MAPS ON COMPACT

. 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 (cid:28)xed point theory and in particular we compute the Lefschetz numbers of period m for large m ’s.


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.More precisely, let f : X → X be a continuous map, a point x ∈ X is periodic of period k ∈ N if f k (x) = x and f j (x) = x for j = 1, . . ., k − 1.The set {x, f (x), . . ., f k−1 (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 understanding the periodic structure of continuous self-maps on manifolds on dimensions greater than 1, because the topology of the manifold plays an important role, in particular we use the Lefschetz xed point theory.
In this article we study the periodic structure of non-constant holomorphic 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 be an ndimensional topological manifold and f a continuous selfmap on X.The map f induces a homomorphism on the kth rational homology group of The Lefschetz Fixed Point Theorem states that if L(f ) = 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 (2) where the sum is taken over all divisors r of m and µ is the Möbius

By the Möbius inversion formula
Observe that L(f m ) and (f m ) are integer numbers for all m.
The Lefschetz zeta function of f is dened as 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 of f .This function is rational, moreover its expression is Using formal computations we can write the Lesfchetz zeta function as a formal innite product that involves the (f d ), for details see [1]: The following result characterizes when the product in the identity (4) has a nite number of nontrivial factors, i.e. when (f d ) = 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 ) = 0 for only nitely many m.
Proof.The only if part.Let m 1 , . . ., m k be the only values of m such that (f m ) = 0. Accoding to (4), Therefore the roots and poles of ζ f (t) are roots of unity.
The if part.Since ζ f (t) is a rational function it has only a nite number of poles and zeros, i.e. let ω 1 , . . ., ω l be such zeros and poles.So they are of the form ω j = e 2πir j /k j , for some positive integer k j and 0 ≤ r j ≤ k j − 1.According to Gauss's lemma the minimal polynomial of ω j over the integers is the k j -th cyclotomic polynomial, i.e.Φ k j (t).Hence Φ k 1 , . . .Φ k l are factors either in the numerator or denominator of ζ f (t).By basic properties of the cyclotomic polynomials (cf.[7]): Therefore there exist non-zero integers c i , with 1 ≤ i ≤ s, for some s, The relationship between the numbers (f m ) and the periodic structure 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 ) = 0 if and only if p ∈ Per(f ).

Compact Riemann surfaces of genius g
Let X = M g be a compact surface of genus g endowed with a complex structure and f : M g → M g be a non-constant holomorphic map.The homology groups of X with rational coecients are: The identity (3) allows to write the Lefschetz zeta function of f as ( 5) where D is the degree of f and p(t) = det(Id − tf * 1 ), which is a polynomial of degree at most 2g.Proof.For continuous self-maps on M g 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 ζ f (t) has zeros which are not roots of unity, according to Theorem 1, there are innitely many m's such that (f m ) = 0.
an integer, we have that ζ f (t) has a root or pole which is not a root of unity.Therefore (f m ) = 0 for innitely many m's.
From Proposition 3 and Theorem 2 it yields the following result.Theorem 4. Let M g be an orientable surface of genus g endowed with a complex structure and f : M g → M g be a nonconstant holomorphic map such that its Leftschetz zeta function is given in (5).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 M g be an orientable surface of genus g endowed with a complex structure and f : M g → M g be a nonconstant holomorphic map.Let λ 1 , . . .λ 2g be the eigenvalues of f * 1 and D the degree of f .If any of these conditions hold 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 For p prime, according to (2) If conditions (a) or (b) are held, it can be easily checked that (f p ) = 0 for suciently large prime p.Hence, by Theorem 2 these prime numbers belong to Per(f).
are arbitrary small for suciently large p. Hence | (f p )| > 0 for large prime numbers p.So by Theorem 2, these numbers p belong to P er(f ).
This proves statement (c).are small for arbitrary large p. Therefore we can conclude (f p ) = 0, for large p.This proves statement (d).
We note that the minimal periods of holomorphics maps on surfaces has been studied in [11].

Proposition 3 .
Let f : M g → M g be a continuous map.Then (a) If p(t) is not a product of cyclotomic polynomials, then (f m ) = 0 for innitely many m.(b) If |D| = 0, 1 and D −1 is not an eigenvalue of f * 1 of multiplicity 1, then (f m ) = 0 for innitely many m.
(a) If p(t) is not a product of cyclotomic polynomials.(b) If |D| = 0, 1 and D −1 is not an eigenvalue of f * 1 of multiplicity 1.
a nite dimensional vector space over Q and f * k is a linear map whose matrix has integer entries.