Periodic point free continuous self-maps on graphs and surfaces

Article history: Received 5 February 2012 Received in revised form 29 February 2012 Accepted 6 March 2012 MSC: 37C05 37C25 37C30


Introduction and statement of the main results
In general the periodic orbits play a main role in the dynamics of a discrete dynamical system, for studying them we can use topological information. Probably the best known results in this direction are the results contained in the seminal paper entitle Period three implies chaos for continuous self-maps on the interval, see [12].
All the non completely standard notions which will appear in the rest of the introduction will be defined in section 2.
In this note M will be either a connected compact graph, or a connected compact surface with or without boundary, orientable or not. Our aim is to study the continuous maps f : M → M having their Per(f ) = ∅, i.e. periodic point free continuous self-maps of M.
Since the word "graph" has several different meanings in mathematics, we shall define here our graphs. A graph is a union of vertices (points) and edges, which are homeomorphic to the closed interval, and have mutually disjoint interiors. The endpoints of the edges are vertexes (not necessarily different) and the interiors of the edges are disjoint from the vertices.
An orientable connected compact surface without boundary of genus g ≥ 0, M g , is homeomorphic to the sphere if g = 0, to the torus if g = 1, or to the connected sum of g copies of the torus if g ≥ 2. An orientable connected compact surface with boundary of genus g ≥ 0, M g,b , is homeomorphic to M g minus a finite number b > 0 of open discs having pairwise disjoint closures. In what follows M g,0 = M g .
A non-orientable connected compact surface without boundary of genus g ≥ 1, N g , is homeomorphic to the real projective plane if g = 1, or to the connected sum of g copies of the real projective plane if g > 1. A non-orientable connected compact surface with boundary of genus g ≥ 1, N g,b , is homeomorphic to N g minus a finite number b > 0 of open discs having pairwise disjoint closures. In what follows N g,0 = N g .
We denote by H k (M, Q) the homological spaces with coefficients in Q. Of course, k = 0, 1 if M is a graph, or k = 0, 1, 2 if M is a surface. Each of these spaces is a finite dimensional linear space over Q. Given a continuous map f : Our main result is the following one.
Theorem 1. The following three statements hold.
(a) Let G be a connected compact graph such that dim Q H 1 (G, Q) = r, and let f : G → G be a continuous map. If Per(f ) = ∅, then the eigenvalues of f * 1 are 1 and 0, this last with multiplicity r − 1.
(b) Let M g,b be an orientable connected compact surface of genus g ≥ 0 with b ≥ 0 boundary components, and f : then the eigenvalues of f * 1 are 1, d and 0, this last with multiplicity 2g − 2 if b = 0; and 1 and 0, this last with multiplicity 2g then the eigenvalues of f * 1 are 1 and 0, this last with multiplicity g + b − 2. Theorem 1 is proved in section 2. The way that we prove this theorem can be used for studying the periodic point free continuous self-maps of many other compact polyhedra, or absolute neighborhood retract spaces which support the existence of Lefschetz numbers (see for more details [5]).
Note that Theorem 1 provide necessary conditions in order that a continuous self-map of M be periodic point free, but such conditions are not sufficient as it is well-known, because for instance we can construct easily continuous self-map of the circle S 1 and of the 2-dimensional torus T 2 having fixed or periodic points and satisfying the necessary conditions of Theorem 1 in order to have Per(f ) = ∅.
From Theorem 1 we obtain the following well-known corollary. For different proofs of the ones that we will provide here of statement (a) of Corollary 2 see for instance [2], and of statement (b) see [1].
There are several papers studying different classes of periodic point free self-maps on the annulus see [8,10], or on the 2-dimensional torus see [4,9,11], or in some connected compact manifolds see [7], . . . But in general to characterize the periodic point free maps on a connected compact polyhedron M is a very hard problem. Here our results are easy to obtain because we only provide necessary conditions for having periodic point free continuous self-maps, which in general are not sufficient.   [3].
The Lefschetz zeta function Z f (t) simplifies the study of the periodic points of f , and also facilitates the determination of the maps f which are Lefschetz periodic point free. The function Z f (t) is a generating function for all the Lefschetz numbers of all iterates of f . More precisely the Lefschetz zeta function of f is defined 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 . From its definition the Lefschetz zeta function is only a formal power series. When M is a polyhedron there is the following alternative way to compute it where n = dim M and Id k is the identity map of H k (M, Q), and by convention det(Id k − tf * k ) = 1 if n k = 0, for more details see [6]. Note that from (3)   We shall use Proposition 4 as a key point for proving our Theorem 1.
Proof of statement (a) of Theorem 1. Let G be a connected compact graph such that dim Q H 1 (G, Q) = r, and let f : G → G be a continuous map. Then, f * 1 is an r × r matrix, and f * 0 is the 1 × 1 matrix (1) because G is connected (for more details see [13,14]). Therefore, if p(λ) is the characteristic polynomial of the matrix f * 1 , from (2) we have If Per(f ) = ∅, by Proposition 4 we must have Z f (t) = 1. Therefore, from the previous equalities the characteristic polynomial must be p(λ) = (−1) r λ r−1 (λ−1). Clearly the zeros of this characteristic polynomial are 1 and 0, this last with multiplicity r−1. Hence the statement is proved.
Proof of statement (b) of Theorem 1. Let M g,b be an orientable connected compact surface of genus g ≥ 0 with b ≥ 0 boundary components, and f : M g,b → M g,b be a continuous map. The degree of f is d if b = 0. We recall the homological spaces of M g,b with coefficients in Q, i.e.
Assume b = 0. If p(λ) is the characteristic polynomial of the matrix f * 1 , from (2) we have Since Per(f ) = ∅, by Proposition 4 we must have Z f (t) = 1. Therefore, from the previous equalities the characteristic polynomial must be p(λ) = λ 2g−2 (λ − 1)(λ − d). Then, the zeros of this characteristic polynomial are 1, d and 0, this last with multiplicity 2g − 2. Hence the statement is Clearly, the rest of the proof follows as in the proof of statement (a), obtaining that the roots of the characteristic polynomial are 1 and 0, this last with multiplicity 2g + b − 2. This completes the proof of statement (b).
Proof of statement (c) of Theorem 1. Let N g,b be a non-orientable connected compact surface of genus g ≥ 1 with b ≥ 0 boundary components, and let f : N g,b → N g,b be a continuous map. We recall the homological spaces of N g with coefficients in Q, i.e.
H k (N g,b , Q) = Q ⊕ n k . . . ⊕ Q, where n 0 = 1, n 1 = g + b − 1 and n 2 = 0; and the induced linear map f * 0 = (1) (see again for additional details [13,14]). So, the action on homology for the surface N g,b is the same than the action on the homology of a graph G with dim Q H 1 (G, Q) = g + b − 1. Therefore, the rest of the proof follows as in the proof of statement (a).