof of this Lemma is in fact an extension of the arguments presented in [Hom96a] §3.3, and was provided in full detail in [AGS11]. We provide it here for completeness.
Proof. If f has periodic orbit of period n ≥ 2, we necessary have f R ((0, µ L ]) ∩ [a n , a n−1 ] = ∅, which can be given due to one of the next three situations (see Figs. 4.11 and 4.12) S.1 a n−1 ∈ f R ((0, µ L ]) S.2 f R ((0, µ L ]) ⊂ (a n , a n−1 ) S.3 a n ∈ f R ((0, µ L ]) If S.1 holds, b n−1 ∈ (0, µ L ] and are continuous contracting functions which must have a unique (stable) fixed point. Therefore, two stable periodic orbits of type L n R and L n−1 R coexist. Note that for n = 2 this proves also the existence of a LR-periodic orbit. In the second case (S.2), b n−1 / ∈ (0, µ L ] ([0, µ L ] ⊂ (b n−1 , b n )) and f n L f R : (0, µ L ] −→ (0, µ L ] is a continuous contracting function which also must have a unique (stable) fixed point. In this case, there exists a unique periodic orbit of type L n R which is the unique attractor in (0, µ L ]. Finally, if S.3 holds, replacing n by n − 1 and arguing as in S.1, one has that a stable periodic orbit of type L n R coexists with a stable L n+1 R-periodic one.
decreasing γ, S.3 holds, a L n+1 R-periodic orbit bifurcates and coexists with the L n R-periodic orbit.
As this occurs for all n, this argument proves the bifurcation scenario described in §2.3