This is a preprint of: “Periods of holomorphic maps on compact Riemann surfaces and product of spheres”, Jaume Llibre, Vı́ctor F. Sirvent, J. Difference Equ. Appl., 2020. PERIODS OF HOLOMORPHIC MAPS ON COMPACT RIEMANN SURFACES AND PRODUCT OF SPHERES JAUME LLIBRE1 AND VÍCTOR F. SIRVENT2 Dedicated to O.M. Sharkovsky in his 82nd birthday. Abstract. In this article we consider non-constant holomorphic maps on Riemann surfaces and product of Riemann spheres, we give conditions on the maps in order that they have arbitrary large prime numbers as periods. We use Lefschetz xed point theory and in particular we compute the Lefschetz numbers of period m for large m's. 1. Introduction In the theory of the dynamical systems and mainly in the study of the iteration of self-maps on a topological manifold X , the periodic orbits play an important role. f : X → X be a More precisely, let continuous map, a point x ∈ X is periodic of period k ∈ N if f (x) = x k j k−1 and f (x) 6= x for j = 1, . . . , k − 1. The set {x, f (x), . . . , f (x)} is the periodic orbit of the periodic point x of period k . If k = 1 then the periodic point x is called a xed point. We shall denote by P er(f ) the set of periods of the map f . A natural question is to ask for all the possible periods that the map f can exhibit. In some situations the knowledge of some possible periods of the map gives understanding of some global properties of the dynamics of the map, as in the case for continuous selfmaps on the interval. If a continuous map on the interval has a periodic orbit of period three, then the map has orbits of all possible periods (cf. [8, 12]). The dierential topological methods are very useful for understand- ing the periodic structure of continuous self-maps on manifolds on di- mensions greater than 1, because the topology of the manifold plays an important role, in particular we use the Lefschetz xed point theory. Date : September 6, 2019. 2010 Mathematics Subject Classication. 37C25,37C30,37E15. Key words and phrases. Lefschetz numbers, holomorphic maps, periodic point, product of spheres, torus. 1 2 J. LLIBRE AND V.F. SIRVENT In this article we study the periodic structure of non-constant holo- morphic maps on Riemann surfaces and product of Riemann spheres, in particular we give conditions on homology such that the maps have arbitrary large prime numbers as periods; the corresponding results are Theorems 4, 5 and Corollary 8. In section 2 we consider the self-maps on Riemann surfaces and in section 3 the maps on product of Riemann spheres. Let X ndimensional topological manifold and f a continuous be an selfmap on X . The map f induces a homomorphism 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 Lefschetz number of f is dened as n X (1) L(f ) := (−1)k trace(f∗k ). k=0 The Lefschetz Fixed Point Theorem states that if L(f ) 6= 0 then f has a xed point (cf. [3] or [9]). The Lefschetz numbers of period m, introduced in [4] and [10], are dened by X (2) `(f m ) := µ(r)L(f m/r ), r|m where the sum is taken over all divisors r of m and µ is the Möbius function dened by   1 if m = 1; µ(m) := 0 if there is a k such that k 2 divides m;  (−1)s if m = p1 · · · ps with pi distinct primes. By the Möbius inversion formula X L(f m ) = `(f r ). r|m m m Observe that L(f ) and `(f ) are integer numbers for all m. The Lefschetz zeta function of f is dened as ! X L(f m ) ζf (t) = exp tm . m≥1 m This function keeps the information of the Lefschetz number for all the iterates of f , so this function gives information about the set of periods PERIODS OF TRANSVERSAL MAPS ON THE TORUS 3 of f. This function is rational, moreover its expression is n Y k+1 (3) ζf (t) = det(Id∗k − tf∗k )(−1) , k=0 where n = dim M , nk = dim Hk (M, Q), Id := Id∗k is the identity map on Hk (M, Q), and by convention det(Id∗k − tf∗k ) = 1 if nk = 0 (cf. [6]). Using formal computations we can write the Lesfchetz zeta function as a formal innite product that involves the `(f d ), for details see [1]: ! X L(f m ) Y `(f d ) (4) ζf (t) = exp tm = (1 − td )− d . m≥1 m d≥1 The following result characterizes when the product in the identity (4) has a nite number of nontrivial factors, i.e. when `(f d ) 6= 0 for only nitely many m. Theorem 1. Let X be an ndimensional topological manifold and f a continuous selfmap on X . The zeros and poles of the Lefschetz zeta function of f are roots of unity if and only if `(f m ) 6= 0 for only nitely many m. Proof. The only if part. Let m1 , . . . , mk be the only values of m such that`(f m ) 6= 0. Accoding to (4), m1 )/m mk )/m ζf (t) = (1 − tm1 )−`(f 1 · · · (1 − tmk )−`(f k . Therefore the roots and poles of ζf (t) are roots of unity. The if part. Since ζf (t) is a rational function it has only a nite number of poles and zeros, i.e. let ω1 , . . . , ωl be such zeros and poles. 2πirj /kj So they are of the form ωj = e , for some positive integer kj and 0 ≤ rj ≤ kj − 1. According to Gauss's lemma the minimal polynomial of ωj over the integers is the kj -th cyclotomic polynomial, i.e. Φkj (t). Hence Φk1 , . . . Φkl are factors either in the numerator or denominator of ζf (t). By basic properties of the cyclotomic polynomials (cf. [7]): Y Φkj (t) = (1 − td )µ(kj /d) . d|kj Therefore there exist non-zero integers ci , with 1 ≤ i ≤ s, for some s, such that s Y ζf (t) = (1 − tdi )ci i=1 where di is a divisor of the least common multiple of k1 · · · kl .  4 J. LLIBRE AND V.F. SIRVENT The relationship between the numbers `(f m ) and the periodic struc- ture for holomorphic maps was given in [5] in the following result. Theorem 2 (Theorem A of [5]). Let M be a compact complex manifold and f : M → M be a nonconstant holomorphic map. Then, there exists M > 0 such that for all p ∈ N prime and p > M , l(f p ) 6= 0 if and only if p ∈ Per(f ). 2. Compact Riemann surfaces of genius g X = Mg be a compact surface of genus g endowed with a complex Let structure andf : Mg → Mg be a non-constant holomorphic map. The homology groups of X with rational coecients are: H0 (Mg , Q) = H2 (Mg , Q) = Q, H1 (Mg , Q) = Q ⊕ · · · ⊕ Q . | {z } 2g The identity (3) allows to write the Lefschetz zeta function of f as p(t) (5) ζf (t) = , (1 − t)(1 − Dt) where D is the degree of f and p(t) = det(Id − tf∗1 ), which is a poly- nomial of degree at most 2g . Proposition 3. Let f : Mg → Mg be a continuous map. Then (a) If p(t) is not a product of cyclotomic polynomials, then `(f m ) 6= 0 for innitely many m. (b) If |D| = 6 0, 1 and D−1 is not an eigenvalue of f∗1 of multiplicity 1, then `(f m ) 6= 0 for innitely many m. Proof. For continuous self-maps on Mg an expression of its Lefschetz zeta function is given in (5), which is written in irreducible form, i.e. 1−t and 1 − Dt are not factors of p(t). If p(t) is not a product of cyclotomic polynomials then ζf (t) has zeros which are not roots of unity, according to Theorem 1, there are innitely many m's such that `(f m ) 6= 0. If D 6= 0 and D−1 f∗1 of multiplicity 1, then is not an eigenvalue of −1 D is a root or pole of ζf (t), because p(t) = det(Id − tf∗1 ). So if D 6= −1, 1 since D is an integer, we have that ζf (t) has a root or pole m which is not a root of unity. Therefore `(f ) 6= 0 for innitely many m's.  From Proposition 3 and Theorem 2 it yields the following result. PERIODS OF TRANSVERSAL MAPS ON THE TORUS 5 Theorem 4. Let Mg be an orientable surface of genus g endowed with a complex structure and f : Mg → Mg be a nonconstant holomorphic map such that its Leftschetz zeta function is given in (5). (a) If p(t) is not a product of cyclotomic polynomials. (b) If |D| = 6 0, 1 and D−1 is not an eigenvalue of f∗1 of multiplicity 1. Then there exists an integer N > 0 such that for all prime p > N , p is a period of the map f . Theorem 5. Let Mg be an orientable surface of genus g endowed with a complex structure and f : Mg → Mg be a nonconstant holomorphic map. Let λ1 , . . . λ2g be the eigenvalues of f∗1 and D the degree of f . If any of these conditions hold (a) |D| > max{1, |λ1 |, . . . , |λ2g |}. (b) There exists |λj | > max{1, |D|, |λi | : i 6= j}. (c) There exists i1 , . . . , ik such that λi1 = · · · = λik and |λi1 | > max{1, |D|, |λj | : j 6= i1 , . . . , ik }. (d) There exists i1 , . . . , ik such that |λi1 | = · · · = |λik | > max{1, |D|, |λj | : j 6= i1 , . . . , ik }. Then there exists an integer N > 0 such that for all prime p > N , p is a period of the map f . Proof. From the denition of the Lefschetz numbers (1) and for all m ∈ N we have L(f m ) = 1 − (λm m m 1 + · · · + λ2g ) + D . For p prime, according to (2) `(f p ) = L(f p ) − L(f )  = 1 − (λp1 + · · · + λp2g ) + Dp − (1 − (λ1 + · · · + λ2g ) + D) = (Dp − D) − (λp1 − λ1 ) − · · · − (λp2g − λ2g ). If conditions (a) or (b) are held, it can be easily checked that `(f p ) 6= 0 for suciently large prime p. Hence, by Theorem 2 these prime numbers belong to Per(f). If λi1 = · · · = λik , the number `(f p ) can be written as X `(f p ) = −k(λpi1 − λi1 ) + (Dp − D) − (λpj − λj ). j6=i1 ,...ik 6 J. LLIBRE AND V.F. SIRVENT Hence X |`(f p )| ≥ k|λpi1 − λi1 | − |Dp − D| − |λpj − λj | j6=i1 ,...ik ! |Dp − D| X |λpj − λj | = |λpi1 − λi1 | k − p − . |λi1 − λi1 | j6=i ,...i |λpi1 − λi1 | 1 k If |λi1 | > max{1, |D|, |λj | : j 6= i1 , . . . , ik }, the ratios |Dp − D| |λpj − λj | , |λpi1 − λi1 | |λpi1 − λi1 | are arbitrary small for suciently large |`(f p )| > 0 for large p. Hence prime numbers p. So by Theorem 2, these numbers p belong to P er(f ). This proves statement (c). If |λi1 | = · · · = |λik | > 1 then X `(f p )| ≥ (|λpi1 − λi1 + · · · + λpik − λik |) − |Dp − D| − |λpj − λj | j6=i1 ,...ik X = |λi1 | p−1 ((λi1 − 1) + · · · + (λik − 1)) − |D − D| − p |λpj − λj | j6=i1 ,...ik X ≥ C1 |λi1 |p − |Dp − D| − |λpj − λj |, j6=i1 ,...ik for some constant C1 > 0. Hence ! |Dp − D| X |λpj − λj | `(f p )| ≥ |λi1 |p C1 − − . |λi1 |p j6=i ,...i |λi1 |p 1 k If |λi1 | > max{1, |D|, |λj | : j 6= i1 , . . . , ik } then the ratios |Dp − D| |λpj − λj | , . |λi1 |p |λi1 |p are small for arbitrary large p. Therefore we can conclude `(f p ) 6= 0, for large p. This proves statement (d).  We note that the minimal periods of holomorphics maps on surfaces has been studied in [11]. PERIODS OF TRANSVERSAL MAPS ON THE TORUS 7 3. Product of Riemann spheres In this section we consider the product of Riemann spheres, which have a natural complex structure. We dene X(n) := S2 × · · · × S2 . | {z } n−times Theorem 6. Let f : X(n) → X(n) be a continuous map. Let λ1 , · · · , λn be the eigenvalues of f∗2 . (a) If |λi | > 1 for 1 ≤ i ≤ n, then `(f p ) 6= 0 for all prime number p. (b) If λi = −1 for some i then `(f p ) = 0, for all odd prime numbers. (c) If λi = −1 for some i and λj 6= 0 for 1 ≤ j ≤ n then `(f 2 ) 6= 0. Before proving Theorem 6 we present the following result that de- termines the value of the Lefschetz numbers for the maps on X(n). Theorem 7 ([2]). Let f : X(n) → X(n) be a continuous map. Then, for all m > 0, L(f m ) = det(Id + f∗2 m ), where Id is the identity map on Qn . In particular, L(f m ) = (−1)n Cf∗2 m (−1), where Ch (t) := det(t Id − h) denotes the characteristic polynomial of h. Proof of Theorem 6. Due to Theorem 7, we have L(f m ) = (−1)n (1 + λm m 1 ) · · · (1 + λn ). If p is a prime number then (6) `(f p ) = L(f p ) − L(f ) (7) = (−1)n ((1 + λp1 ) · · · (1 + λpn ) − (1 + λ1 ) · · · (1 + λn )). If λi > 1 for 1≤i≤n then λpi > λi so (−1)n `(f p ) > 0 for every prime p. λi < −1 for 1 ≤ i ≤ n then λpi < λi for every prime p > 2. If n is If p p even then `(f ) > 0 and if n is odd `(f ) < 0. Similarly if p = 2 then (−1)n `(f p ) > 0 for all n. This proves statement (a). If λi = −1 for some i, then it follows from (7) that `(f p ) = 0 for p an odd prime. Moreover if λj 6= 0 for all 1 ≤ j ≤ n then, from (7) we 2 have `(f ) 6= 0. This proves statements (b) and (c), respectively.  8 J. LLIBRE AND V.F. SIRVENT The following corollary is a consequence of Theorem 6 and Theo- rem 2. Corollary 8. Let f : X(n) → X(n) be a non-constant holomorphic map. Let λ1 , · · · , λn be the eigenvalues of f∗2 . If |λi | > 1, for 1 ≤ i ≤ n then there exists an integer N such that all prime numbers p > N are periods of f . Theorem 9. Let f : X(n) → X(n) be a non-constant holomorphic map. If f∗2 has an eigenvalue bigger than one in modulus and that it does not have root of unity as eigenvalues. Then there exists N such that all prime numbers p > N are periods of f . Theorem 9 is a particular case of the following more general result proved in [13]. Theorem 10. Let f : X(n) → X(n) be a continuous map. We assume that f∗2 has an eigenvalue bigger than one in modulus and that it does not have root of unity as eigenvalues. Then there exists N such that `(f m ) 6= 0 for m ≥ N . Acknowledgements The rst author is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER), the Agència de Gestió d'Ajuts Univer- sitaris i de Recerca grant 2017SGR1617, and the H2020 European Re- search Council grant MSCA-RISE-2017-777911. References [1] I. K. Babenko and S. A. Bogatyi, The behaviour of the index of periodic points under iterations of a mapping, Math. USSR Izv., 38 (1992), 126. [2] P. Berrizbeitia, M.J. González, and V.F. Sirvent, On the Lefschetz zeta function and the minimal sets of Lefschetz periods for MorseSmale dif- feomorphisms on products of `-spheres, Topology Appl. 235 (2018) 428444. [3] R.F. Brown, The Lefschetz xed point theorem, Scott, Foresman and Com- pany, Glenview, IL, 1971. [4] A. Dold, Fixed point indices of iterated maps, Invent. math. 74 (1983), 419 435. [5] N. Fagella and J. 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Sirvent A note on the periodic structure of transversal maps on the torus and the product of spheres, Pre-print (submitted). 1 Departament de Matemàtiques. Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Catalunya, Spain Email address : jllibre@mat.uab.cat 2 Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile Email address : victor.sirvent@ucn.cl