On the preservation of combinatorial types for maps on trees

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Introduction and statement of the main result.
We deal with a classical problem in combinatorial dynamics: Given a topological space X, a continuous map f : X → X and a periodic orbit A ⊂ X of f , what can be said about the dynamics (periodic orbits, topological entropy-see [1]-, etc.) of f in terms of f A ?
A well known case is when X is a closed interval I ⊂ R. Indeed, if f : I → I is a continuous map then intrinsic information can be obtained by considering the "pattern" of A which is characterized by the permutation π A induced by f A (see [7] or [9]). To each pattern π we may associate a map f π which has a finite invariant set B such that the permutation induced by f π B is π and f π is monotone between consecutive points of B (a "connect-the-dots" map). Such a map is called a π-monotone model and its existence has some important consequences. During the 1980's and the early 1990's, monotone models for interval maps (and a trivial generalization to star maps) were used by several authors to tackle a wide variety of problems (see for instance [4], [5], [7], [8], [9] and [10]).
We are interested in continuous maps defined on trees (from now on, such a map will be called a tree map). In [2], the authors introduce a notion of pattern of a finite invariant set of a tree map, and prove that for any pattern π there exist π-monotone models which minimize the topological entropy. In this paper, which is a continuation and a natural extension of [2], we want to investigate whether these π-monotone models also minimize the dynamics (measured by the set of periodic orbits). Let us start by introducing some notation.
Given any subset X of a topological space, we will denote by Int(X) and Cl(X) the interior and the closure of X, respectively. The boundary of X will be denoted by Bd(X). For a finite set A we will denote its cardinality by |A|.
By an interval we mean any space homeomorphic to [0, 1] ⊂ R. A tree is a uniquely arcwise connected space that is either a point or a union of finitely many intervals. If T is a tree and x ∈ T , we define the valence of x to be the number of connected components of T \ {x}. Each point of valence 1 will be called an endpoint of T and the set of such points will be denoted by En(T ). A point of valence different from 2 will be called a vertex of T , and the set of vertices of T will be denoted by V (T ). The closure of each connected component of T \ V (T ) will be called an edge of T . Any tree which is a union of r > 1 intervals whose intersection is a unique point x of valence r will be called an r-star, and x will be called its central point.
Given a tree T and P ⊂ T we will define the convex hull of P , denoted by P T or simply by P , as the smallest closed connected subset of T containing P . When P = {x, y}, we will write x, y or [x, y] to denote P . The notations (x, y), (x, y] and [x, y) will be understood in the natural way.
Next we recall the definitions of pattern and monotone model, which are the central notions used in this paper. For further discussion on these ideas, we refer the reader to [2]. Let T be a tree and let A ⊂ T be a finite subset of T . The pair (T, A) will be called a pointed tree. A set Q ⊂ A is said to be a discrete component of (T, A) if either |Q| > 1 and there is a connected component C of T \ A such that Q = Cl(C) ∩ A, or |Q| = 1 and Q = A. We say that two pointed trees (T, A) and (T , A ) are equivalent if there exists a bijection φ : A → A which preserves discrete components. The equivalence class of a pointed tree (T, A) by this relation will be denoted by [T, A].
Let (T, A) and (T , A ) be equivalent pointed trees, and let θ : A → A and θ : A → A be maps. We will say that θ and θ are equivalent if θ = ϕ • θ • ϕ −1 for a bijection ϕ : A → A which preserves discrete components. The equivalence class of θ by this relation will be denoted by [θ]. If [T, A] is an equivalence class of pointed trees and [θ] is an ANNALES DE L'INSTITUT FOURIER equivalence class of maps then the pair ([T, A], [θ]) will be called a pattern. If in addition θ is a cycle over A, then the pattern is said to be periodic.
. This pattern will be denoted by [T, A, f ]. Now we extend the notion of a "connect-the-dots" interval map to the setting of trees. Let f : T → T be a tree map. Given x, y ∈ T we say that f [x,y] is monotone if either f ([x, y]) is a point or it is an interval and, given two homeomorphisms φ : In this case we will say that the model (T, P, f) is monotone. [2] that the image of each vertex z is uniquely determined and is either a vertex or belongs to P . In fact, if we take three different The next theorem, which is the main result of [2], states the existence and minimality of the topological entropy for monotone models of tree maps. THEOREM 1.2 (Theorem A of [2]). -Let (T ,Θ) be a pattern. Then the following statements hold.
(a) There exists a monotone model exhibiting the pattern (T ,Θ).
(b) The topological entropy of any monotone model is the minimum within the class of models which exhibit the pattern (T ,Θ).
As we have said, the goal of this paper is to show that the dynamics of monotone models is also minimal from the point of view of the set of periodic orbits. To be more precise we recall some more notions and results from [2].
Let f : T → T be a tree map, and let x, y ∈ T be fixed points of f n for some n ∈ N. We say that x and y are f -monotone equivalent if either x = y or f n x,y is monotone. It is not difficult to see that the periods of x and y are either equal, or one is a multiple of the other. Observe that, since x and y are fixed points of f n , the f -monotone relation is an equivalence relation.
Remark 1.3. -It is easy to see that if x and y are f -monotone equivalent then f i (x) and f i (y) are also f -monotone equivalent, for each i ≥ 0.
Given a model (T, A, f ), we say that a periodic point of f is significant if it is not f -monotone equivalent to any element of A ∪ V (T ) and its period is minimal within its f -monotone equivalence class. The significant periodic points of a monotone model of a pattern P can be essentially put in a oneto-one correspondence with the loops of a certain combinatorial graph, which is uniquely associated to P. Let us introduce these notions in detail.
Let P = [T, A, f ] be a pattern. Any (unordered) binary subset of a discrete component will be called a basic path of (T, A). The P-path graph is the oriented graph whose vertices are in one-to-one correspondence with the basic paths π i of (T, A) and there is an arrow from the vertex i to the vertex j if the corresponding basic paths satisfy π j ⊂ f (π i ) . Note that this definition is independent of the particular choice of the model (T, A, f ). Let π 0 → π 1 → · · · → π n−1 → π 0 be a loop α of length n in the P-path graph. The length of such a loop α will be denoted by |α|. The loop α and a point x ∈ T are said to be associated if f i (x) ∈ π i mod n T for each i ≥ 0. We note that if in addition x is a periodic point, then the period of x is a divisor of |α|. Recall that a loop is called simple if it is not an n-repetition (with n ≥ 2) of any other loop. Now we are ready to state the two results of [2] which describe the sense in which the monotone models have minimal dynamics. THEOREM 1.4 (Theorem C of [2]). -Let (T ,A,f) be a monotone model exhibiting a pattern (T ,Θ). Then the following statements hold.   Theorem A below is a stronger version of the above results for periodic patterns. It says that, when (T , Θ) is periodic, there is essentially a periodpreserving injective map from the set of (almost all) significant periodic points of f into the set of periodic points of each model exhibiting (T , Θ). Moreover, the corresponding orbits are associated to the same loop in the path graph. Therefore, the relative position of the points inside the respective trees is essentially preserved.
Before stating in detail this result, we remark that, for simplicity, we will deal with a particular kind of monotone models, which will be called Alinear models. An A-linear model is essentially a special kind of "simplified" monotone model which does not exhibit unnecessary duplicated orbits. An A-linear model (T, A, f ) behaves (with respect to some natural metric) like an affine map of the real line (with respect to the Lebesgue metric) on each connected component of T \ (A ∪ V (T )). This condition is not restrictive: in Section 2 we prove that any pattern admits an A-linear model. We also prove that the monotone equivalence relation defined before Remark 1.3 is particulary simple in A-linear models.
Let (T, A, f ) be a monotone model and let x be a significant n-periodic point of f . In particular, x / ∈ V (T ). The point x will be called positive if f n preserves orientation in a neighborhood of x, and negative if f n reverses orientation.
Finally, for any pointed tree (S, P ) we define M (S, P ) as the number of basic paths π such that Int( π S ) ∩ V (S) = ∅ multiplied by the maximum number of vertices of S contained in the interior of the convex hull of a basic path. It is easy to see that the number M (S, P ) can be upper bounded by | En(S)| 2 · |V (S)| 2 .
This paper is devoted to prove: Consider a periodic pattern P and take an A-linear model (T, A, f ) of P. As it has been said, the significant periodic orbits of f can be easily obtained by studying the loops of the P-path graph (see Theorem 1.4 and Proposition 3.4). Therefore, Theorem A (a) can be used to obtain relevant information about the periodic orbits of any particular model (S, P, g) which exhibits the pattern P. For instance, from Theorem A it follows that, for any n > M(S, P ), the number of n-periodic points of g is lower bounded by the number of significant n-periodic points of the A-linear model. Moreover, the fact that the µ-corresponding orbits are associated to the same loop allows us localizing the orbits of g in S with a certain precision.
On the other hand, (b) of Theorem A tells us that the complement of Λ f (that is, the set of periodic points of f which are not controlled by (a) of Theorem A) consists of a finite set Ω ∪ Γ together with a set C contained in a cycle of k connected components, each of them intersecting A. The periods of the points in Ω, Γ and C are respectively bounded by M (S, P ), |V (T ) \ A| and 8|P | − 4. It is easy to check that |V (T ) \ A| < M(S, P ), so each point in Ω ∪ Γ has period not greater than M (S, P ). The cardinality of Γ is at most |V (T ) \ A| while, on the other hand, it is not difficult to calculate an explicit upper bound for the cardinality of Ω in terms of |P | (see Remark 3.5). For simplicity, this bound has not been made explicit in x 1 x 2 x 3 x 4 Since there are only three convex hulls of basic paths whose interior contains one vertex of S and this vertex is unique, it follows that M (S, P ) = 3. By checking the loops of the P-path graph one easily gets that Γ = ∅, C = A (see the proof of Theorem A (b) for explicit definitions of Γ, C and Ω) and the set of periods of the points in Λ f is N \ {1, 2, 3}. Moreover, Ω consists of three negative periodic points {y, w, w }, where y is a fixed point associated to the loop {x 1 , Hence, from Theorem A it follows that the set of periods of each model (S, P, g) of the pattern P contains N \ {1, 2, 3}.
In the previous paper [3], two of the authors and their coauthors proved a minimality result for graph maps which has the flavour of Theorem A (existence of a pattern-preserving injective map from the set of periodic orbits of a canonical model into that of a given graph map, with at most a finite number of exceptions). Since any tree is in particular a graph, one would expect that this result can be applied to tree maps. This idea does not work straightforwardly, mainly because the notion of pattern used in [3] for graph maps differs greatly from the corresponding notion used in [2] for tree maps. This difference arises from the distinct aims of both papers. However, by using some tools and results from [3], in a forthcoming paper we will prove that there is a period-preserving injective map from the set of all negative significant periodic points of TOME 55 (2005), FASCICULE 7 the A-linear model of a pattern P into the set of periodic points of any representative of P. It follows that the period-preserving injective map µ in Theorem A can be extended to the whole set of significant periodic points of the A-linear model. But observe that in this case we do not claim that the µ-corresponding orbits are associated to the same loop in the P-path graph. Hence, this result fills the gap of negative significant periods between 2 and M (S, P ), but is weaker than Theorem A in the sense that it does not give any information about the localization of the corresponding orbits.
Thus one may wonder whether the hypotheses which define the set Λ f in Theorem A (positivity or period greater than M (S, P )) are necessary in order to conclude that the µ-corresponding orbits are associated to the same loop in the path graph. Consider the pattern P and the fixed point y of the corresponding A-linear model (T, A, f ) given in Example 1.6. The significant and negative fixed point y is associated to the loop {x 1 , This paper is organized as follows. In Section 2 we use the notion of a canonical model, first introduced in [2], to define a particular class of "simplified" monotone models called linear models. We also prove some minimality properties of the linear models. Finally in Section 3 we use these results to prove Theorem A.
Acknowledgements. -We thank an anonymous referee for his clever and detailed remarks, that have greatly improved the readability of this paper.

Linear models.
In this section we prove that any pattern admits a particular type of monotone model, which will be called an A-linear model. We also study the f -monotone equivalence classes of periodic points for A-linear models, which turn out to be specially simple.

ANNALES DE L'INSTITUT FOURIER
We start by defining the notion of a canonical model, first introduced in [2]. Let (T, A, f ) be a monotone model. We will say that v 1 A monotone model (T, A, f ) will be called a canonical model if it has no pairs of f -identifiable vertices. By Theorem B of [2], each pattern admits a canonical model. Moreover, it can be shown that this canonical model is unique (up to homeomorphisms of trees and conjugacy of maps). The procedure to obtain it from a generic monotone model (S, B, g) essentially consists of contracting the convex hull of each pair of f -identifiable vertices to a point. In doing it, besides keeping intact the B-monotonicity of the map and the pattern [S, B, g], we eliminate some non-significant periodic orbits of vertices.
An A-linear model will be defined to be a canonical model which exhibits a certain piecewise linearity property. To introduce this condition, we must first consider an appropriate metric on trees. Given a tree T , a metric d : for each x, y ∈ T and z ∈ [x, y] will be called a proper metric (some authors use the term taxicab metric). Since a tree is a uniquely arcwise connected space, it admits a proper metric. For completeness, next we outline the basic ideas of the construction of a proper metric d on T . Let I 1 , I 2 , . . . , I n be the edges of T . For each of them choose a homeomorphism h i : I i → [0, 1] and let φ be the Lebesgue measure on [0, 1]. Take x, y ∈ T and let I i1 , I i2 , . . . , I i k be the set of edges of T whose interior intersects [x, y]. If k = 1, then x, y ∈ I i1 and we define d(x, y) = φ(h i1 (x), h i1 (y)). Otherwise, assume that x ∈ I i1 and y ∈ I i k and define d( It is easy to check that f is well defined and satisfies the required properties. The following remark and lemma state the basic properties of the pre-images of the sets A ∪ V (T ) and Although V k is not necessarily finite, the A-monotonicity of f implies that it has finitely many connected components, each of them being a point or a subtree on which f k is constant. In addition, each connected component of T \ V k is an interval on which f k is linear.
canonical. If f is not constant on K, then f is linear with a positive slope on K, since (T, A, f ) is A-linear. In both cases, it easily follows that K ∩ f −1 (X \ A) is finite or empty for each finite set X ⊂ T . In particular, K ∩ f −1 (V (T ) \ A) is finite or empty. Since in addition this intersection is disjoint from A, the lemma easily follows from a simple inductive argument.
x 2 y Figure 2. A non-linear canonical model.
The linearity condition in canonical models allows us removing most of the non-significant periodic orbits (not necessarily associated to orbits of vertices). For instance, Example 2.4 shows a non A-linear canonical model exhibiting a proper interval of non-significant 2-periodic points: these points will be removed when one constructs the corresponding A-linear model. for each w ∈ A and each k ∈ N). It follows that y is a significant point of f (because its period is minimal and its class contains no points of A) whose f -monotone equivalence class coincides with [x 1 , As a consequence of the fact that the linearity condition eliminates most of the non-significant periodic orbits in canonical models, the fmonotone equivalence classes are particulary simple in an A-linear model. In particular, from the following technical lemma it follows that the significant periodic points of A-linear models are alone in their monotone equivalence classes.

LEMMA 2.5. -Let (T ,A,f) be an A-linear model and let x be a periodic point of f which is not f -monotone equivalent to any point of A. Then the f -monotone equivalence class of x reduces to {x}.
Proof. -During this proof, let us write a ∼ b to denote that a and b are f -monotone equivalent. First we claim that, for each i > 0, the f -monotone equivalence class of f i (x) contains no points of A. Assume now that there exist y ∈ A and i > 0 such that f i (x) ∼ y. Let n ∈ N be such that f n (x) = x. Then Thus, by Remark 1.3, x ∼ f n−(i mod n) (y) ∈ A. This contradiction proves the claim.
Next we prove that is monotone and f i (a) and in contradiction with the claim above. This proves (2.1). Now assume that there exists some z = x such that z ∼ x. This will lead us to a contradiction. Let m ∈ N such that f m (x) = x, f m (z) = z and f m is monotone on [x, z]. Choose e, e ∈ En(T ) and an order ≤ in [e, e ] with e < x < z < e .
Since the f -monotone equivalence class of x does not contain any point of A, it follows that a, b ∈ V (T ). By (2.1), a and b are f -identifiable, in contradiction with the fact that (T, A, f ) is a canonical model. The following remark will be used several times in the rest of this section. It easily follows from the definition of the f -monotone equivalence relation. The rest of this section is devoted to state and prove Proposition 2.10, which describes the structure of the set of non-significant periodic points of A-linear models and will be used in the proof of statement (b) of Theorem A. To prove it, we need the next two technical lemmas. Proof. -Assume that x, y and z are f -monotone equivalent and this will lead us to a contradiction. From the definition of the f -monotone relation and the fact that y ∈ (x, z) it easily follows that Set X = Orb(x) T . Since Orb(x) is a periodic orbit containing y and En(X) ⊂ Orb(x), there exists some i ≥ 0 such that f i (y) ∈ En(X), a contradiction with (2.2).
Proof. -By definition of the f -monotone equivalence relation, all the points in a class are periodic. This proves that f is injective on each class. Let us call any f -monotone equivalence class intersecting A an A-class. From Remark 1.3, the f -image of a class is contained in a class. Moreover, the f -image of an A-class is contained in an A-class because A is f -invariant. Since all the points in a class are periodic and A is a single periodic orbit, it easily follows that there exists k ∈ N and a labelling C 0 , C 1 , . . . , C k−1 of the A-classes such that f (C i ) ⊂ C i+1 mod k for 0 ≤ i ≤ k − 1. Let x be an n-periodic point which belongs to some C i . Then f n−1 (x) is a pre-image of x which must belong to C i−1 . Therefore, f is onto and thus bijective on each C i . Consequently, f (C i ) = C i+1 mod k . Observe that k is a divisor of the period of each periodic point contained in an A-class. In particular, since A is a periodic orbit, k divides |A|.
We claim that each A-class is a closed set. For each n ∈ N, let P n denote the set of fixed points of f i for 1 ≤ i ≤ n. It is clear that P n is a finite union of closed sets and thus is closed. Moreover, since (T, A, f ) is an A-linear model, it easily follows that P n has finitely many connected components. Let C be an A-class and set n = 8|A| − 4. By Lemma 2.9, the period of each point in C is not greater than n. Hence, C ⊂ P n . To prove the claim it is enough to show that C ∩ K is closed for each connected component K of P n such that C ∩ K = ∅. By Remark 2.7, the set C ∩ K is convex. From the definition of the f -monotone equivalence relation, if two periodic points x and y belong to an A-class then there exists some m ∈ N such that f m (x) = x, f m (y) = y and f m is monotone on [x, y]. Recall that the periods of x and y are either equal, or one is a multiple of the other. Therefore, all the points in C ∩ K are pairwise f -monotone equivalent and fixed points of f m for some m ≤ n. Thus f m is the identity map on C ∩ K. In consequence, each accumulating point of a sequence of points in C ∩ K belongs to C ∩ K, and the claim follows. Let x and y be different points belonging to an A-class, and let m ∈ N be such that f m (x) = x, f m (y) = y and f m is monotone on [x, y]. In Finally we prove that C i ∩ C j = ∅ whenever i = j. This is obvious when each A-class reduces to one point. Thus from now on assume that some A-class contains at least two points. Hence, each A-class contains at least two points, because f is a bijection over the A-classes. Assume that there are 0 ≤ i < j ≤ k − 1 such that X := C i ∩ C j = ∅. This will lead us to a contradiction. Observe that X is a subtree of T . Moreover, by the preceding paragraph f (X) ⊂ C i+1 ∩ C j+1 , and iterating k times this argument we get f k (X) ⊂ X. Thus, f k X is a tree map and so it has fixed points. Hence, there exists a periodic point z ∈ X of f . By the claim above, each A-class is closed and thus En( C i ) ⊂ C i . Since z ∈ C i and each A-class contains at least two points, there exist x, y ∈ En(

Proof of Theorem A.
We start this section with four preliminary results which will be used in the proof of Theorem A. They are three technical lemmas and ANNALES DE L'INSTITUT FOURIER Proposition 3.4, which is a refinement of Theorem 1.4 and plays a central role in the proof of Theorem A. LEMMA 3.1. -Let P be a pattern. For 1 ≤ j ≤ s, let π 0 → π j 1 → . . . → π j n−1 → π 0 be s different loops of length n in the P-path graph, and let g : S → S be a tree map exhibiting P. Then there exist subsets J 1 ,J 2 , . . . ,J s of π 0 such that, for each j = 1,2, . . . ,s, we have: Proof. -Statement (a) is simply a particular instance of Lemma 3.2 of [2], and statements (b.1)-(b.4) follow immediately from its proof. Lemma 3.2 states that from a closed sequence of coverings of sets contained in discrete components we can derive the existence of a periodic orbit of vertices.
Observe that if the claim holds then, since V (T ) is finite, the lemma follows.  (f (a), f(c)). Thus, f (a) and f (c) belong to two different discrete components, in contradiction with (b).
. Hence, f (v) is also the central point of the 3-star {a , b , c } , which is contained in T i+1 mod n , and the claim follows.
Theorem A (a) will essentially be a corollary of the next lemma. To state this result, we need to introduce a few more notions.
To each loop in a (T , [θ])-path graph we can associate a sign as follows. First we endow each basic path π of (T , [θ]) with an ordering ≤ π . Clearly, each ≤ π induces a linear ordering on π T and θ(π) T . Let π be the linear ordering induced by ≤ π on θ(π) T and let π → σ be an arrow in the (T , [θ])-path graph. Set π = {a, b} and σ = {c, d} and assume that a < π b and c < σ d. Then the sign of the arrow π → σ is +1 if θ(a) π c ≺ π d π θ(b) and -1 if θ(a) π d ≺ π c π θ(b). A loop π 0 → π 1 → · · · → π n−1 → π 0 is said to be positive if the product of the signs of the arrows π 0 → π 1 , π 1 → π 2 , . . . , π n−1 → π 0 is +1 and negative if it is −1. It is easy to see that the sign of a loop is independent from the particular choice of the set of orderings ≤ π . If (T, A, f ) is a model of (T , [θ]) and there is a periodic point x of f associated to a positive (negative) loop, then we also say that x is positive (respectively, negative). Now we are ready to state and prove Lemma 3.3. By Theorem 1.4, each significant periodic point of a monotone model is associated to a unique simple loop in the path graph. Therefore, the loop β in the statement of Lemma 3.3 is unique. Recall also that the constant M (S, P ) has been defined in just before Theorem A. Consequently, if either β is positive or n > M(S ,P ) then there is an n-periodic point of g in S \ P associated to β.
Proof. -We start by claiming that if a fixed point z of g n is associated to β then z / ∈ P . Indeed: if z ∈ P then Orb g (z) = P and, given a bijection θ : P → A which preserves discrete components, θ(z) is a point of A associated to β. Since β is simple and x is also associated to β, from Theorem 1.4(b) it follows that x and θ(z) are f -monotone equivalent, a contradiction with the fact that x is significant. This proves the claim.
Let π 0 → π 1 → · · · → π n−1 → π 0 be the loop β. By the claim above, to prove (a) it is enough to show that there exists a fixed point of g n associated to β. We will consider two cases.
Since β is positive, g n {a1,b1,...,am,bm} is increasing. An easy argument, analogous to the one used in Lemma 3 of [6], shows that there exists z ∈ J such that g n (z) = z. By (i), z and β are associated. This ends the proof of (a) in this case.
For s ∈ {0, 1, . . . , k − 1}, we define D s = π∈Πs π. In view of (3.2), each D s is contained in a discrete component. Therefore, (i) of Lemma 3.2 is satisfied because l ≥ 2, while (ii) is obviously verified. Hence, by Lemma 3.2 there is a periodic point w of f such that f s (w) ∈ Int( D s mod k T ) ∩ V (T ) for each s ≥ 0. In particular, there exists 0 ≤ i < l such that w ∈ π ik T . Note that, for each j ≥ 0, π ik+j+1 mod n is the only basic path in D j+1 mod k contained in f ( π ik+j mod n ). Consequently, w is associated to a shift of the loop β starting with π ik . Therefore, there is a point w ∈ Orb f (w) associated to β and, by Theorem 1.4, x and w are f -monotone equivalent. Since w ∈ V (T ), this is a contradiction with the fact that x is significant. There exists an injective map β from the set of significant periodic points x of f into the set of simple loops in the [S ,P ,g]-path graph such that x is associated to β(x) and the length of β(x) equals the period of x.
Proof. -We start by defining the map β. For each significant periodic point x of f , by Theorem 1.4(a) there is a unique simple loop in the [S, P, g]path graph, which we define to be β(x), of length | Orb f (x)| and associated to x. Let us see that the map β is injective. Take a significant periodic point x = x and assume that β(x ) = β(x). By Theorem 1.4 (b), x and x are f -monotone equivalent, in contradiction with Lemma 2.5. This ends the proof of the proposition.
Remark 3.5. -For a periodic pattern P of period n, the number of vertices of the P-path graph coincides with the number of basic paths of P, which is not greater than n 2 . On the other hand, any finite combinatorial graph with r vertices has at most r k loops of length k. Therefore, by Proposition 3.4, an A-linear model f of P has at most n 2 k significant periodic points of period k. Consequently, if (S, P, g) is any model of P then the number of significant periodic points of f with period not greater than any positive integer (in particular, M (S, P )) is finite.
Finally we are ready to prove Theorem A.
Proof of Theorem A. -We start by proving (a). We have to define the map µ on Λ f . Let x ∈ Λ f and set n = | Orb f (x)|. Let β be the injective map from the set of significant periodic points of f into the set of simple loops in the [S, P, g]-path graph given by Proposition 3.4, so that x and β(x) are associated and the length of β(x) is n. Since x ∈ Λ f , either β(x) is positive or n > M(S, P ). Hence, by Lemma 3.3, we can choose an n-periodic point of g in S \ P , denoted by µ(x), associated to β(x).