On the periods of a continuous self-map on a graph

Let G be a graph and f be a continuous self-map on G. We present new and known results (from another point of view) on the periods of the periodic orbits of f using mainly the action of f on its homology, or the shape of the graph G.


Introduction and statement of the main results
A discrete dynamical system (G, f ) is formed by a continuous map f : G → G, where G is a topological space.
A point x ∈ G is periodic of period k if f k (x) = x and f i (x) = x if 0 < i < k. If k = 1, then x is called a fixed point. Per( f ) denotes the set of periods of all the periodic points of f .
The orbit of the point x ∈ G is the set {x, f (x), f 2 (x), . . . , f n (x), . . .}, whereby f n we denote the composition of f with itself n times. To knowledge the behavior of all different kinds of orbits of f is to study the dynamics of the map f .
Many times the periodic points play an important role for understanding the dynamics of a discrete dynamical system. One of the best known results in this direction is the paper Period three implies chaos for continuous interval maps, see Li and Yorke (1975).
Here, a graph G is a compact connected space containing a finite set V , such that G\V has finitely many open connected components, each one homeomorphic to the interval (0, 1), called edges of G, and the points of V are called the vertexes of G. The edges are disjoint from the vertexes, and the vertexes are at the boundary of the edges.
In this paper, we shall work with a graph G. Our goal is to study the periods of the periodic points of the continuous maps f : G → G. Independently of the fact that to study the set of periods of these kinds of graph maps is relevant by itself for understanding their dynamics. The graph maps are relevant for studying the dynamics of some different kinds of surface maps, see, for instance (Handel and Thurston 1985;Mendes de Jesus 2017).
The degree of a vertex V of a graph G is the number of edges having V in its boundary if an edge has both boundaries in V, then we count this edge twice. An endpoint of a graph G is a vertex of degree one. A branching point of a graph G is a vertex of degree at least three.
The homological spaces of G with coefficients in Q are denoted by H k (G, Q). Since G is a graph k = 0, 1. A continuous map f : G → G induces linear maps f * k : H k (G, Q) → H k (G, Q). We only work with graphs, so H 0 (G, Q) ≈ Q and f * 0 is the identity map. A subset of G homeomorphic to a circle is a circuit. It is known that H 1 (G, Q) ≈ Q m being m the number of the independent circuits of G in the sense of the homology. Here f * 1 is a m × m matrix A with integer entries. For more details on this homology see, for instance (Spanier 1981).
If A is a m × m matrix, then a submatrix lying in the same set of k rows and columns is a k × k principal submatrix of A. The determinant of a principal submatrix is a k × k principal minor. The sum of the m (1) The biggest modulus of the eigenvalues of the matrix A is called the spectral radius of A and it is denoted by sp(A).
Our main results are the following ones.
Theorem 1 Let G be a graph, f : G → G be a continuous map, and A be the integral matrix of the endomorphism f * 1 : H 1 (G, Q) → H 1 (G, Q) induced by f on the first homology group of G. The following statements hold.

. . , m then the intersection of Per( f ) and the set of all the divisors of k is not empty.
Theorem 1 is proved in Sect. 2 using the Lefschetz fixed point theory. The next result is an immediate consequence of Theorem 1.

Corollary 2 Under the assumptions of Theorem 1, if the characteristic polynomial of the matrix
While Theorem 1 provides information about periods of a continuous map from a graph into itself, in Llibre (2012), there is a characterization of these maps without periodic points.
Let k be a positive integer we denote by god(k) the greatest odd divisor of k. Let S be a subset of positive integer, the pantheon of S is the set {god(k) : k ∈ S}.
Theorem 3 Let G be a graph, f : G → G be a continuous map, and f * 1 : H 1 (G, Q) → H 1 (G, Q) be the endomorphism induced by f on the first homology group of G. If sp( f * 1 ) > 1, then f has infinitely many periods. More precisely, there is an n ∈ N such that {kn : k ∈ N} ⊂ Per( f ) and the pantheon of Per( f ) is infinite. From Theorem 4, we can deduce many results similar to the one given in the seminal paper Period three implies chaos for self-continuous maps on the interval in the sense of having infinitely many periods.

Corollary 5
The following map f have infinitely many periods if: (a) f is a continuous self-map on the graph having the shape of the letter Y with the branching point fixed and having a period n, such that god(n) > 3; (b) f is a continuous self-map on the graph having the shape of the number 8 or on the graph having the shape of the letter θ with the branching points fixed and having a period n such that god(n) > 4; (c) f is a continuous self-map on the glasses graph having the shape of the graph described in Fig. 1 with the branching points fixed and having a period n such that god(n) > 6.
Theorems 3, 4 and Corollary 5 are proved in Sect. 3. We note that this paper is a kind of survey with new results. That is, Theorem 1 is completely new, but Theorems 3 and 4 essentially follow combining known results on the continuous self-maps on graphs as we will see in their proofs.
Note that our graphs are compact spaces by definition. Therefore, the unique graphs which are manifolds are the closed interval and the circle. Since the homology of a closed interval reduces to H 0 (G) = Q, the same than the homology of a point, then f * 1 is not defined. Therefore, our technique does not provide any information about the dynamics of a continuous self-map on a closed interval. For a continuous self-map f of the circle, we have that f * 1 is the degree of the map f , and Theorem 3 says that if the degree is different from − 1, 0 and 1, then f has infinitely many periods.

Proof of Theorem 1
Let f : G → G be a continuous map on the graph G. The Lefschetz number of f is defined by The Lefschetz Fixed Point Theorem states: If L( f ) = 0 then f has a fixed point [see, for instance (Brown 1971)].
The results which provide information about the periodic points of a self-continuous map f : G → G using information of the action of f on the homology of G, are based on the Lefschetz Fixed Point Theorem, which does not use information related with the torsion which appear in the homology of G if instead of working with rational homology we work, for instance, with real homology. This is the reason that we take homology with coefficients in Q instead of coefficients in R or Z.
To control the whole sequence of the Lefschetz numbers of the iterates of f , i.e., {L( f n )} n≥1 , we use the formal Lefschetz zeta function of f defined by It is known that for a continuous self-map of a graph G, the Lefschetz zeta function is the rational function: where A is the integer matrix defined by f * 1 ; for a proof, see Franks (1982).
From (2) and (3), we obtain Therefore, we have that Therefore Since a graph G is a subset of R 3 , we consider the distance between two points of G as the distance of these two points in R 3 . Now, we define the distance d n on G by A finite set S is called (n, ε)-separated with respect to f if for different points x, y ∈ S we have d n (x, y) > ε. We denote by S n the maximal cardinality of an (n, ε)-separated set. Define Then is the topological entropy of f . We have given the definition of Bowen because probably is the shorter one, the classical definition was due to Adler et al. (1965), see, for instance, the book of Hasselblatt and Katok (2002) and Balibrea (2016) for other equivalent definitions and properties of the topological entropy.
The next result is due to Manning (1975).
Theorem 6 Let f : G → G be a continuous map on the graph G, then log max{1, sp( f * ,1 )} ≤ h( f ).
There are two different proofs for the next result, see Llibre and Misiurewicz (1993) and Blokh (1991): Theorem 7 Let f : G → G be a continuous map on the graph G. Then the following statements are equivalent: (a) There is an m ∈ N such that {mn : n ∈ N} ⊂ Per( f ).
The following result can be found in Llibre and Saghin (2012).
Theorem 8 Let f : G → G be a continuous map on the graph G having e endpoints, s edges, v vertexes and at least one branching point. Assume that f has all branching points fixed. Then, god(n) > e + 2s − 2v + 2 for some period n of f if and only if h( f ) > 0.

Proof of Theorem 4
Under the assumptions of Theorem 4 we have that god(n) > e + 2s − 2v + 2 for some period n of f , so h( f ) > 0 by Theorem 8. Again, by Theorems 4 and 7 is proved.

Proof of Corollary 5
The proof is a direct consequence of the application of Theorem 4 taking account that e + 2s − 2v + 2 is respectively equal to 3 (e = 3, v = 4 and s = 3) for the graph Y ; equal to 4 for the graph 8 (e = 0, v = 1 and s = 2) and for the graph θ (e = 0, v = 2 and s = 3), and 6 for the glasses graph (e = 2, v = 6 and s = 7).