Entropy and periods for continuous graph maps

For continuous self-maps on topological graphs, we provide new relationships between their topological entropy, their homology and their periods.


Introduction
In the field of dynamical systems, an important question is: what kind of topological conditions implies positive topological entropy? In the present article, we deal with this question for continuous self-maps on graphs, not homotopic to a point or a circle. We explore some relationships between the topological entropy of a graph map, the induced map on homology and its periodic structure. In particular, we give sufficient conditions on a graph map (in terms of its Lefschetz numbers, Lefschetz zeta function and/or the characteristic polynomial of the induced map on homology) to have positive topological entropy (Theorem 3). All these concepts are defined in Sect. 2. In Theorem 4, we show that these conditions are not necessary conditions, in particular, we prove that given any product of cyclotomic polynomial of total degree s, there exists a graph map on G s (a bouquet of s circles), so that the characteristic polynomial of the induced map on homology is the given polynomial. Theorems 7 and 8 and Corollaries 9 and 10 show how the periodic structure of a graph map on G s is determined only by first s Lefschetz numbers of the map. The article is structured as follows: the basic notions, definitions and the statements of the main results are presented in Sect. 2. In the first part of this section, we consider the topological entropy of graph maps and in the second part the periodic structure of these maps. In Sect. 3, we present the proofs of the main results, i.e. Theorems 3, 8 and Corollaries 9, 10. Whereas Theorem 4, is proved in Sect. 4, moreover, we give explicit examples of the construction in which the proof is based. Furthermore, we present some open questions related to these matters.

Basic notions and statement of the main results
A topological graph or simply a graph G is a compact connected space having a finite set of points V such that G \ V consists of finitely many connected components each of them homeomorphic to an open interval. Some graphs are homotopic to particular cases of wedge sums of circles, which we shall define later on. However, not all graphs can be obtained in this way, e.g. the interval, in general any tree, the topological space with the shape of the capital letter sigma. We say that a graph is trivial if it is homotopic to a circle or to a point, e.g. intervals and trees are trivial graphs.
Given topological spaces X and Y with chosen points x 0 ∈ X and y 0 ∈ Y , then the wedge sum X ∨ Y is the quotient of the disjoint union X and Y obtained by identifying x 0 and y 0 to a single point (for details, see Hatcher 2002[pp. 10]). The wedge sum is also known as "one point union". For example, S 1 ∨ S 1 is homeomorphic to the figure of shape "8", two circles "touching" at a point. Some graphs can be obtained as particular cases of wedge sums of S 1 , and a graph X such that dim(H 1 (X , Q)) = s is homotopic to S 1 ∨ s−times · · · ∨ S 1 , as usual here H 1 (X , Q) denotes the first homology group of the topological space X with coefficients in Q. These spaces are also called bouquet of circles, we denote by G s := S 1 ∨ s−times · · · ∨ S 1 . Since our techniques rely on homology, we shall mainly consider bouquet of circles, i.e. graphs of the type G s , for some integer s > 1.
We have that the homology spaces for G s are: H 0 (G s , Q) = Q, since G s is connected; and H 1 (G s , Q) = Q ⊕ · · · ⊕ Q s , by elementary properties of the wedge sum of spaces (Hatcher 2002[pp. 126]). For trivial graphs, their H 1 (G s , Q) is trivial (when there are homotopic to a point) or Q (when there are homotopic to a circle).
A loop of a graph G is a subset of G homeomorphic to the circle S 1 . These loops generate the fundamental group of the graph. The first homology group H 1 (G, Q) of the topological graph G with coefficients in Q is generated by the number of independent loops of the graph G, independent as elements of the fundamental group.
A continuous map from a graph into itself is called a graph map. Let f be a graph map on G. The action of f * 1 induced by f on H 1 (G, Q) is the following. Let γ i for i = 1, . . . , n be the independent loops. Then, if the n × n matrix defined by f * 1 has in its row i and column j the integer a i j which is the number of oriented times that f (γ j ) covers the loop γ i .

Topological entropy of graph maps
For a definition of topological entropy of a continuous map of a topological space into itself see for instance Adler et al. 1965;Alsedà et al. 2000;Llibre 2015.
A well-known result that relates the topological entropy h( f ) of a continuous map f : X → X with the homology of X is the following one due to Manning.
However, for working on graphs-maps (with the graph different from a closed interval or the circle), we cannot use Theorem 1 since it is valid for compact manifolds. There are particular results that deals the entropy for maps on graphs and the spectra of the induced maps on homology.
The Lefschetz number of a graph map f is defined as We note that f * 1 : is a linear transformation that can be represented by a s × s matrix with integer entries (cf. Hatcher 2002).
The Lefschetz Fixed Point Theorem states that if L( f ) = 0, then f has a fixed point (cf. Brown 1971or Lefschetz 1926. From (2), we get The asymptotic Lefschetz number L ∞ ( f ) is defined to be the growth rate of the Lefschetz number of the iterates of f , i.e.
The asymptotic Lefschetz number allows to obtain a lower bound for the topological entropy of a continuous graph map.
The spectral radius sp(T) of a linear transformation T : U → U on a finite dimensional vector space U is defined as the maximum of the norm of its eigenvalues, i.e. Guaschi and Llibre (1995), and statement (b) is due to Jiang (1993Jiang ( ,1996.

Statement (a) of Theorem 2 is proved in
From Theorem 2, it follows that if a graph map f has zero topological entropy then the spectral radius of f * 1 is either 0 or 1. Therefore is natural to state the following question: Open question: Are there some conditions on f * 1 with sp( f * 1 ) = 1, which force that h( f ) > 0?
The Lefschetz zeta function of f is defined as Since ζ f (t) is the generating function of all the Lefschetz numbers, L( f m ), it keeps the information of the Lefschetz number for all the iterates of f . There is an alternative way to compute the Lefschetz zeta function of a graph map where I d is the identity map on H 1 (G, Q) (cf. Franks 1982). Our first main result is the following one.
Theorem 3 Let G be a graph, homotopic to G s , for some s > 1, and f : G → G be a continuous map.
Theorem 3 is proved in Sect. 3. The next result shows that there are continuous maps of G s with positive entropy, spectral radius equal to 1, and having infinitely many periods, the definition is given in Sect. 2.2.
We recall Gauss' Lemma which states that the cyclotomic polynomials are the irreducible polynomials (over the integers) whose roots are roots of unity. See Lang 2005 for definition and basic properties of cyclotomic polynomials.
Theorem 4 Let G be a graph homotopic to G s , where s is a positive integer. Let p(t) be a polynomial of degree s which is a product of cyclotomic polynomials. Then, there is a continuous map f : G → G such that h( f ) > 0, the characteristic polynomial of f * 1 is p(t) and having infinitely many periods. The proof of this theorem is based in the fact the cyclotomic polynomials have even degree (with the exception of the first two), so the hypotheses force that the characteristic polynomial of f * 1 has odd degree, so it has an eigenvalue of norm greater than 1.
Theorem 4 shows that the conditions given in Theorems 3 and 5 are not necessary conditions in order that a map has positive topological entropy.

Periods of graphs maps
In this subsection, we provide some results which relate the periodic structure of a graph map with its homology and its topological entropy.
Assume that f : G → G is a graph map. A point x ∈ G is periodic of period k if f k (x) = x and f j (x) = x for j = 1, . . . , k − 1. We denote by Per( f ) the set of periods of all periodic points of f .
First, we recall a well-known result that relates the topological entropy and the periods of the map. This theorem was proved in Blokh (1992) and Llibre and Misiurewicz (1993).
Theorem 6 Let f : G → G be a graph map. Then, the entropy of f is positive if and only if there is an m ∈ N such that {km | k ∈ N} ⊂ Per( f ).
In Llibre and Sirvent (2013), it was shown the following relation between the periods of f and its homology on the graph G.

Theorem 7 Let G be a graph, homotopic to G s and f : G → G be a continuous map such that it does not have periodic points of period k for
The following result is a refinement of Theorem 7, and it is proved in Sect. 3 using the same tool: Newton's formulae for symmetric polynomials.

Corollary 10 Let G be a graph, homotopic to G s and f : G → G be a continuous map. If f * 1 is invertible, then f is not Lefschetz periodic point free.
Corollaries 9 and 10 are proved in section 3. Some other results on the periods of graph maps can be found in Alsedà et al. (2000); García Guirao and Llibre (2019); Llibre and Sá (1994).
We recall the Vieta's formulae: If we suppose that h( f ) = 0, by statement (a) of Theorem 2 we have that |λ i | ≤ 1 for all 1 ≤ i ≤ s. From (4), it follows For the general term: This proves statement (a).
This proves statement (b). The Lefschetz zeta function can be written as a rational function, the expression (3) in our context is If p(t) is the characteristic polynomial of f * 1 , i.e. p(t) = det( f * 1 − t I d 1 ), then q(t) = (−1) s t s p(t −1 ). If r is the multiplicity of 0 as eigenvalue of f * 1 , we set r = 0 if 0 is not an eigenvalue of f * 1 , then If we suppose that h( f ) = 0, then all the roots of p(t) have modulus smaller than or equal to one. We apply the argument based on the Vieta's formulae used in statement (a), to the polynomial u(t) = t s−r − a 1 t s−r −1 + · · · + (−1) r a s−r , if all its roots have norm smaller than or equal to 1, then |b k | ≤ s−r k , because the degree of u(t) is s −r . Therefore, statement (c) follows from statement (a).
Proof (Proof of Corollary 9) By Theorem 8 and its proof, if L( f ) = · · · = L( f s ) = 0 then a s = · · · = a 2 = 0 and a 1 = 1, i.e. the characteristic polynomial of f * 1 is t s−1 (t − 1). Therefore, all the eigenvalues of f * 1 are zero except one eigenvalue which is equal to 1.

Some graph maps on G s and proof of Theorem 4
Before proving Theorem 4, we introduce some examples that illustrate the construction in which is based the proof of Theorem 4. Let g : S 1 → S 1 be a continuous circle map of degree 1 with positive topological entropy, infinite Per(g) and a fixed point p, see (Alsedà et al. 2000) for the existence of this kind of circle maps, in particular, the map shown on the left of Figure 1. Letg(x) := 1 − g(x), with x ∈ S 1 = [0, 1] (mod 1); which shares these properties but its degree is −1. Let G 2 = S 1 ∨ S 1 the wedge sum of two circles, by identifying a point, we choose p such point. We denote the two circles of G 2 by S 1 and S 2 . We identify G 2 with the interval [0, 2] identifying the points 0, 1 and 2 with the point p. Then, we identify S 1 and S 2 with the intervals [0, 1] and [1, 2], respectively.
Let f : G 2 → G 2 be the map defined as follows: Fig. 1. Since g has positive topological entropy and an infinite set of periods, the map f defined in this way on G 2 , has positive topological entropy and an infinite set of periods, specifically (1)), it follows f * 1 (a 1 ) = a 2 and f * 1 (a 2 ) = −a 1 , i.e.
consequently its characteristic polynomial is 4 (t) = t 2 + 1, the 4-th cyclotomic polynomial, see Lang (2005) or Washington (1982. Similarly, we can get maps f : G 2 → G 2 with positive topological entropy, infinite Per( f ) with the characteristic polynomials of f * 1 equal to the 3-th and 6-th cyclotonic polynomial, i.e. 3 (t) = t 2 + t + 1 and 6 (t) = t 2 − t + 1, respectively (see Figure 2) because Fig. 1 On the right the graphic of the circle-map g, and on the left the graphic of the G 2 -map f .

Fig. 2
Maps f : G 2 → G 2 such that the characteristic polynomials of f * 1 are 3 (t) and 6 (t) on the right and left, respectively.
Let f be a self-map on G 3 whose graphs is in Fig. 3. We remark that f ( p) = p, f (S 1 ) = S 2 , f (S 2 ) = S 3 and if consider the following partition of [2, 3] = I 1 ∪ I 2 ∪ I 3 in closed intervals of the same length; we have f (I 1 ) = S 1 , f (I 2 ) = S 2 and f (I 3 ) = S 3 .
Since f maps I i onto S i piece-wise linearly, its entropy is given by logarithm of the maximum of the absolute value of its slopes. In this case, the slope is −6, for 2 ≤ i ≤ 3, hence h( f ) > 0.
Proof (Proof of Theorem 4) Let p(t) = t s + c s−1 t s−1 + · · · + c 1 t + c 0 be a polynomial whose factors are cyclotomic polynomials. Let G s = S 1 ∨ · · · ∨ S 1 s , we consider it as s circles: S 1 , S 2 , . . . S s identified by a common point p. We identify G s with the interval [0, s], the points 0, 1, . . . , s with the point p, and the circle S j with the interval [ j − 1, j], with 1 ≤ j ≤ s. We define f : G s → G s as follows: If x ∈ S i then f (x) := x + i, for 1 ≤ i ≤ s − 1. We consider the partition of S s = I 1 ∪ · · · ∪ I s , where where ϕ is the affine map that sends I 1 into [0, 1]. We recall that c 0 = ±1, since p(t) is a product of cyclotomic polynomials. Let 2 ≤ j ≤ s, if c j−1 = 0 then f (x) := s − 1, for x ∈ I j . If c j−1 = −d = 0 then we consider the partition of I j in |d| subintervals of the same length, i.e. I j = I 1 j ∪ · · · ∪ I d j , where I k j := [β j,k−1 , β j,k ], with β j,k := s − 1 + j − 1 s + k s|d| , for 1 ≤ k ≤ |d|.
For x ∈ I k j , we define f (x) := sd(x − β j,k−1 ) + j − 1 if d > 0 and f (x) = sd(x − β j,k−1 ) + j if d < 0, i.e. f maps in an affine way I j onto S j with slope sd.
note that this matrix is the Frobenius companion matrix of the polynomial p(t).
Since the map g has positive topological entropy and has infinite periods, it follows that the map f also shares these properties.
We would like to remark that the spectral radius of f * 1 , its characteristic polynomial, and the Lefschetz numbers depend only on f * 1 , so they will not change if we replace f by a map homotopic to f . On the other hand, positive topological entropy and large set of periods can be achieved locally. Thus, for every f we can fix a small arc ν, then "drag" its image back to ν via a homotopy, and then make ν invariant for a new map g (homotopic to f ), with g restricted to I equal to any prescribed continuous map. This fact allows to have, every f is homotopic to a map with positive topological entropy and an infinite set of periods.