This is a preprint of: “Entropy and periods for continuous graph maps”, Jaume Llibre, Vı́ctor F. Sirvent, Comput. Appl. Math., vol. 40, 300:1–11, 2021. DOI: [10.1007/s40314-021-01677-2] ENTROPY AND PERIODS FOR CONTINUOUS GRAPH MAPS JAUME LLIBRE1 AND VÍCTOR F. SIRVENT2 Abstract. For continuous maps from a topological graph into it- self we provide new relationships between their topological entropy, their homology and their periods. 1. Introduction and statement of the main results A topological graph or simpliy 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 x0 ∈ X and y0 ∈ Y , then the wedge sum X ∨ Y is the quotient of the disjoint union X and Y obtained by identifying x0 and y0 to a single point (for details, see [9, pp. 10]). The wedge sum is also known as “one point union”. For example, S1 ∨ S1 is homeomorphic to the figure of shpe “8”, two circles touching at a point. Some graphs can be obtained as particular cases of wedge sums of S1 , and a compact connected graph X such s−times that dim(H1 (X, Q)) = s is homotopic to S1 ∨ · · · ∨ S1 , as usual here H1 (X, Q) denotes the first homology group of the topological space X with coefficients in Q. These spaces are also called bouquet of circles, s−times we denote by Gs := S1 ∨ · · · ∨ S1 . Since our techniques rely on homology, we shall mainly consider bou- quet of circles, i.e. graphs of the type Gs , for some integer s > 1. 2010 Mathematics Subject Classification. 37C25, 37C30, 37E15, 55M20. Key words and phrases. Topological entropy, periods, Lefschetz numbers, Lef- schetz zeta function, graphs. 1 2 J. LLIBRE AND V.F. SIRVENT We have that the homology spaces for Gs are: H0 (Gs , Q) = Q, since Gs is connected; and H1 (Gs , Q) = Q ⊕ · · · ⊕ Q, by elementary    s properties of the wedge sum of spaces ([9, pp. 126]). For trivial graphs their H1 (Gs , Q) is trivial (when there are homotopic to a point) or Q (when there are homotopic to a circle). The spectral radius sp(T) of a linear transformation T : U → U on a finite dimensional vector space U is defined as the maximun of the norm of its eigenvalues, i.e. sp(T ) := max {| λ| : λ is an eigenvalue of T } . A continuous map from a graph into itself is called a graph map. In the present article we explore some relationships between the topo- logical 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 order to have positive topological entropy (Theorem 4). In Proposition 5, we show that these conditions are not necessary con- ditions. Theorems 8 and 10 and Corollaries 9 and 11 show how the first s Lefschetz numbers of a graph map on Gs determine its periodic struc- ture. In section 3 we extend the construction given in Proposition 5, so that we can have maps on Gs with positive topological entropy, all possible periods and the characteristic polynomial of the induced map on homology consists of products of cyclotomic polynomials of total degree s. Furthermore we present some open questions related to these matters. 1.1. Topological entropy of graph maps. For a definition of topo- logical entropy of a continuous map of a topological space into itself see for instance [1, 2, 14]. 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 ([18]). Theorem 1 (Manning). Let f be a C 0 self-map on a compact manifold. Then h(f ) ≥ log(sp(f∗1 )). However for working on graphs-maps 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. ENTROPY AND PERIODS FOR CONTINUOUS GRAPH MAPS 3 The Lefschetz number of a graph map f is defined as (1) L(f ) := 1 − trace(f∗1 ). We note that f∗1 : H1 (Gs , Q) → H1 (Gs , Q) is a linear transformation that can be represented by an s × s matrix with integer entries. The Lefschetz Fixed Point Theorem states that if L(f ) = 0 then f has a fixed point (cf. [4] or [13]). From (1) we get L(f m ) = 1 − trace(f∗1 m ) = 1 − (λm m 1 + · · · + λs ), where λ1 , . . . , λs are the eigenvalues of f∗1 . The asymptotic Lefschetz number L∞ (f ) is defined to be the growth rate of the Lefschetz number of the iterates of f , i.e. L∞ (f ) := max{1, lim sup |L(f m )|1/m }. n→∞ The asymptotic Lefschetz number allows to obtain a lower bound for the topological entropy of a continuous graph map. Theorem 2. Let f : G → G be a graph map. (a) L∞ (f ) = max{1, sp(f∗1 )}. (b) The topological entropy of f satisfies h(f ) ≥ log L∞ (f ). Statement (a) of Theorem 2 is proved in [8], and statement (b) is due to Jiang [10, 11]. The following result is well known, but since its proof is easy we shall provide it in section 2. Corollary 3. If the topological entropy of a graph map f is zero, then all the roots of the characteristic polynomial of f∗1 are zero or roots of unity. Moreover sp(f∗1 ) is either 0, or 1. Related with Theorem 2 and Corollary 3 we have the following ques- tion. Open question 1. 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    L(f m ) ζf (t) := exp tm . m≥1 m 4 J. LLIBRE AND V.F. SIRVENT Since ζf (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 det(Id − tf∗k1 ) (2) ζf (t) = , 1−t where Id is the identity map on H1 (G, Q) (cf. [5]). Our first main result is the following one. Theorem 4. Let G be a graph, homotopic to Gs , for some s > 1, and f : G → G be a continuous map. (a) Let p(t) = ts −a1 ts−1 +· s· ·+(−1) as be the characteristic polyno- s mial of f∗1 . If |ak | > k for some 1 ≤ k ≤ s, then the entropy of the map f is positive. (b) If L(f m ) > 1 + s for some m ≥ 1, then h(f ) > 0. (c) Let ζf (t) be the Lefschetz zeta function of f . Then ζf (t) = q(t)(1 − t)−1 , where q(t) = b0 ts−r + · · · + bs−r, where r is the s−r multiplicity of 0 as eigenvalue of f∗1 . If |bk | > s−r−k for some 0 ≤ k ≤ s − r, then h(f ) > 0. Theorem 4 is proved in section 2. The next result shows that there are continuous maps of Gs with positive entropy, spectral radius equal to 1, and having all periods Proposition 5. Let G be a graph homotopic to Gs being s an arbitrary positive integer. Then there are continuous maps f : G → G such that h(f ) > 0, sp(f∗1 ) = 1, L(f k ) = 0 for all positive integer k and Per(f ) = N, being N the set of all positive integers. Proposition 5 is proved in section 2. In [7] was proved the following result, however stated in a different manner: Theorem 6 ([7]). Let G be a graph, homotopic to Gs and f : G → G be a continuous map with f∗1 the induced map on homology and p(t) the characteristic polynomial of f∗1 . (a) If s is odd and the number of roots of p(t) equal to 0 or ±1, considering their multiplicity is even, then h(f ) > 0. (b) If s is even and the number of roots of p(t) equal to 0 or ±1, considering their multiplicity is odd, then h(f ) > 0. ENTROPY AND PERIODS FOR CONTINUOUS GRAPH MAPS 5 The proof of this theorem is based in the fact the cyclotomic poly- nomials have even degree (with the exception of the first two), so the hypotheses force that the characteristic polynomial of f∗1 has odd de- gree, so it has an eigenvalue of norm greater than 1. Proposition 5 shows that the conditions given in Theorems 4 and 6 are not necessary conditions in order that a map has positive topolog- ical entropy. 1.2. 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. If 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 en- tropy and the periods of the map. This theorem was in proved [3] and [15]. Theorem 7. 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 ). From its proof it is not clear how the number m is related with the homology of the graph G, more precisely we have the following question. Open question 2. Let f : Gs → Gs be a graph map with positive entropy. We wonder if there is a relationship between the values h(f ), m, s (as in the statement of Theorem 7) and f∗1 . In [17] it was shown the following relation between the periods of f and its homology on the graph G. Theorem 8 ([17]). Let G be a graph, homotopic to Gs and f : G → G be a continuous map such that it does not have periodic points of period k for 1 ≤ k ≤ n, with n ≥ 2. Assume that the induced map in the first homology space f∗1 is invertible. Then n < s. A map f is Lefschetz periodic point free if L(f k ) = 0 for all positive integers k. From Theorem 8 it follows the next result. Corollary 9. Let G be a graph, homotopic to Gs and f : G → G be a continuous map. If f∗1 is invertible, then f is not Lefschetz periodic point free. 6 J. LLIBRE AND V.F. SIRVENT Corollary 9 is proved in section 2. The following result is a refinement of Theorem 8, and it is proved in section 2 using the same tool: Newton’s formulae for symmetric polynomials. Theorem 10. Let G be a graph, homotopic to Gs and f : G → G be a continuous map such that the characteristic polynomial of f∗1 is ts − a1 ts−1 + · · · + (−1)s as . Then L(f ) = · · · = L(f j ) = 0 if and only if aj = 0 with j ∈ {2, . . . , s}. Corollary 11. Let G be a graph homotopic to Gs and f : G → G be a continuous map. If L(f ) = · · · = L(f s ) = 0 then L(f k ) = 0 for all positive integer k. Corollary 11 is proved in section 2. Some other results on the periods of graph maps can be found in [2, 6, 16]. In section 3 we show how to get graph maps of Gs with positive entropy, an infinite set Per(f ), spectral radius equal to 1 and with a characteristic polynomial p(t) of f∗1 formed by an arbitrary product of cyclotonic polynomials and the factor tr , where r is any non-negative integer. Without loss of generality we shall take s = 2, the arguments extend to any positive integer s. 2. Proofs of the results Proof of Corollary 3. If all the eigenvalues of f∗1 are zero, then sp(f∗1 ) = 0, and the corollary is proved. Let λ1 , . . . , λr with 1 ≤ r ≤ s be the non-zero eigenvalues of f∗1 . By Theorem 2 we get that |λj | ≤ 1 for j = 1, . . . , r. Then the characteristic polynomial of f∗1 is of the form r ts−r (t − λj ) = ts−r (tr − a1 tr−1 + · · · + (−1)r ar ). j=1 r Since ar = λj is a non-zero integer and |λj | ≤ 1 for j = 1, . . . , r, j=1 we have that |ar | = rj=1 |λj | = 1. Hence |λj | = 1 for j = 1, . . . , r. It is known that if a polynomial has integer coefficients, constant term equal to one and all its roots have modulus one, then all its roots are roots of unity, see for instance [20]. Therefore, clearly sp(f∗1 ) = 1 and the corollary is proved.  ENTROPY AND PERIODS FOR CONTINUOUS GRAPH MAPS 7 Proof of Theorem 4. Let λ1 , · · · , λs be the eigenvalues of f∗1 , i.e. s p(t) = (t − λi ) = ts − a1 ts−1 + · · · + (−1)s as . i=1 We recall the Vieta’s formulae: as = λ1 · · · λs , as−1 = λi1 · · · λis−1 , i1 <···