This is a preprint of: “On topological entropy, Lefschetz numbers and Lefschetz zeta functions”, Jaume Llibre, Vı́ctor F. Sirvent, Topology Appl., vol. 268(106906), 1–14, 2019. DOI: [10.1016/j.topol.2019.106906] ON TOPOLOGICAL ENTROPY, LEFSCHETZ NUMBERS AND LEFSCHETZ ZETA FUNCTIONS JAUME LLIBRE1 AND VÍCTOR F. SIRVENT2 Abstract. In the present article we give sucient conditions for C ∞ self-maps on some connected compact manifold manifolds in order to have positive entropy. The conditions are given in terms of the Lefschetz numbers of the iterates of the map and/or its Lef- schetz zeta function. We consider the cases where the manifold is a compact orientable and non-orientable surface, the n-dimensional torus, the product of n spheres of dimension ℓ and the product of spheres of dierent dimensions. 1. Introduction The topological entropy of a topological dynamical system is a non- negative real number which measures the complexity of the system. Topological entropy was rst introduced in 1965 by Adler, Konheim and McAndrew [1]. Later on Dinaburg [8] and Bowen [7] provided a distinct, weaker denition inspired in the Hausdor dimension. This new denition claried the meaning of the topological entropy. Thus for a system dened by the iteration of a function, the topological en- tropy is the exponential growth rate of the number of distinguishable orbits of the iterates. Here we provide sucient conditions in order that the topological entropy of a map in a compact manifold be positive in terms of the Lefschetz numbers of the iterates of the map or in function of the Lefschetz zeta function of the map. This paper is organized as follows: In section 2 we introduce basic notions and denitions, as entropy, Lefschetz numbers and Lefschetz zeta functions, we also describe the entropy conjecture. Later in this section we prove Theorem 6 which gives sucient conditions for a C ∞ Date : May 21, 2018. 2010 Mathematics Subject Classication. 37C25,37C30,37E15. Key words and phrases. Entropy, Lefschetz zeta function, Lefschetz numbers, product of spheres, surfaces, torus. 1 http://www.gsd.uab.cat 2 J. LLIBRE AND V.F. SIRVENT self-maps on a manifold to have positive topological entropy in terms of its Lefschetz zeta function. In section 3 we consider the case of compact and connected surfaces, either orientable and non-orientable, with and without boundary. We give sucient conditions for C 0 , as well as C ∞ , self-maps on a surface in order to have positive entropy. The case of orientable surfaces are considered in Theorems 7 and 8 and non-orientable surfaces in Theorems 9 and 3. In Theorems 7 and 9 the conditions are given in terms of the Lefschetz zeta functions of the map, these two theorems are based on Theorem 6 and in Theorems 8 and 3 the conditions are given in terms of their Lefschetz numbers. In section 4 we consider the case of the n-dimensional torus, since the entropy conjecture is true for continuous self-maps on Tn , we consider in this section continuous maps on this manifold. In Theorem 11 we give the sucient conditions for a continuous map to have positive entropy in terms of their Lefschetz numbers. In section 5 we consider the case of the product of spheres. First we study the product of spheres of the same dimension. In Theorem 14 a criteria is given in terms of the Lefschetz numbers of the map. We also consider the case of product of spheres of dierent dimensions, which is more elaborated from the combinatorial point of view. In Theorem 16 the conditions are given in terms of the Lefschetz numbers, where as Corollary 17 deals with the Lefschetz zeta functions of the maps. All maps considered in this section are C ∞ . In section 6, we give some remarks about the relationship between the entropy of a map and the factorization of its Lefschetz zeta function in factors of the form (1 ± tdi )±1 , with positive integers di . Some results related with the ones of this paper can be found in [12, 15, 17, 19, 24]. 2. Basic Notions, definitions and general results Let X be a topological space and f a continuous self-map on X . We give the Bowen's denition of the topological entropy (cf. [7]). We suppose that X is a compact metric space, with a metric d. We dene the metric dm on X as dm (x, y) := max d(f j (x), f j (y)). 0≤j≤m A nite set S is called (m, ϵ)-separated with respect to f if for any x, y ∈ S we have dm (x, y) > ϵ. We denote by S(m, ϵ) the maximal ON TOPOLOGICAL ENTROPY AND LEFCHETZ ZETA FUNCTIONS 3 cardinality of an (m, ϵ)separated set. The topological entropy of f is dened by ( ) 1 h(f ) := lim lim sup log S(m, ϵ) . ϵ→0 m→∞ m We suppose that X is ndimensional. The map f induces a homo- morphism on the k th rational homology group of X for 0 ≤ k ≤ n, i.e. f∗k : Hk (X, Q) → Hk (X, Q). The Hk (X, Q) is a nite dimensional vector space over Q and f∗k is a linear map whose matrix has integer entries. The spectral radii of f∗k are dened as sp(f∗k ) := max {|λ| : λ is an eigenvalue of f∗k } , and we call the spectral radius of f∗ (sometimes it s also called the spectral radius of f ): sp(f∗ ) := max sp(f∗k ). 0≤k≤n There is a famous and long standing conjecture called the entropy conjecture that states that if f is a C 1 self-map on a compact manifold then log(sp(f∗ )) ≤ h(f ). This conjecture was stated by Shub in 1974 (cf. [23]). Yomdin proved it for C ∞ maps (cf. [25]): Theorem 1 (Yomdin). If f is a C ∞ self maps on a compact manifold, then h(f ) ≥ log(sp(f∗ )). In the case of self-maps on the n-dimensional torus the conjecture was proved for continuous maps by Misiurewicz and Przytycki (cf. [22]). The conjecture is also true for continuous self-maps on nilmanifolds (cf. [21]). Other important result related to this matter is the following theo- rem, due to Manning (cf. [20]): Theorem 2 (Manning). Let f be a C 0 self-map on a compact manifold then h(f ) ≥ log(sp(f∗1 )). The Lefschetz number of f is dened as n ∑ L(f ) = (−1)k trace(f∗k ). k=0 The Lefschetz Fixed Point Theorem states that if L(f ) ̸= 0 then f has a xed point (cf. [6] or [16]). 4 J. LLIBRE AND V.F. SIRVENT The Lefschetz zeta function of f is dened as ( ) ∑ L(f m ) ζf (t) = exp tm . m≥1 m Since ζf (t) is the generating function of 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 it: n ∏ k+1 (1) ζf (t) = det(Idk − tf∗k )(−1) , k=0 where n = dim X , mk = dim Hk (X, Q), Idk is the identity map on Hk (X, Q), and by convention det(Idk − tf∗k ) = 1 if mk = 0 (cf. [10]). We mention an elementary fact about the Lefschetz zeta function that we use in this article. Proposition 3. Let f : X → X be a continuous map. If λ ̸= 0 is a zero or pole of ζf (t) then λ−1 is an eigenvalue of f∗k , for some k. Moreover 0 is never a zero or pole of ζf (t). In particular if the eigenvalues of f∗ are roots of unity then the zeroes and poles of ζf (t) are also roots of unity. Proof. Let qk (t) = det(Idk − tf∗k ). Note that qk (0) = 1, for all k; so from (1), we get ζf (0) = 1. If λ ̸= 0 is a zero or pole of ζf (t) then qk (λ) = 0 for some k . Hence det(λ−1 Id∗k − f∗k ) = 0, i.e. λ−1 is an eigenvalue of f∗k . Note that ω is a root of unity if and only if ω −1 is a root of unity. Hence if all eigenvalues of f∗ are root of unity then the zeroes or poles are also roots of unity.  From (1) it follows the general expression of a continuous self-map on X is either ζf (t) = 1 or of the form N ∏ ζf (t) = qi (t)si , i=1 where qi (t) = ci,mi t + · · · + ci,0 is a polynomial not necessary monic, mi however ci,0 = 1, for all 1 ≤ i ≤ N , we will always assume that the polynomials qi (t) do not have common factors, and the numbers si are integers. We remark that if pi (t) is the characteristic polynomial of f∗i , its degree is ni = dim(Hi (X, Q)), and qi (t) = det(Idi − tf∗i ). Then ON TOPOLOGICAL ENTROPY AND LEFCHETZ ZETA FUNCTIONS 5 qi (1/t) = t−ni p(t), if not all the eigenvalues of f∗i are zero. Note that qi (t) is a polynomial of degree at most ni , and it is equal to ni if and only if zero is not a root of pi (t). First we show two propositions that relate the sizes of the coecients of the polynomials with their roots. We shall use these facts throughout the article. Proposition 4. Let s(t) = tm + αm−1 tm−1 + · · · + α1 t + α0 be a monic polynomial with integers coecients. Suppose that all its roots are zero or roots of unity. ( ) (a) If α0 ̸= 0 then |αm−k | ≤ mk for 1 ≤ k ≤ m − 1. ( ) (b) If α0 = α1 = · · · = αr−1 = 0 and αr ̸= 0 then |αm−k | ≤ m−r k for 1 ≤ k ≤ m − r. Proof. Vieta's formulae allow to express the coecient of the poly- nomial s(t) in term of its roots. Let λ1 , . . . , λm be the roots of the polynomial s(t). The Vieta's formulae are α0 = λ1 ·∑ · · λm , α1 = λi1 · · · λim−1 , i1 <··· m−r j for some 1 ≤ j ≤ m − r − 1. Then there exists λ root of s(t) satisfying |λ| > 1 and moreover λ−1 is a root of q(t). Proof. If the polynomials s(t) and q(t) satisfy the relation q(t) = tr s(t−1 ) then cm−j = αj for 0 ≤ j ≤ r. From Proposition 4 it follows that if(all the ) roots of s(t) are zero or they have norm equal 1 then |αm−j | ≤ m−r for 1 ≤ j ≤ m − r. Hence (m−r) j ( ) |cj | ≤ j . Therefore the condition |cj | > m−r j implies that s(t) has a root λ, with |λ| ̸= 1. Since α0 is an integer and from formula (2) we can conclude that |λ| > 1. Clearly λ−1 is a root of q(t).  Theorem 6. Let f be a C ∞ self-map on compact connected dieren- tiable manifold X , whose Lefschetz zeta function is of the form N ∏ (3) ζf (t) = qi (t)si , i=1 where N a positive integer, the si are integers and qi (t) are polynomials of the form: (4) qi (t) = ci,mi tmi + · · · + ci,0 , where the coecients ci,j are integers, and in particular ci,0 = 1. If the polynomials qi (t) do not have common factors and for some (i, j), i.e. ( )qi (t) and qk (t) are co-prime, for every i ̸= k . there exists |ci,j | > mj i , then the map f has positive topological entropy. Proof. Suppose that in the expression of the Lefschetz zeta function of f as in (3), there is a polynomial qi (t) as in (4), with ci,0 = 1. Hence qi (t) = ci,mi tmi + · · · + ci,0 . ( ) By Proposition 5, if |ci,j | > mj i then there exists λ a root of qi (t), such that 0 < |λ| < 1. Hence λ−1 is an eigenvalue of f∗k , for some k . From Theorem 1, we can conclude that the map f has positive entropy.  ON TOPOLOGICAL ENTROPY AND LEFCHETZ ZETA FUNCTIONS 7 3. Surfaces Let X = X + (g, b) be a compact connected orientable surface of genus g ≥ 0 with b ≥ 0 boundary components, its homology groups are:    Q if k = 0,       2g−times   z }| {    Q ⊕ · · · ⊕ Q if k = 1 and b = 0, Hk (X + (g, b), Q) =   (2g+b−1)−times   z }| {     Q ⊕ · · · ⊕ Q if k = 1 and b > 0,     Q if k = 2 and b = 0,  0 if k = 2 and b > 0. Let f be a continuous self-map on X + (g, b). If b = 0 its Lefschetz zeta function is q(t) ζf (t) = , (1 − t)(1 − Dt) where D is the degree of f , i.e. the map f∗2 = (D) and q(t) = det(Id − f∗1 t). Let p(t) be the characteristic polynomial of f∗1 , its degree is 2g , then q(1/t) = t−2g p(t), if not all the eigenvalues of f∗1 are zero. Note that q(t) is a polynomial of degree at most 2g , and it is equal to 2g if and only if zero is not a root of p(t). In the case b > 0 the Lefschetz zeta function is q(t) ζf (t) = , 1−t where q(t) = det(Id − f∗1 t), it is a polynomial of degree at most 2g + b − 1. As it can be seen in the previous remark, the eigenvalues of the induced map on the rst homology play a very important role in the expression of the Lefschetz zeta function. Therefore Theorem 2 is very relevant in this situation. For this reason we can consider C 0 maps. The adapted version of Theorem 6 for self-maps on X + (g, b) is the following statement: Theorem 7. Let f be a C 0 -self map on X + (g, b). (a) Let b = 0 and the Lefschetz zeta function of f : q(t) ζf (t) = , (1 − t)(1 − Dt) 8 J. LLIBRE AND V.F. SIRVENT where D is the degree of f and the polynomial q(t) is of the form: q(t) = cm−r tm−r + · · · + c0 , ( ) with c0 = 1, m ≤ 2g and 0 ≤ r ≤ m − 1. If |cj | > m−r j for some j such that 1 ≤ j ≤ m−r −1, then the map f has positive topological entropy. Moreover, if f is C ∞ and D ̸= 0, 1, −1 then the topological entropy of f is positive. (b) Let b > 0 and the Lefschetz zeta function of f : q(t) ζf (t) = , 1−t where the polynomial q(t) is of the form: q(t) = cm−r tm−r + · · · + c0 , ( ) with c0 = 1, m ≤ 2g + b − 1 and 0 ≤ r ≤ m − 1. If |cj | > m−r j for some 1 ≤ j ≤ m − r − 1, then the map f has positive topological entropy. Proof. Since the polynomial q(t) appears in the Lefschetz zeta funtion, is q(t) = det(Id − tf∗1 ). Hence p(t) the characteristic polynomial of f∗1 is of the form p(t) = tr (tm−r + am−1 tm−r−1 + · · · + ar ), where cj = am−j for 1 ≤ j ≤ m − r. We apply Proposition 4 to the polynomial s(t) = tm−r +am−1 tm−r−1 + · · · + ar . Hence if r = 0, i.e. a0 ̸= 0, and all the roots of s(t) are roots of unity then |am−1 | ≤ m. So the condition |am−1 | > m, which is |c1 | > m, implies that there exits at least one eigenvalue of f∗1 of modulus greater than( 1); hence, by Theorem ( ) 2 we conclude h(f ) > 0. Similarly if |am−j | > mj , i.e. |cj | > mj , implies h(f ) > 0. If r > 0 then a0 = a1 = · · · = ar−1 = 0. By Proposition 4 |am | ≤ m − r if all the roots of s(t) are roots of unity. Hence the condition |am | > m − r implies that there exits an eigenvalue of f∗1 outside the unit circle, by Theorem ( )2 the map f(m−r has)positive topological entropy. Similarly |am−j | > m−r j , i.e. |c j | > j , implies h(f ) > 0. On the other hand, since f∗2 = (D) the condition D ∈ / {0, 1, −1}, implies that the spectral ratio of f∗ is greater than 1. If f is C ∞ then h(f ) > 0, due to Theorem 1. This proves statement (a). ON TOPOLOGICAL ENTROPY AND LEFCHETZ ZETA FUNCTIONS 9 Statement (b) is proved in a similar manner as in statement (a), therefore we omit its proof.  We can also express a sucient condition for having positive entropy in terms of the Lefschetz numbers. Theorem 8. Let f be a C ∞ self-map on X + (g, b). If b = 0 and |L(f )| > 2(g + 1) then f has positive entropy. Moreover, if f is C 0 and |L(f )| > 2g + 1 + |D| then f has positive entropy. If f is a C 0 self-map on X + (g, b) with b > 0 and |L(f )| > 2g + b then f has positive topological entropy. Proof. From the denition of the Lefschetz number, we have L(f ) = trace(f∗0 ) − trace(f∗1 ) + trace(f∗2 ) = 1 − (λ1 + · · · + λm ) + D. where m = dim(H1 (X + (g, b))), i.e. m = 2g if b = 0 and m = 2g + b − 1 if b > 0, and D = 0 if b > 0. If all the eigenvalues of f∗1 are root of unity or zero then |L(f )| ≤ 1 + |λ1 | + · · · + |λm | + |D| ≤ 1 + m + |D|. Hence in the case b = 0, if |L(f )| > 1+2g +|D| then the spectral radius of f∗1 is greater than 1. From Theorem 2 it follows f has positive topological entropy. However if f is a C ∞ map and |D| > 1 then h(f ) > 0; therefore the condition |L(f )| > 2(1 + g) implies that one of the eigenvalues of f∗ is outside of the unit circle, hence h(f ) > 0. In the case b > 0, we have D = 0, so the condition |L(f )| > 2g + b implies the existence of an eigenvalue of f∗1 of modulus greater then 1, hence it has positive topological entropy.  Let X = X − (g, b) be a compact, connected, nonorientable surface of genus g ≥ 1 with b ≥ 0 boundary components, its homology groups are:    Q if k = 0;    (g+b−1)−times Hk (X − (g), Q) = z }| {     Q ⊕ · · · ⊕ Q if k = 1.  0 if k = 2. 10 J. LLIBRE AND V.F. SIRVENT Let f be a continuous self-map on X − (g, b), its Lefschetz zeta func- tion is given by: q(t) ζf (t) = , 1−t where q(t) = det(Id − f∗1 t). Let p(t) be the characteristic polynomial of f∗1 , its degree is g + b − 1, then q(1/t) = t−(g+b−1) p(t). If not all the eigenvalues of f∗1 are zero. As in the orientable case q(t) is a polynomial of degree at most g + b − 1, and it is equal to g + b − 1 if and only if zero is not a root of p(t). Theorem 9. Let f be a C 0 self-map on X − (g, b) with Lefschetz zeta function q(t) ζf (t) = , 1−t where q(t) is a polynomial of the form: q(t) = cm−r tm−r + · · · + c0 , ( ) with c0 = 1, m ≤ g + b − 1 and 0 ≤ r ≤ m − 1. If |cj | > m−r j for some j such that 1 ≤ j ≤ m − r − 1, then the map f has positive topological entropy. The proof of this theorem follows the same arguments as the proof of Theorem 7, for this reason we omit it here. Theorem 10. Let f be a C 0 self-map of X − (g, b). If |L(f )| > g + b then f has positive entropy. Proof. From the denition of the Lefschetz number, we have L(f ) = trace(f∗0 ) − trace(f∗1 ) + trace(f∗2 ) = 1 − (λ1 + · · · + λg+b−1 ) + 0, where the λi 's are the eigenvalues of f∗1 . If |L(f )| > g + b then there exits one λi with |λi | > 1. Hence the topological entropy of f is positive.  4. The n-dimensional torus: Tn In this section we set X = S1 · · × S}1 = Tn , the n dimensional | × ·{z n−times torus. As it was mention in section 2, in the entropy conjecture is true for continuous self-maps on Tn (cf. [22]). Therefore, in this section we consider continuous maps, i.e. C 0 maps. ON TOPOLOGICAL ENTROPY AND LEFCHETZ ZETA FUNCTIONS 11 Theorem 11. Let f : Tn → Tn be a continuous map. If |L(f )| > 2n then h(f ) > 0. ∏ Proof. The Lefschetz numbers for f satisfy L(f ) = ni=1 (1 − λi ), where the λi 's are the eigenvalues of f∗1 (cf. [5]). If all the eigenvalues of f are zero or root of unity then ∏n n ∏ |L(f )| ≤ |1 + λi | ≤ (1 + |λi |) ≤ 2n . i=1 i=1 Hence if |L(f )| > 2 there exits i ∈ {1, . . . , n} such that |λi | > 1. n Therefore, from Theorem 1, it follows the topological entropy of f is positive.  Now we will consider criteria for a continuous self-map on Tn to have positive entropy in terms of the Lefschetz zeta function. It is known that the Lefschetz zeta function for a map whose eigenvalues are not zero, is a rational function such that the degree of the polynomial on the numerator and the degree of the polynomial on the denominator are the same (cf. [2]). Furthermore if the eigenvalues of f∗1 are root of unity then the Lefschetz zeta function of f is of the form (cf. [3]): N ∏ ζf (t) = (1 ± tdi )ri , i=1 or ζf (t) = 1, with positive integers di and integers ri . Moreover in [3] there are explicit formulae for the values of di and ri in terms of the characteristic polynomial of f∗1 . Proposition 12. Let f be a continuous self-map on Tn if its Lefschetz zeta function is not constant and it cannot be factorized in the form N ∏ (5) ζf (t) = (1 ± tdi )ri , i=1 where N , di are positive integers and ri integers. Then h(f ) > 0. Proof. If h(f ) = 0 then all the eigenvalues of f∗1 are roots of unity. It is well known that 1 is an eigenvalue of f∗1 if and only if ζf (t) = 1 (cf. [13]). So if 1 is not an eigenvalue of f∗1 then the Lefschetz zeta function of f can be factorized in the form of ∏N ζf (t) = (1 ± tdi )ri , i=1 with N , di positive integers and ri integers, ([3]).  12 J. LLIBRE AND V.F. SIRVENT Theorem 6 is valid for continuous self maps on the n-dimensional torus. In the following example we show the importance of the fact that the polynomials qi (t) in the Lefschetz zeta function in the expression (3), do not have common factors. Let f be a map on T3 given by the integer matrix in R3 :   R 0 0 A =  0 −1 0  , 0 0 1 where R > 1 is a integer. The map f has positive entropy since the eigenvalues of f∗1 are R, 1 and −1. However its Lefschetz zeta function is: ζf (t) = 1, as many maps of topological entropy zero, for example if R = ±1. As we re- marked before the Lefschetz zeta function, of a continuous self-map on Tn , is equal 1 if and only if 1 is an eigenvalue of f∗1 (cf. [13]). 5. The product of spheres Let X = Sℓ1 ×· · ·×Sℓn . Using the Künneth Theorem (see for instance [14]), we compute the homology groups of X over the rational numbers Q. They are given by   |Q ⊕ ·{z· · ⊕ Q if bk ̸= 0, } Hk (X, Q) = bk  {0} if bk = 0, with 0 ≤ k ≤ ℓ1 + · · · + ℓn ; where bk is the number of ways that k can be obtained as by summing up subsets of (ℓ1 , . . . , ℓn ), i.e. bk is the cardinality of the set { } ∑ (6) S ⊂ {1, . . . , n} : ℓj = k . j∈S The numbers bk are the Betti numbers of X . First we consider the product of spheres of the same dimension: Let X = X(ℓ, n) = S ℓ · · × S}ℓ . If ℓ1 = · · · = ℓn = ℓ, then | × ·{z n times  (n)  j if k = jℓ, bk =  0 if k is not a multiple of ℓ. ON TOPOLOGICAL ENTROPY AND LEFCHETZ ZETA FUNCTIONS 13 For the Lefschetz numbers of self-maps on X(ℓ, n), we recall the fol- lowing theorem: Theorem 13 ([4]). Let f : X(ℓ, n) → X(ℓ, n) be a continuous map. Then, for all m > 0, L(f m ) = det(Id + (−1)ℓ f∗ℓ m ), where Id is the identity map on Qn . In particular, { L(f m ) = Cf∗ℓ m (1) if ℓ is odd, L(f ) = (−1) Cf∗ℓ (−1) if ℓ is even, m n m where CH (t) := det(t Id − H) denotes the characteristic polynomial of the linear map H . Theorem 14. Let f : X(ℓ, n) → X(ℓ, n) be a C ∞ map. If |L(f )| > 2n then h(f ) > 0. Proof. The proof goes along the same idea as in Theorem 11. By Theorem 13, n ∏ n ∏ |L(f )| = | det(Id + (−1)ℓ f∗ℓ )| = | (1 + (−1)ℓ λi )| ≤ (1 + |λ|i |), i=1 i=1 where the λi 's are the eigenvalues of f∗ℓ So if L(f ) > 2n there exists λi , for some 1 ≤ i ≤ n, such that |λi | > 1. Hence h(f ) > 0.  As in the case of the n-dimensional torus if the eigenvalues of f∗ℓ are root of unity then the Lefschetz zeta function of f is of the form (cf. [4]): n ∏ ζf (t) = (1 ± tdi )ri , i=1 or ζf (t) = 1, with di positive integers and ri integers. So we have the following result, which is similar to Proposition 12. Proposition 15. Let f be a C ∞ self-map on X(ℓ, n) if its Lefschetz zeta function is not constant and it cannot be factorized in the form n ∏ ζf (t) = (1 ± tdi )ri , i=1 with di are positive integers and ri integers. Then h(f ) > 0. In the following lines we will consider the product of spheres with dierent dimensions, i.e. X = Sℓ1 × · · · × Sℓn , with ℓ1 < · · · < ℓn . 14 J. LLIBRE AND V.F. SIRVENT Let n ∪ M = M (ℓ1 , . . . , ℓn ) := {ℓi1 + · · · + ℓis : i1 < · · · < is }. s=1 We suppose that ℓ1 < · · · < ℓn . By elementary combinatorics we have that the cardinality of the set M is at most 2n − 1. We shall assume that the cardinality of M is exactly 2n − 1, i.e. the numbers ℓ1 , . . . , ℓn are such that all the sums dened in the set M are dierent. In this condition is satised we say that M is a sum free set. In this case the Betti numbers of X are bk = 1 if k ∈ M and bk = 0 otherwise, i.e. n ∪ Hk (X, Q) = Q if k ∈ {0} {ℓi1 + · · · + ℓis | i1 < · · · < is }, s=1 and trivial otherwise. Let f : X → X be a continuous map, since the homology groups are either 1-dimensional or trivial, its induced maps on homology are f∗k = (ak ) if k ∈ M and f∗k = 0 otherwise. Theorem 16. Let X = Sℓ1 × · · · × Sℓn and f be a C ∞ self-map on X . We assume that the set of partial sums of the indexes of the dimension of the spheres, M (ℓ1 , . . . , ℓn ), is a sum free set. If L(f ) > 2n then the topological entropy of f is positive. Proof. The Lefschetz numbers for f are: ∑ L(f m ) = 1 + (−1)k am k , k∈M ∑ for all integer m > 0. So |L(f )| ≤ 1 + k∈M |ak |. Since the cardinality of the set M is 2n − 1, if the condition |L(f )| > 2n holds, then there exist j ∈ M , such that |aj | > 1. Then the positive entropy of f is positive.  According to (1) the Lefschetz zeta function of the map f in the case of the sum-free condition is ∏ k+1 (1 − tak )(−1) ∏ k+1 k∈M ζf = (1 − a0 t)−1 (1 − tak )(−1) = . k∈M 1−t Clearly if ak are dierent from 0, 1 or −1 then f has positive entropy. In the following lines we describe the Lefschetz zeta function of f , when the sum-free condition is not satised. For this we need to give ON TOPOLOGICAL ENTROPY AND LEFCHETZ ZETA FUNCTIONS 15 a partition of the set M . We recall that the Betti number bk is the cardinality of the set described in (6). If r ̸= 0, we introduce { } ∑ Mr := k = ℓj : S ⊂ {1, . . . , n}, |S| = r . j∈S In other words the set Mr consists of the numbers k 's such that can be obtained as r dierent sums of elements of the set {ℓ1 , . . . , ℓn }. Clearly they form a disjoint partition of M : M = ∪R r=1 Mr , where R = max1≤k≤L bk and L = ℓ1 + · · · + ℓn . The set M is sum-free if and only if M = M1 . From the denition follows that k ∈ Mr if and only if bk = r. We will give an example: Let ℓ1 = 1, ℓ2 = 3, ℓ3 = 5, ℓ4 = 6, them M1 = {1, 3, 4, 5, 7, 8, 9, 10, 11, 14, 15}, M2 = {6, 9}; since 6 = ℓ4 = ℓ1 + ℓ3 and 9 = ℓ2 + ℓ4 = ℓ1 + ℓ2 + ℓ3 . From (1) the Lefschetz zeta function of the map f , ζf (t) is of the form: ∏L ∏ k+1 k+1 det(Idk − tf∗k )(−1) = (1 − t)−1 det(Idk − tf∗k )(−1) = k=0 k∈M R ∏ ∏ k+1 = (1 − t)−1 det(Idk − tf∗k )(−1) = r=1 k∈Mr ∏ ∏ k+1 ∏ k+1 k∈M1 (1 − tak ) det(Idk − tf∗k )(−1) · · · k∈MR det(Idk − tf∗k )(−1) k∈M2 . 1−t Let qk (t) := det(Idk − tf∗k ). Note that det(Idk − tf∗k ) is a polynomial of degree at most bk if k ∈ Mbk . So qk (t) = ck,mk −rk tmk −rk + · · · + ck,0 , with ck,0 = 1 and mk − rk ≤ bk , if k ∈ Mbk . In the example of X = S1 × S3 × S5 × S6 , here L = 15 and R = 2, since bk = 1 for k ∈ M1 , bk = 2 for k ∈ M2 , and bk = 0 otherwise. The Lefschetz zeta function of a self map f , ζf (t), on this space is of the form: ∏15 k+1 det(Idk − tf∗k )(−1) = k=0 ∏ k+1 ∏ k+1 (1 − t)−1 det(Idk − tf∗k )(−1) det(Idk − tf∗k )(−1) = k∈M1 k∈M2 (1 − ta1 )(1 − ta3 )(1 − ta5 )(1 − ta7 )(1 − ta9 )(1 − ta11 )(1 − ta15 ) q9 (t) · , (1 − t)(1 − ta4 )(1 − ta8 )(1 − ta10 )(1 − ta14 ) q6 (t) 16 J. LLIBRE AND V.F. SIRVENT where q6 (t) = c6,2 t2 + c6,1 t + c6,0 , q9 (t) = c9,2 t2 + c9,1 t + c9,1 , with c6,0 = c9,0 = 1. Theorem 6 for self-maps on X = Sℓ1 × · · · × Sℓn is the following statement: Corollary 17. Let f be a C ∞ self-map on X = Sℓ1 × · · · × Sℓn . (a) If |ak | > 1, for some k ∈ M1 then h(f ) > 0. (b) If the polynomials qk (t) do not have( common ) roots and there ex- its a coecient ck,j , with |ck,j | > mkj−rk . Then the topological entropy of f is positive. 6. Remarks For self-maps on Tn and X(ℓ, n), we have seen in Propositions 12 and 15, that if its Lefschetz zeta functions are not constant and they cannot be factorized on the form (5), i.e. N ∏ ζf (t) = (1 ± tdi )ri , i=1 with positive integers N , di and integers ri ; then the maps have positive entropy. In general if the Lefschetz zeta function is factorized as in (5), i.e. all its zeros and poles are roots of unity, it does not mean that the spectral radius of f∗ is greater than 1. Since there could be eigenvalues λi of f∗k , for some k , such that the corresponding factors 1−λi t in ζf (t) cancel out. The maps whose Lefschetz zeta function is of the form (5) or constant, are called almost quasi-unipotent, and they were studied in [18]. The map shown in the example given in section 4 is an almost quasi-unipotent map, with an eigenvalue of modulus bigger than 1. We would like to remark the fact that the factorization of the Lef- schetz zeta function in the form (5) is a necessary condition for a map to have a nite number of periodic points (cf. [9, 11]), provided that the space is a dierentiable manifold and the map dierentiable. As for necessary conditions for a map to have positive entropy, we wonder if the reciprocal of Proposition 12 is true. Question: Let f be a continuous self-map on Tn with positive topo- logical entropy and 1 is not an eigenvalue of f∗1 . Does ζf (t) have a pole or zero which is not root of unity? ON TOPOLOGICAL ENTROPY AND LEFCHETZ ZETA FUNCTIONS 17 Acknowledgements The rst author is supported by the Ministerio de Economía, Indus- tria y Competitividad, Agencia Estatal de Investigación grant MTM2016- 77278-P (FEDER), the Agència de Gestió d'Ajuts Universitaris i de Recerca grant 2017 SGR 1617, and the European project Dynamics- H2020-MSCA-RISE-2017-777911. Part of this work was done during a visit of the second author to Centre de Recerca Matèmatica (CRM), in the framework of the Lluis Santaló fellowship. References [1] R.L. Adler, A.G. Konheim and M.H. McAndrew, Topological Entropy. Trans. Amer. Math. Soc. 114 (1965), 309319. [2] P. Berrizbeitia and V.F Sirvent, On the Lefschetz zeta function for quasi- unipotent maps on the n-dimensional torus. J. Dierence Equ. Appl. 20 (2014), no. 7, 961972. [3] P. Berrizbeitia, M.J. González, A. Mendoza and V.F. Sirvent, On the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus. 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Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Catalunya, Spain E-mail address : jllibre@mat.uab.cat 2 Departamento de Matemáticas, Universidad Simón Bolívar, Apartado 89000, Caracas 1086-A, Venezuela E-mail address : vsirvent@usb.ve