Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Fixed Point Theory and Algorithms https://doi.org/10.1186/s13663-020-00687-9 for Sciences and Engineering RESEARCH Open Access Minimal set of periods for continuous self-maps of the eight space Jaume Llibre1* and Ana Sá2 * Correspondence: jllibre@mat.uab.es Abstract 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, Let Gk be a bouquet of circles, i.e., the quotient space of the interval [0, k] obtained by 08193 Bellaterra, Barcelona, Spain identifying all points of integer coordinates to a single point, called the branching Full list of author information is point of Gk . Thus, G1 is the circle, G2 is the eight space, and G3 is the trefoil. Let available at the end of the article f : Gk → Gk be a continuous map such that, for k > 1, the branching point is fixed. If Per(f ) denotes theset of periods of f , the minimal set of periods of f , denoted by MPer(f ), is defined as gf Per(g) where g : Gk → Gk is homological to f . The sets MPer(f ) are well known for circle maps. Here, we classify all the sets MPer(f ) for self-maps of the eight space. MSC: 37E15 Keywords: Continuous maps; Periodic orbit; Period; Minimal period; The space in shape of eight 1 Introduction and statement of the results In dynamical systems it is often the case that topological information can be used to study qualitative or quantitative properties of the system. This work deals with the problem of determining the set of periods of the periodic orbits of a map given the homology class of the map. A finite graph (simply a graph) G is a topological space formed by a finite set of points V (points of V are called vertices) and a finite set of open arcs (called edges) in such a way that each open arc is attached by its endpoints to vertices. An open arc is a subset of G homeomorphic to the open interval (0, 1). Note that a finite graph is compact since it is the union of a finite number of compact subsets (the closed edges and the vertices). Notice that a closed edge is homeomorphic either to the closed interval [0, 1] or to the circle. It may be either connected or disconnected, and it may have isolated vertices. The valence of a vertex is the number of edges with the vertex as an endpoint (where the closed edges homeomorphic to a circle are counted twice). The vertices with valence 1 of a connected graph are endpoints of the graph and the vertices with valence larger than 2 are branching points. Suppose that f : G → G is a continuous map, in what follows a graph map. A fixed point of f is a point x in G such that f (x) = x. We will call x a periodic point of period n if x is a © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 2 of 26 fixed point of f n but it is not fixed by any f k for 1 ≤ k < n. We denote by Per(f ) the set of natural numbers corresponding to periods of the periodic points of f . Let G be a connected graph, and let f be a graph map. Then f induces endomorphisms f∗k : Hk (G) → Hk (G) for k = 0, 1 on the integral homology groups of G, where H0 (G) ≈ Z k (because G is connected), and H1 (G) ≈ Z ⊕ · · · ⊕ Z, where k is the number of indepen- dent circuits or loops of G as elements of H1 (G). A circuit of G is a subset of G homeo- morphic to the circle. The endomorphisms f∗0 and f∗1 are represented by integer matrices. Furthermore, since G is connected, f∗0 is the identity. The endomorphism f∗1 will play the main role in our analysis of the minimal sets of periods for graph maps on G. In what follows f∗1 will be denoted by f∗ . For example, if H1 (G) ≈ Z ⊕ Z and   a b f∗ = , c d this means that the graph G has two independent oriented circuits. Moreover, if the first circuit covers itself exactly a1 times following the same orientation (not necessarily in a consecutive way) and exactly a2 times following the converse orientation (not necessarily in a consecutive way), then a = a1 – a2 . Similarly, if the first circuit covers the second one exactly b1 times following the same orientation (not necessarily in a consecutive way) and exactly b2 times following the converse orientation (not necessarily in a consecutive way), then b = b1 – b2 . An analogous explanation can be given with the second independent circuit and with b and d instead of a and c, respectively. Let Gk be a bouquet of k circles, that is, the quotient space of [0, k] obtained by iden- tifying all points of integer coordinates to a single point. Notice that G1 is the circle and that G2 is usually called the eight space. For the Gk graph, we have H0 (Gk ) ≈ Z, k H1 (Gk ) ≈ Z ⊕ · · · ⊕ Z, f∗0 ≈ id, and f∗1 = f∗ = A, where A is a k × k integral matrix. For more details on graph maps, see [16] or [18]. Our main goal is to study the set Per(f ) for graph maps. More explicitly, we want to pro- vide a description of the minimal set of periods (see below) attained within the homology class of a given graph map. When the map g : G → G is homological to f (i.e., g induces the same endomorphisms as f on the homology groups of G), we shall write g  f . We define the minimal set of periods of f to be the set  MPer(f ) = Per(g). gf From its definition MPer(f ) is the maximal subset of periods contained in Per(g) for all g f. Our main objective is to characterize the minimal sets of periods MPer(f ) for graph maps f : Gi → Gi with the branching point a fixed point for i = 2, 3. So, always 1 ∈ MPer(f ). Even for circle maps f : G1 → G1 the characterization of all minimal sets of periods MPer(f ) is interesting and nontrivial, see Theorem A. This result was stated by Efremova [12] and Block et al. [9] without giving a complete proof. As far as we know, the first complete proof was given in [4]. We denote by N the set of all natural numbers and by kN the set {kl : l ∈ N}. Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 3 of 26 Theorem A Let f : G1 → G1 be a circle map such that the endomorphism induced by f on the first homology group is f∗ = (d) (i.e., d is the degree of f ). Then the following statements hold: (a) If d ∈/ {–2, –1, 0, 1}, then MPer(f ) = N. (b) If d = –2, then MPer(f ) = N \ {2}. (c) If d ∈ {–1, 0}, then MPer(f ) = {1}. (d) If d = 1, then MPer(f ) = ∅. In the next theorem we characterize the minimal sets of periods for eight maps, i.e., for continuous maps f : G2 → G2 . Theorem B Let f : G2 → G2 be an eight map such that   a b f∗ = . c d Suppose that the branching point is a fixed point. Then the following statements hold: (a) If {a, d} ⊂ {–2, –1, 0, 1}, then MPer(f ) = N. (b) If –2 ∈ {a, d} and {a, d} ⊂ {–2, –1, 0, 1}, then ⎧ ⎨N \ {2} if bc = 0, MPer(f ) = ⎩N if bc = 0. (c) Assume that {a, d} ⊂ {–1, 0, 1}. (c1) If |a| + |d| = 2, then ⎧ ⎪ ⎨{1} ⎪ if bc = 0, MPer(f ) = N \ {2} if bc = 1, ⎪ ⎪ ⎩ N if bc = –1 or |bc| > 1. (c2) If |a| + |d| = 1 and (c21) a = 1, d = 0, then ⎧ ⎪ ⎨{1} ⎪ if bc = 0, MPer(f ) = N \ {2} if (b, c) ∈ R, ⎪ ⎪ ⎩ N otherwise; where R = {(1, 1), (–1, –1), (1, 2), (–1, –2)}. (c22) a = 0, d = 1, then it follows (c21) interchanging b and c. (c23) a = –1, d = 0, then ⎧ ⎪ ⎪{1} if bc = 0, ⎪ ⎪ ⎪ ⎨N \ {2} if (b, c) ∈ R, MPer(f ) = ⎪ ⎪N \ {3} if bc = –1, ⎪ ⎪ ⎪ ⎩ N otherwise. (c24) a = 0, d = –1, then it follows (c23) interchanging b and c. Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 4 of 26 (c3) If |a| + |d| = 0, then ⎧ ⎪ ⎪{1} if bc = 0 or bc = 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨{1, 2} ⎪ if bc = –1, MPer(f ) = {1} ∪ (2N \ {2}) if bc = 2, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ {1} ∪ (2N \ {4}) if bc = –2, ⎪ ⎪ ⎩ {1} ∪ 2N if |bc| > 2. We remark that Theorem B implies Theorem A if f has a fixed point by choosing, for instance, a = b = c = 0. The study of the minimal set of periods of a homotopy class of maps instead of its homol- ogy class is the main objective of the fixed point theory, see for instance the books of Brown [10], Jiang [13], and Kiang [15]. Other extensions from circle maps to n-dimensional torus have been done in [2] and [14], and from circle maps to transversal n-sphere maps in [11]. Some different results on the periods of graph maps have been given in [1, 3, 5–8, 16, 17]. Finally, we recall that the classification of the dynamics on the graph maps helps the clas- sification of the homeomorphisms on the surfaces, see for instance [19] and the references quoted therein. This work is organized as follows. How to obtain a given period for a graph map by using the notion of f -covering is described in Sect. 2. The proof of Theorem B is given in Sect. 3. 2 Periods and f -covering Let f : G → G be a graph map and x ∈ G be a periodic point of period n. The set {x, f (x), . . . , f n–1 (x)} is called the periodic orbit of x. A set I ⊂ G will be called an interval if there is a homeomorphism h : J → I, where J is [0, 1], (0, 1], [0, 1), or (0, 1). The set h((0, 1)) will be called the interior of I. If J = [0, 1], the interval I will be called closed; if J = (0, 1), it will be called open. Notice that it may happen that the above terminology does not coincide with the one used when we think about I as a subset of G (the same applies to the edges of G). For example, if G = I = [0, 1] and h = identity, then for I regarded as a subset of the topological space G, I is both open and closed, and the interior of I is I. Let Ci and Cj be two circuits of Gk . A closed interval I = [u, v] is basic if I ⊂ Ci , f (I) = Cj , where {i, j} ⊂ {1, 2, . . . , k}, f (u) = f (v) = p, where p is the branching point of Gk , and there is no other closed interval K  I such that f (K) = Cj . Of course the definition of basic interval also applies to the particular case that [u, v] = Ci . Let I and J be two basic intervals, K ⊂ I, L ⊂ J be two subintervals. Then we say that K f-covers L, and we write K → L if there exists a closed subinterval M of K such that f (M) = L. If L = J = Cj , we say that K = I f -covers L because either f (K) = L or K = I = Ci and f (K) = L by the definition of basic intervals. Lemma 2.1 Suppose that I1 , I2 , . . . , In are intervals such that I1 → I2 → · · · → In → I1 with I1 different from a circuit. Then there is a fixed point z of f n such that z ∈ I1 , f (z) ∈ I2 , . . . , f n–1 (z) ∈ In . Proof Since In → I1 and I1 is not a circuit, there is a closed interval Jn ⊂ In such that f (Jn ) = I1 . Similarly, there are closed intervals or circuits J1 , . . . , Jn–1 such that, for each k = 1, . . . , n– Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 5 of 26 1, Jk ⊂ Ik and f (Jk ) = Jk+1 . It follows that f n (J1 ) = I1 and since J1 ⊂ I1 and I1 is not a circuit, by Bolzano’s theorem f n has a fixed point z ∈ J1 . Clearly, z ∈ I1 , f (z) ∈ I2 , . . . , f n–1 (z) ∈ In .  A sequence of the form I1 → I2 → · · · → In → I1 is called a loop of length n. Let I1 → I2 → · · · → In → I1 and J1 → J2 → · · · → Jm → J1 be two loops such that I1 = J1 . We define the concatenation of these two loops as the loop I1 → I2 → · · · → In → I1 → J2 → · · · → Jm → I1 . We say that a loop is an m-repetition, m ≥ 2, of a given loop if it is the concatenation of that loop with itself m times. We say that a loop is non-repetitive if it is not an m-repetition of any of its subloops with m ≥ 2. In what follows, a Gk -map f is a continuous map f : Gk → Gk such that f (p) = p, where p is the branching point of Gk . Proposition 2.2 Let f be a Gk -map. Suppose that f has two intervals I1 and I2 such that Int(I1 ) ∩ Int(I2 ) = ∅ and I1 ∩ I2 has no fixed points. If f has the subgraph I1  I2 , then   Per(f ) = N. Proof Clearly, since p ∈/ I1 ∩ I2 , at least one of the intervals, I1 or I2 , is not a circuit. Without loss of generality, we assume that I1 is not a circuit. We consider the non-repetitive loop I1 → I2 → I1 → · · · → I1 of length n ≥ 2. Since Int(I1 ) ∩ Int(I2 ) = ∅ and I1 ∩ I2 has no fixed points, by Lemma 2.1, there is a periodic point z of f with period n ≥ 2. That is, Per(f ) = N.  In what follows when we say “we have two intervals A and B” we are really saying that we have two different intervals A and B. We remark that if we have two basic intervals I1 and I2 such that p ∈/ I1 ∩ I2 , then they satisfy the assumptions of Proposition 2.2. Proposition 2.3 Let f be a Gk -map. Suppose that f has three intervals I1 , I2 , and I3 such that Int(Ii ) ∩ Int(Ij ) = ∅ for all i = j and Ii ∩ Ij has no fixed points for some i = j. If f has the subgraph I1 → I2 → I3 → I1 , then Per(f ) ⊃ N \ {2}. Moreover, if I2 ∩ I3 = ∅ and I3 → I2 ,  then 2 ∈ Per(f ). Proof We consider the non-repetitive loop I1 → I2 → I3 → I1 → · · · → I1 of length n ≥ 3. Since Int(Ii ) ∩ Int(Ij ) = ∅ for all i = j and Ii ∩ Ij has no fixed points for some i = j, by Lemma 2.1, there is a periodic point z of f with period n ≥ 3. Therefore, Per(f ) ⊃ N \ {2}. We suppose now that I2 ∩ I3 = ∅ and I3 → I2 . We consider the non-repetitive loop I2 → I3 → I2 of length 2. By Lemma 2.1 there is a periodic point z of f with period 2.  We remark that if we have three basic intervals I1 , I2 , and I3 such that p ∈/ Ii for some i ∈ {1, 2, 3}, then we are in the assumptions of Proposition 2.3. 3 The eight In this section we shall prove Theorem B. The two circuits of G2 are denoted by C1 and C2 . If f∗ is given as in Theorem B, we consider that the circuit C1 covers itself |a| times and it covers C2 |c| times. Similarly for the circuit C2 . Proof of statement (a) of Theorem B Suppose that {a, d} ⊂ {–2, –1, 0, 1}. Case 1: Assume that {a, d} ⊂ {–2, –1, 0, 1, 2}. Without loss of generality, we may assume that |a| ≥ 3. From the graph of f (see for instance Fig. 1), it is clear that there are two basic Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 6 of 26 Figure 1 Examples of maps with {a, d} ⊂ {–2, –1, 0, 1, 2} Figure 2 I11 = [p, a1 ], I21 = [b0 , b1 ], and I13 = [a1 , a2 ] intervals I1 and I2 in C1 such that p ∈/ I1 ∩ I2 and f has the subgraph of Proposition 2.2, so Per(f ) = N. That is, MPer(f ) = N. Case 2: Suppose that 2 ∈ {a, d} and {a, d} ⊂ {–2, –1, 0, 1, 2}. Without loss of generality, we may assume that a = 2. Since a = 2, this means that f has at least two basic intervals I1 and I2 in C1 such that f has the subgraph of Proposition 2.2. If p ∈/ I1 ∩ I2 , then, by Proposition 2.2, Per(f ) = N. But not always I1 and I2 satisfy that p ∈/ I1 ∩ I2 . In this case let p and a0 be the endpoints of I1 , b0 and p be the endpoints of I2 (see for instance Fig. 2). We establish an ordering in the intervals I1 and I2 in such a way that p is the smallest element of I1 and the greatest of I2 . Set I1 = [p, a0 ] and I2 = [b0 , p]. Notice that we may have a0 = b0 . Consider the subset (f |I1 )–1 (a0 ) of C1 . Let a1 be the infimum of the points in (f |I1 )–1 (a0 ). Consider the subset (f |I2 )–1 (a0 ) of C1 and choose b1 to be the infimum of the points in (f |I2 )–1 (a0 ). Set I11 = [p, a1 ], I12 = [a1 , a0 ], and I21 = [b0 , b1 ]. Now we take the in- terval I13 = [a1 , a2 ], where a2 denotes the infimum of the points in the subset (f |I12 )–1 (b1 ) I11 → I13  I21 → I11 . Since I21 ∩ I13 = ∅, by Proposi-  of C1 . Then f has the subgraph tion 2.3, n ∈ Per(f ) for all n ≥ 1. Therefore, MPer(f ) = N. This proves statement (a).  Proof of statement (b) of Theorem B Suppose that –2 ∈ {a, d} and {a, d} ⊂ {–2, –1, 0, 1}. Without loss of generality, we may assume that a = –2. Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 7 of 26 First we suppose that bc = 0. We always have four basic intervals I1 , I2 , I3 , and I4 , I1 , I2 , I3 ⊂ C1 , and I4 ⊂ C2 such that either p ∈/ I1 ∩ I3 or I2 ∩ I4 = ∅ and f has the subgraph (see for instance Fig. 3). If p ∈/ I1 ∩ I3 , by Proposition 2.2, Per(f ) = N. If I2 ∩ I4 = ∅, by Proposition 2.3, Per(f ) = N. Therefore, if bc = 0, MPer(f ) = N. We suppose now that bc = 0. As it can be deduced from the examples of Fig. 4, 2 ∈/ MPer(f ). Since a = –2, this means that f has at least two basic intervals I1 and I2 in C1 such that f has the subgraph of Proposition 2.2. If p ∈/ I1 ∩ I2 , then by Proposition 2.2 Per(f ) = N. But not always p ∈/ I1 ∩ I2 . In this case let p and a0 be the endpoints of I1 , b0 and p be the endpoints of I2 (see for instance Fig. 5). We consider an ordering in the intervals I1 and I2 in such a way that p is the smallest element of I1 and the greatest of I2 . Write I1 = [p, a0 ] and I2 = [b0 , p]. Notice that we may have a0 = b0 . Consider the subsets (f |I1 )–1 (a0 ) and (f |I2 )–1 (a0 ) of C1 . Let a1 be the infimum of the points in (f |I1 )–1 (a0 ) and b1 be the infimum Figure 3 Examples of maps with a = –2 and bc = 0 Figure 4 Examples of maps with a = –2, d ∈ {–2, –1, 0, 1}, bc = 0, and 2 ∈/ Per(f ) Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 8 of 26 Figure 5 I11 = [p, a1 ], I12 = [a1 , a0 ], and I21 = [b1 , p] Figure 6 Examples of maps with {a, d} ⊂ {–1, 0, 1}, |a| + |d| = 2, and bc = 0 of the points in (f |I2 )–1 (a0 ). Set I11 = [p, a1 ], I12 = [a1 , a0 ], and I21 = [b1 , p]. Then f has the subgraph I12 → I11 → I21 → I12 . Since we are in the assumptions of Proposition 2.3,  n ∈ Per(f ) for all n = 2. Therefore, MPer(f ) = N \ {2}. This proves statement (b).  Proof of statement (c1) of Theorem B Suppose that {a, d} ⊂ {–1, 0, 1} and |a| + |d| = 2. We consider first the case bc = 0. Without loss of generality, we may assume that c = 0. From the examples of Fig. 6 it is clear that n ∈/ MPer(f ) for any n ∈ N larger than 1, so MPer(f ) = {1} since the branching is fixed. We assume now that |bc| > 1. From the graph of f (see for instance Fig. 7) it is easy to see that we always have three basic intervals I1 , I2 , and I3 , with I1 , I2 ⊂ C1 and I3 ⊂ C2 such that p ∈/ Ii for some i ∈ {1, 2, 3} and f has the subgraph of Proposition 2.3, so Per(f ) ⊃ N \ {2}. Now we will prove that 2 ∈ MPer(f ). If {b, c} ⊂ {–2, –1, 1, 2}, that is, if either |b| ≥ 3 or |c| ≥ 3, we can choose I2 in one circuit and I3 in the other circuit in such a way that I2 ∩ I3 = ∅ (see (a), (b), and (c) of Fig. 7) and I3 → I2 . By Proposition 2.3, 2 ∈ Per(f ). If {b, c} ⊂ {–2, –1, 1, 2}, in general there do not exist two basic intervals Ii and Ij , Ii = Ij , such that p ∈/ Ii ∩ Ij and Ii  Ij (see (e) and (f ) of Fig. 7). If they exist, then by Lemma 2.1, considering the non-repetitive loop Ii → Ij → Ii , there is a periodic point z of f with period 2. If they do not exist, we shall find two intervals with empty intersection such that one f -covers the other. We suppose first that |bc| = 2. We may assume, without loss of generality, that |b| = 1 and |c| = 2. We know that f has five basic intervals I1 , I2 , I3 , I4 , and I5 , the first three in C1 and the other two in C2 , such that f (I2 ) = f (I3 ) = f (I5 ) = C2 and f (I1 ) = f (I4 ) = C1 . Let p and a0 be the endpoints of I2 , a0 and a1 be the endpoints of I1 , a1 and p be the endpoints of I3 (see for instance Fig. 8). We consider an ordering in the intervals I1 , I2 , and I3 in such a way that p is the smallest element of I2 and the greatest of I3 . Set I2 = [p, a0 ], I1 = [a0 , a1 ], and I3 = [b0 , p]. We have two possibilities for the interval I4 : either I4 = [p, b0 ] or I4 = [b0 , p]. If I4 = [p, b0 ] and b = 1, let b1 be the supremum of the points in (f |I4 )–1 (a1 ) and I42 = [b1 , b0 ]. We have I42  I3 and Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 9 of 26 Figure 7 Examples of maps with {a, d} ⊂ {–1, 0, 1}, |a| + |d| = 2, and |bc| > 1 Figure 8 Examples of maps with {a, d} ⊂ {–1, 0, 1}, |a| + |d| = 2 and |bc| = 2 I42 ∩ I3 = ∅, so, by Lemma 2.1, 2 ∈ Per(f ). If I4 = [p, b0 ] and b = –1, set b1 = sup{(f |I4 )–1 (a0 )} and I42 = [b1 , b0 ]. Then I42  I2 and I42 ∩ I2 = ∅ so, by Lemma 2.1, 2 ∈ Per(f ). If I4 = [b0 , p] and b = 1, write b1 = inf{(f |I4 )–1 (a0 )} and I41 = [b0 , b1 ]. Then I41  I2 and I41 ∩ I2 = ∅, so, by Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 10 of 26 Figure 9 Examples of maps with {a, d} ⊂ {–1, 0, 1}, |a| + |d| = 2, and |bc| = 4 Figure 10 Examples of maps with {a, d} ⊂ {–1, 0, 1}, |a| + |d| = 2, bc = 1, and 2 ∈/ Per(f ) Lemma 2.1, 2 ∈ Per(f ). If I4 = [b0 , p] and b = –1, take b1 = inf{(f |I4 )–1 (a1 )} and I41 = [b0 , b1 ]. Then I41  I3 and I41 ∩ I3 = ∅, so, by Lemma 2.1, 2 ∈ Per(f ). Suppose now that |bc| = 4. We know that f has six basic intervals I1 , I2 , I3 , I4 , I5 , and I6 , the first three in C1 and the other three in C2 , such that f (I2 ) = f (I3 ) = f (I5 ) = C2 and f (I1 ) = f (I4 ) = f (I6 ) = C1 (see for instance Fig. 9). Using the same ordering as the above set I2 = [p, a0 ], I1 = [a0 , a1 ], I3 = [b0 , p], I4 = [p, b0 ], I5 = [b0 , b1 ], and I6 = [b1 , p]. If b = 2, set b2 = inf{(f |I6 )–1 (a0 )} and I61 = [b1 , b2 ]. Then I61  I2 and I61 ∩ I2 = ∅ so, by Lemma 2.1, 2 ∈ Per(f ). If b = –2, write b2 = inf{(f |I6 )–1 (a1 )} and I61 = [b1 , b2 ]. Then I61  I3 and I61 ∩ I3 = ∅ so, by Lemma 2.1, 2 ∈ Per(f ). Therefore, if |bc| > 1, MPer(f ) = N. We suppose that |bc| = 1. We assume that b = c = 1. As it can be seen from examples (a), (c), and (e) of Fig. 10, 2 ∈/ MPer(f ). Now we will prove that Per(f ) = N \ {2}. We know that f has four basic intervals I1 , I2 , I3 , and I4 , the first two in C1 and the other two in C2 , such that f (I1 ) = f (I3 ) = C1 and f (I2 ) = f (I4 ) = C2 . We have four possibilities for these intervals. Let a0 ∈ I1 ∩ I2 and b0 ∈ I3 ∩ I4 (see for instance Fig. 11). First, we take Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 11 of 26 Figure 11 Examples of maps with {a, d} ⊂ {–1, 0, 1}, |a| + |d| = 2, and b = c = 1 the interval I3 to be [p, b0 ]. Set I32 = [b1 , b0 ] where b1 = sup{(f |I3 )–1 (a0 )}. If I1 = [p, a0 ], then f has the subgraph I4 → I32 → I2 → I4 , and by Proposition 2.3, Per(f ) = N \ {2}. If  I1 = [a0 , p], then f has the subgraph I1 → I2 → I32 → I1 , and by Proposition 2.3, Per(f ) =  N \ {2}. Now we take the interval I3 to be [b0 , p]. Set b1 = inf{(f |I3 )–1 (a0 )} and I31 = [b0 , b1 ]. If I1 = [p, a0 ], then f has the subgraph I1 → I2 → I31 → I1 , and by Proposition 2.3, Per(f ) =  N\{2}. If I1 = [a0 , p], then f has the subgraph I4 → I31 → I2 → I4 , and by Proposition 2.3,  Per(f ) = N \ {2}. Therefore, if |a| = |d| = 1 and b = c = 1, then MPer(f ) = N \ {2}. We assume now that b = c = –1. As it can be seen from examples (b), (d), and (f ) of Fig. 10, 2 ∈/ MPer(f ). Now we will prove that Per(f ) = N \ {2}. We know that f has four basic intervals I1 , I2 , I3 , and I4 , the first two in C1 and the other two in C2 , such that f (I1 ) = f (I3 ) = C1 and f (I2 ) = f (I4 ) = C2 . We have four possibilities for these intervals. Let a0 ∈ I1 ∩ I2 and b0 ∈ I3 ∩ I4 (see for instance Fig. 12). First we take I3 to be the interval [p, b0 ]. Consider b1 = sup{(f |I3 )–1 (a0 )} and I32 = [b1 , b0 ]. If I1 = [p, a0 ], then f has the subgraph I1 → I2 → I32 → I1 , and by Proposition 2.3, Per(f ) = N \ {2}. If  I1 = [a0 , p], then f has the subgraph I4 → I32 → I2 → I4 , and by Proposition 2.3, Per(f ) =  N \ {2}. If I3 = [b0 , p], consider b1 = inf{(f |I3 )–1 (a0 )} and I31 = [b0 , b1 ]. If I1 = [p, a0 ], then f has the subgraph I4 → I31 → I2 → I4 , and by Proposition 2.3, Per(f ) = N \ {2}. If I1 = [a0 , p],  then f has the subgraph I1 → I2 → I31 → I1 , and by Proposition 2.3, Per(f ) = N \ {2}.  Therefore, if b = c = –1, then MPer(f ) = N \ {2}. Hence, if |a| + |d| = 2 and bc = 1, MPer(f ) = N \ {2}. Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 12 of 26 Figure 12 Examples of maps with {a, d} ⊂ {–1, 0, 1}, |a| + |d| = 2, and b = c = –1 We consider now the case b = –1 and c = 1. We know that f has four basic intervals I1 , I2 , I3 , and I4 , the first two in C1 and the other two in C2 , such that f (I1 ) = f (I3 ) = C1 and f (I2 ) = f (I4 ) = C2 . We have four possibilities for these intervals. Let a0 ∈ I1 ∩ I2 and b0 ∈ I3 ∩ I4 (see for instance Fig. 13). We suppose first that I2 = [a0 , p]. If I3 = [p, b0 ], choose a1 = inf{(f |I2 )–1 (b0 )} and set I21 = [a0 , a1 ]. Then f has the subgraph I1 → I21  I3 → I1 with  I3 ∩ I21 = ∅, and by Proposition 2.3, Per(f ) = N. If I3 = [b0 , p], denote b1 = inf{(f |I3 )–1 (a0 )} and I31 = [b0 , b1 ]. Then f has the subgraph I4 → I31  I2 → I4 with I2 ∩ I31 = ∅, and by  Proposition 2.3, Per(f ) = N. We consider now I2 = [p, a0 ]. If I3 = [p, b0 ], set b1 = sup{(f |I3 )–1 (a0 )} and I32 = [b1 , b0 ]. Then f has the subgraph I4 → I32  I2 → I4 with I2 ∩ I32 = ∅, and by Proposition 2.3,  Per(f ) = N. If I3 = [b0 , p], write a1 = sup{(f |I2 )–1 (b0 )} and I22 = [a1 , a0 ]. Then f has the sub- graph I1 → I22  I3 → I1 with I3 ∩ I22 = ∅, and by Proposition 2.3, Per(f ) = N. Therefore,  if b = –1 and c = 1, then MPer(f ) = N. We consider now the case b = 1 and c = –1. We know that f has four basic intervals I1 , I2 , I3 , and I4 , the first two in C1 and the other two in C2 , such that f (I1 ) = f (I3 ) = C1 and f (I2 ) = f (I4 ) = C2 . We have again four possibilities for these intervals. Let a0 ∈ I1 ∩ I2 and b0 ∈ I3 ∩ I4 (see for instance Fig. 14). We take the interval I2 to be [a0 , p]. If I3 = [p, b0 ], define b1 = sup{(f |I3 )–1 (a0 )} and I32 = [b1 , b0 ]. It follows that f has the subgraph I4 →  I32  I2 → I4 with I2 ∩ I32 = ∅, and by Proposition 2.3, Per(f ) = N. If I3 = [b0 , p], consider a1 = inf{(f |I2 )–1 (b0 )} and I21 = [a0 , a1 ]. Then f has the subgraph I1 → I21  I3 → I1 with  I3 ∩ I21 = ∅, and we get, by Proposition 2.3, Per(f ) = N. Suppose that I2 = [p, a0 ]. If I3 = [p, b0 ], set a1 = sup{(f |I2 )–1 (b0 )} and I22 = [a1 , a0 ]. Then f has the subgraph I1 → I22  I3 → I1 with I3 ∩ I22 = ∅, and by Proposition 2.3, Per(f ) = N.  Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 13 of 26 Figure 13 Examples of maps with {a, d} ⊂ {–1, 0, 1}, |a| + |d| = 2, b = –1, and c = 1 Figure 14 Examples of maps with {a, d} ⊂ {–1, 0, 1}, |a| + |d| = 2, b = 1, and c = –1 Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 14 of 26 If I3 = [b0 , p], consider b1 = inf{(f |I3 )–1 (a0 )} and I31 = [b0 , b1 ]. Then f has the subgraph I4 → I31  I2 → I4 with I2 ∩ I31 = ∅, and by Proposition 2.3, Per(f ) = N. Therefore, if  b = 1 and c = –1, then MPer(f ) = N. Hence, if |a| + |d| = 2 and bc = –1, then MPer(f ) = N. This completes the proof of statement (c1).  Proof of statement (c21) of Theorem B We assume now that a = 1 and d = 0. If bc = 0, then MPer(f ) = {1} as it can be deduced from the examples of Fig. 15. We suppose that b and c are such that |bc| > 1 and (b, c) ∈/ {(2, 1), (2, –1), (–2, 1), (–2, –1)}. From the graph of f (see for instance Fig. 16) it follows that there are three basic intervals I1 , I2 , and I3 , I1 , I2 ⊂ C1 , I3 ⊂ C2 , such that either p ∈/ I1 ∩ I2 or p ∈/ I1 ∩ I3 and f has the subgraph of Proposition 2.3, so Per(f ) ⊃ N \ {2}. If {b, c} ⊂ {–2, –1, 1, 2}, then we can choose I2 and I3 such that I2 ∩I3 = ∅, and by Proposi- tion 2.3, 2 ∈ Per(f ). If {b, c} ⊂ {–2, –1, 1, 2}, in general there do not exist two basic intervals Ii and Ij , Ii = Ij , such that p ∈/ Ii ∩ Ij and Ii  Ij . If they exist, then by Lemma 2.1, considering the non-repetitive loop Ii → Ij → Ii , there is a periodic point z of f with period 2. If they do not exist (see for instance (c) and (d) of Fig. 16) and (b, c) ∈ {(1, 2), (–1, –2)}, 2 ∈/ Per(f ) as we can see from the examples of Fig. 17. Now we will prove that if (b, c) ∈ {(1, –2), (–1, 2)} or |b| = |c| = 2, then 2 ∈ Per(f ). We suppose first that (b, c) ∈ {(1, –2), (–1, 2)}. We know that f has four basic intervals I1 , I2 , I3 , and I4 , the first three in C1 and I4 = C2 , such that f (I1 ) = f (I4 ) = C1 and f (I2 ) = f (I3 ) = C2 . Let p and a0 be the endpoints of I2 , a0 and a1 be the endpoints of I1 , a1 and p be the endpoints of I3 (see for instance Fig. 18). We consider an ordering in the intervals I1 , I2 , and I3 in such a way that p is the smallest element of I2 and the greatest of I3 . Under these Figure 15 Examples of maps with a = 1, d = 0, and bc = 0 Figure 16 Examples of maps with a = 1, d = 0, (b, c) ∈/ {(2, 1), (2, –1), (–2, 1), (–2, –1)}, and |bc| > 1 Figure 17 Examples of maps with a = 1, d = 0, (b, c) ∈ {(1, 2), (–1, –2)}, and 2 ∈/ Per(f ) Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 15 of 26 Figure 18 Examples of maps with a = 1, d = 0, and (b, c) ∈ {(1, –2), (–1, 2)} Figure 19 Examples of maps with a = 1, d = 0, and |b| = |c| = 2 assumptions, set I2 = [p, a0 ], I1 = [a0 , a1 ], and I3 = [a1 , p]. Define b0 = sup{(f |I4 )–1 (a0 )}, I41 = [p, b0 ], and I42 = [b0 , p]. Set a2 = inf{(f |I3 )–1 (b0 )} and I31 = [a1 , a2 ]. If (b, c) = (1, –2), we have I42  I31 and I42 ∩ I31 = ∅. If (b, c) = (–1, 2), we get I41  I31 and I41 ∩ I31 = ∅. So, by Lemma 2.1, 2 ∈ Per(f ). Suppose now that |b| = |c| = 2. We know that f has five basic intervals I1 , I2 , I3 , I4 , and I5 , the first three in C1 and the other two in C2 , such that f (I2 ) = f (I3 ) = C2 and f (I1 ) = f (I4 ) = f (I5 ) = C1 . Taking an ordering similar to the previous case, define the intervals I2 = [p, a0 ], I1 = [a0 , a1 ], I3 = [a1 , p], I4 = [p, b0 ], and I5 = [b0 , p] (see for instance Fig. 19). Set a2 = sup{(f |I2 )–1 (b0 )} and I22 = [a2 , a0 ]. If c = 2, we have I22  I5 and I22 ∩ I5 = ∅. If c = –2, we have I22  I4 and I22 ∩ I4 = ∅. So, by Lemma 2.1, 2 ∈ Per(f ). Therefore, if |bc| > 1 and Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 16 of 26 Figure 20 Examples of maps with a = 1, d = 0, and (b, c) ∈ {(2, 1), (–2, 1)} (b, c) ∈/ {(2, 1), (2, –1), (–2, 1), (–2, –1)}, we have MPer(f ) = N \ {2} if (b, c) ∈ {(1, 2), (–1, –2)} and MPer(f ) = N otherwise. We assume that |bc| > 1 and (b, c) ∈ {(2, 1), (2, –1), (–2, 1), (–2, –1)}. We know that f has four basic intervals I1 , I2 , I3 , and I4 , the first two in C1 and the others in C2 , such that f (I1 ) = f (I3 ) = f (I4 ) = C1 and f (I2 ) = C2 . Let p and a0 be the endpoints of I1 and I2 , and b0 and p be the endpoints of I3 and I4 (see for instance Figs. 20 and 21). For each pair (b, c), we have two possibilities for the intervals I1 and I2 . If (b, c) ∈ {(2, 1), (–2, 1)} and I2 = [a0 , p], write a1 = inf{(f |I2 )–1 (b0 )} and I21 = [a0 , a1 ]. Then f has the subgraph I1 → I21  I3 → I1 with  I3 ∩ I21 = ∅, and by Proposition 2.3, Per(f ) = N. If I2 = [p, a0 ], consider a1 = sup{(f |I2 )–1 (b0 )} and I22 = [a1 , a0 ]. Then f has the subgraph I1 → I22  I4 → I1 with I4 ∩ I22 = ∅, and by  Proposition 2.3, Per(f ) = N. If (b, c) ∈ {(–2, –1), (2, –1)} and I2 = [a0 , p], set a1 = inf{(f |I2 )–1 (b0 )} and I21 = [a0 , a1 ]. Then f has the subgraph I1 → I21  I4 → I1 with I4 ∩ I21 = ∅, and by Proposition 2.3,  Per(f ) = N. If I2 = [p, a0 ], consider a1 = sup{(f |I2 )–1 (b0 )} and I22 = [a1 , a0 ]. Then f has the subgraph I1 → I22  I3 → I1 with I3 ∩ I22 = ∅, and by Proposition 2.3, Per(f ) = N. There-  fore, if |bc| > 1 and (b, c) ∈ {(2, 1), (2, –1), (–2, 1), (–2, –1)}, MPer(f ) = N. We consider the case |bc| = 1. First assume that bc = 1. As we can see from the examples of Fig. 22, 2 ∈/ MPer(f ). Now we will prove that Per(f ) = N \ {2}. We know that f has three basic intervals I1 , I2 , and I3 , the first two in C1 and I3 = C2 , such that f (I1 ) = f (I3 ) = C1 and f (I2 ) = C2 . We have two possibilities for the intervals I1 and I2 : either p is the smallest element of I1 and the greatest of I2 or p is the smallest element of I2 and the greatest of I1 (see for instance Fig. 23). In the assumption that b = c = 1, if I1 = [p, a0 ], write b0 = inf{(f |I3 )–1 (a0 )}, I31 = [p, b0 ], a1 = inf{(f |I2 )–1 (b0 )}, and I21 = Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 17 of 26 Figure 21 Examples of maps with a = 1, d = 0, and (b, c) ∈ {(–2, –1), (2, –1)} Figure 22 Examples of maps with a = 1, d = 0, bc = 1, and 2 ∈/ Per(f ) [a0 , a1 ]. Then f has the subgraph I1 → I21 → I31 → I1 and by Proposition 2.3, Per(f ) ⊃  N \ {2}. If I1 = [a0 , p], define b0 = sup{(f |I3 )–1 (a0 )}, I32 = [b0 , p], a1 = sup{(f |I2 )–1 (b0 )}, and I22 = [a1 , a0 ]. Then f has the subgraph I1 → I22 → I32 → I1 , and by Proposition 2.3,  Per(f ) ⊃ N \ {2}. If b = c = –1, we consider first the case I1 = [p, a0 ]. Set b0 = sup{(f |I3 )–1 (a0 )}, I32 = [b0 , p], a1 = inf{(f |I2 )–1 (b0 )}, and I21 = [a0 , a1 ]. Then f has the subgraph I1 → I21 → I32 → I1 ,  and by Proposition 2.3, Per(f ) ⊃ N \ {2}. If I1 = [a0 , p], write b0 = inf{(f |I3 )–1 (a0 )}, I31 = [p, b0 ], a1 = sup{(f |I2 )–1 (b0 )}, and I22 = [a1 , a0 ]. Then f has the subgraph I1 → I22 →  I31 → I1 , and by Proposition 2.3, Per(f ) ⊃ N \ {2}. Therefore, if a = 1, d = 0, and bc = 1, MPer(f ) = N \ {2}. Assume now that bc = –1. We know that f has three basic intervals I1 , I2 , and I3 , the first two in C1 and I3 = C2 , such that f (I1 ) = f (I3 ) = C1 and f (I2 ) = C2 . We have two possibilities for the intervals I1 and I2 : either p is the smallest element of I1 and the greatest of I2 or p is the smallest element of I2 and the greatest of I1 (see for instance Fig. 24). Define b0 = inf{(f |I3 )–1 (a0 )}, I31 = [p, b0 ], I32 = [b0 , p], and a1 = inf{(f |I2 )–1 (b0 )}. Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 18 of 26 Figure 23 Examples of maps with a = 1, d = 0, and bc = 1 Figure 24 Examples of maps with a = 1, d = 0, and bc = –1 Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 19 of 26 If I1 = [a0 , p], let I21 = [p, a1 ] and I22 = [a1 , a0 ]. Consider a2 = inf{(f |I1 )–1 (a1 )}. We write I11 = [a0 , a2 ] and I12 = [a2 , p]. If b = 1 and c = –1 (see (a) of Fig. 24), f has the subgraph We consider the non-repetitive loops I11 → I21 → I32 → I11 and I12 → I11 → I21 → I32 → I12 → · · · → I12 of lengths 3 and n ≥ 4, respectively. From the first loop and by Lemma 2.1, there is a periodic point z of f with period 3; from the second loop and by Lemma 2.1, there is a periodic point z of f with period n ≥ 4. Moreover, I31  I22 and I31 ∩ I22 = ∅, so, by Lemma 2.1, 2 ∈ Per(f ). Hence, Per(f ) = N. If b = –1 and c = 1 (see (b) of Fig. 24), f has the subgraph Now from the non-repetitive loops I11 → I21 → I31 → I11 and I12 → I11 → I21 → I31 → I12 → · · · → I12 of lengths 3 and n ≥ 4, respectively, and I32  I22 and I32 ∩ I22 = ∅, it follows that Per(f ) = N. If I1 = [p, a0 ], let I21 = [a0 , a1 ], I22 = [a1 , p]. Define a2 = sup{(f |I1 )–1 (a1 )}, I11 = [a0 , a2 ], and I12 = [a2 , p]. If b = 1 and c = –1 (see (c) of Fig. 24), f has the subgraph Again from the non-repetitive loops I12 → I22 → I31 → I12 and I11 → I12 → I22 → I31 → I11 → · · · → I11 of lengths 3 and n ≥ 4, respectively, I32  I21 and I32 ∩ I21 = ∅, Per(f ) = N. If b = –1 and c = 1 (see (d) of Fig. 24), f has the subgraph We consider the non-repetitive loops I12 → I22 → I32 → I12 and I11 → I12 → I22 → I32 → I11 → · · · → I11 of lengths 3 and n ≥ 4, respectively, I31  I21 and I31 ∩ I21 = ∅. We obtain Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 20 of 26 that Per(f ) = N. Therefore, if a = 1, d = 0 and bc = –1, MPer(f ) = N. This completes the proof of statement (c21).  Proof of statement (c22) of Theorem B If a = 0 and d = 1, by using the same kind of argu- ments as those in the case a = 1 and d = 0, and interchanging b and c, we obtain statement (c22).  Proof of statement (c23) of Theorem B We suppose that a = –1 and d = 0. If bc = 0, then MPer(f ) = {1} as it can be seen from the examples of Fig. 25. The cases in which MPer(f ) is either N \ {2} or N can be proved following exactly the same kind of arguments as those in the proof of statement (c21). Assume now that bc = –1. From the examples of Fig. 26 we can see that 3 ∈/ MPer(f ). We know that f has three basic intervals I1 , I2 , and I3 , the first two in C1 and I3 = C2 , such that f (I1 ) = f (I3 ) = C1 and f (I2 ) = C2 . We have two possibilities for the intervals I1 and I2 : either p is the smallest element of I1 and the greatest of I2 or p is the smallest element of I2 and the greatest of I1 (see for instance Fig. 27). Denote b0 = inf{(f |I3 )–1 (a0 )}, I31 = [p, b0 ] I32 = [b0 , p], and a1 = inf{(f |I2 )–1 (b0 )}. If I1 = [a0 , p], let I21 = [p, a1 ] and I22 = [a1 , a0 ]. Consider a2 = inf{(f |I1 )–1 (a1 )}. Write I11 = [a0 , a2 ] and I12 = [a2 , p]. If b = 1 and c = –1 (see (a) of Fig. 27), f has the subgraph I11 → I12 → I21 → I32 → I11 . We consider the non-repetitive loop I11 → I12 → I21 →  I32 → I11 → · · · → I11 of length n ≥ 4. By Lemma 2.1 there is a periodic point z of f with period n ≥ 4. Moreover, I31  I22 and I31 ∩ I22 = ∅, so, by Lemma 2.1, 2 ∈ Per(f ). Hence, Per(f ) = N \ {3}. If b = –1 and c = 1 (see (b) of Fig. 27), f has the subgraph I11 → I12 → I21 → I31 → I11 . We consider the non-repetitive loop I11 → I12 → I21 →  I31 → I11 → · · · → I11 of length n ≥ 4. By Lemma 2.1 there is a periodic point z of f with Figure 25 Examples of maps with a = –1, d = 0, and bc = 0 Figure 26 Examples of maps with a = –1, d = 0, bc = –1, and 3 ∈/ Per(f ) Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 21 of 26 Figure 27 Examples of maps with a = 1, d = 0 and bc = –1 period n ≥ 4. Moreover, I32  I22 and I32 ∩ I22 = ∅, so, by Lemma 2.1, 2 ∈ Per(f ). Hence, Per(f ) = N \ {3}. If I1 = [p, a0 ], let I21 = [a0 , a1 ] and I22 = [a1 , p]. Consider a2 = sup{(f |I1 )–1 (a1 )}. Write I11 = [p, a2 ] and I12 = [a2 , a0 ]. If b = 1 and c = –1 (see (c) of Fig. 27), f has the subgraph I12 → I11 → I22 → I31 → I12 . From the non-repetitive loop I12 → I11 → I22 → I31 →  I12 → · · · → I12 of length n ≥ 4, I32  I21 , and I32 ∩ I21 = ∅, we obtain that Per(f ) = N \ {3}. If b = –1 and c = 1 (see (d) of Fig. 27), f has the subgraph I12 → I11 → I22 → I32 → I12 .  Using the non-repetitive loop I12 → I11 → I22 → I32 → I12 → · · · → I12 of length n ≥ 4, I31  I21 and I31 ∩ I21 = ∅, we get that Per(f ) = N \ {3}. Therefore, if a = –1, d = 0, and bc = –1, MPer(f ) = N \ {3}. This completes the proof of statement (c23).  Proof of statement (c24) of Theorem B If a = 0 and d = –1, by using the same kind of argu- ments as those in the case a = –1 and d = 0, and interchanging b and c, we obtain statement (c24).  Proof of statement (c3) of Theorem B We suppose that a = d = 0. If bc = 0 or bc = 1, we can deduce from the examples of Fig. 28 that MPer(f ) = {1}. If bc = –1, then MPer(f ) = {1, 2} (see for instance Fig. 29). We assume now that |bc| = 2. Since a = d = 0, we may assume without loss of generality that |b| = 1 and |c| = 2. We consider first the case bc = –2. Clearly, {1, 2} ⊂ Per(f ), no other odd number belongs to MPer(f ) and 4 ∈/ MPer(f ) as it can be deduced from Fig. 30. Now we will prove that n ∈ Per(f ) for any n even larger than 4. We know that f has three basic intervals I1 , I2 , and I3 , the first two in C1 and I3 = C2 , such that f (I1 ) = f (I2 ) = C2 and f (I3 ) = C1 (see for instance Fig. 31). Consider Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 22 of 26 Figure 28 Examples of maps with a = d = 0 and either bc = 0 or bc = 1 Figure 29 Examples of maps with a = d = 0 and bc = –1 Figure 30 Examples of maps with a = d = 0, bc = –2, and 4 ∈/ Per(f ) Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 23 of 26 Figure 31 Examples of maps with a = d = 0 and bc = –2 Figure 32 Examples of maps with a = d = 0, bc = 2, and 2 ∈/ Per(f ) b0 = inf{(f |I3 )–1 (a0 )}, a1 = inf{(f |I1 )–1 (b0 )}, b1 = inf{(f |I3 )–1 (a1 )}. Set I11 = [p, a1 ], I12 = [a1 , a0 ], I31 the interval with endpoints b1 and p, I32 the interval with endpoints b1 and b0 , and I33 the interval with endpoints b0 and p. Then f has the subgraph We consider the non-repetitive loops I32 → I12 → I32 and I2 → I32 → I12 → I31 → I11 → I33 → I2 → · · · → I33 → I2 of lengths 2 and n even, n ≥ 6, respectively. We have I32 ∩ I12 = ∅, so, from the first loop and by Lemma 2.1, there is a periodic point z of f with period 2; from the second loop and by Lemma 2.1, there is a periodic point z of f with period n even n ≥ 6. Therefore, if bc = –2, then MPer(f ) = {1} ∪ (2N \ {4}). We suppose that bc = 2. No odd number other than 1 belongs to MPer(f ), as it can be seen from the examples of Fig. 32. Also from Fig. 32 we can deduce that 2 ∈/ MPer(f ). Now we will prove that n ∈ Per(f ) for any n even larger than 2. We know that f has three basic intervals I1 , I2 , and I3 , the first two in C1 and I3 = C2 , such that f (I1 ) = f (I2 ) = C2 and f (I3 ) = C1 (see for instance Fig. 33). Denote b0 = inf{(f |I3 )–1 (a0 )}, a1 = inf{(f |I1 )–1 (b0 )}, and b1 = inf{(f |I3 )–1 (a1 )}. Write I11 = [p, a1 ], I12 = [a1 , a0 ], I32 the in- Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 24 of 26 Figure 33 Examples of maps with a = d = 0 and bc = 2 Figure 34 Examples of maps with a = d = 0 and either |b| > 2 or |c| > 2 terval with endpoints b1 and b0 , and I33 the interval with endpoints b0 and p. Then f has the subgraph I32 → I12 → I33  I2 → I32 . We take the non-repetitive loop I2 → I32 → I12 → I33 → I2 → · · · → I33 → I2 of length n even, n ≥ 4. By Lemma 2.1, there is a periodic point z of f with period n even n ≥ 4. Therefore, if bc = 2, then MPer(f ) = {1} ∪ (2N \ {2}). We consider now the case |bc| > 2. We must separate the case |b| = |c| = 2 from the others. If |b| > 2 or |c| > 2, then there are three basic intervals I1 , I2 , and I3 such that I2 ∩ I3 = ∅ and I1  I3  I2 (see for instance Fig. 34). By Lemma 2.1, the non-repetitive loop I2 → I3 → I2 gives a periodic point z of f with period 2, and the non-repetitive loop I1 → I3 → I2 → I3 → · · · → I2 → I3 → I1 of length n even larger than 2 gives a periodic point z of f with period n even. No odd number other than 1 belongs to MPer(f ). Therefore, if |b| > 2 or |c| > 2, then MPer(f ) = {1} ∪ 2N. We suppose that |b| = |c| = 2. Clearly, no odd number other than 1 belongs to MPer(f ). Now we will prove that n ∈ Per(f ) for any n even. We know that f has four basic intervals I1 , I2 , I3 , and I4 , the first two in C1 and the others in C2 , such that f (I1 ) = f (I2 ) = C2 and f (I3 ) = f (I4 ) = C1 (see for instance Fig. 35). Consider b1 = inf{(f |I3 )–1 (a0 )} and a1 = inf{(f |I1 )–1 (b1 )}. Denote I11 = [p, a1 ], I12 = [a1 , a0 ], I2 = [a0 , p], I31 = [p, b1 ], I32 = [b1 , b0 ], and I4 = [b0 , p]. If (b, c) ∈ {(2, 2), (–2, 2)}, then f has the subgraph I2  I4  I12 . We take the non-repetitive loops I4 → I12 → I4 and I2 → I4 → I12 → I4 → · · · → I12 → I4 → I2 of lengths 2 and n even larger than 2, re- spectively. By Lemma 2.1, the first loop gives a periodic point z of f with period 2, and the second loop gives a periodic point z of f with period n even larger than 2. Hence, if (b, c) ∈ {(2, 2), (–2, 2)}, Per(f ) = {1} ∪ 2N. If (b, c) = (–2, –2), then f has the subgraph I4  I11  I32 . We consider the non-repetitive loops I32 → I11 → I32 and I4 → I11 → I32 → I11 → · · · → I32 → I11 → I4 of lengths 2 and n even larger than 2, respectively. By Lemma 2.1, the first loop gives a periodic point z of f Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 25 of 26 Figure 35 Examples of maps with a = d = 0 and |b| = |c| = 2 with period 2, and the second loop gives a periodic point z of f with period n even larger than 2. Hence, if (b, c) = (–2, –2), Per(f ) = {1} ∪ 2N. If (b, c) = (2, –2), then f has the subgraph I4  I2  I32 . We consider the non-repetitive loops I2 → I32 → I2 and I4 → I2 → I32 → I2 → · · · → I32 → I2 → I4 of lengths 2 and n even larger than 2, respectively. By Lemma 2.1, the first loop gives a periodic point z of f with period 2, and the second loop gives a periodic point z of f with period n even larger than 2. Hence, if (b, c) = (–2, –2), Per(f ) = {1} ∪ 2N. Therefore, if |b| = |c| = 2, then MPer(f ) = {1} ∪ 2N. This completes the proof of statement (c3).  Acknowledgements We thank our universities for the facilities that they gave us during the elaboration of this work. Funding The first author has been partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. Availability of data and materials No materials are necessary. Competing interests The authors declare that they have no competing interests. Authors’ contributions Both authors have contributed approximately in one half to the paper. Both authors read and approved the final manuscript. Author details 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain. 2 Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal. Llibre and Sá Fixed Point Theory Algorithms Sci Eng (2021) 2021:3 Page 26 of 26 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 11 December 2019 Accepted: 2 December 2020 References 1. Abdulla, U.G., Abdulla, R.U., Abdulla, M.U., Iqbal, N.H.: Second minimal orbits, Sharkovski ordering and universality in chaos. Int. J. Bifurc. Chaos Appl. Sci. Eng. 27, 1–24 (2017) 2. 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