Partially periodic point free self-maps on surfaces, graphs, wedge sums and products of spheres

Let be a topological discrete dynamical system. We say that it is partially periodic point free up to period n, if f does not have periodic points of periods smaller than . When X is a compact connected surface, a connected compact graph, or , we give conditions on X, so that there exist partially periodic point free maps up to period n. We also introduce the notion of a Lefschetz partially periodic point free map up to period n. This is a weaker concept than partially periodic point free up to period n. We characterize the Lefschetz partially periodic point free self-maps for the manifolds , with , , and .


Introduction
Let X be a topological space and let f : X ! X be a continuous map. A ðdiscreteÞ topological dynamical system is formed by the pair ðX; f Þ.
We say that x [ X is a periodic point of period k if f k ðxÞ ¼ x and f j ðxÞx for j ¼ 1; . . . ; k 2 1. We denote by Per( f ) the set of all periods of f.
The set {x; f ðxÞ; f 2 ðxÞ; . . . ; f n ðxÞ; . . . } is called the orbit of the point x [ X. To study the dynamics of a map f is to study all the different kinds of orbits of f. If x is a periodic point of f of period k, then its orbit is {x; f ðxÞ; f 2 ðxÞ; . . . ; f k21 ðxÞ}, and it is called a periodic orbit.
Often the periodic orbits play an important role in the dynamics of a discrete dynamical system, and for studying them we can use topological tools. One of the bestknown results in this direction is the result contained in the well-known paper entitled 'Period three implies chaos' for continuous self-maps on the interval, see [19].
If Perð f Þ ¼ B then we say that the map f is periodic point free. There are several papers studying different classes of periodic point free self-maps on the annulus, see [12,16], or on the two-dimensional torus, see [2,14,18].
If Perð f Þ > {1; 2; . . . ; n} ¼ B then we say that the map f is partially periodic point free up to period n. If n ¼ 1, we say that f is fixed point free. Different classes of partially periodic point free self-maps are studied in [6,25,28].
Let n be the topological dimension of a compact polyhedron X. We denote by H k ðX; QÞ, for 0 # k # n, the homology groups of X with coefficients over the rational numbers. They are finite dimensional vector spaces over Q. Given a continuous map f : X ! X, it induces linear maps f *k : H k ðX; QÞ ! H k ðX; QÞ, for 0 # k # n. All the entries of the matrices f *k are integer numbers.
The Lefschetz number Lð f Þ is defined as ð21Þ k traceðf *k Þ: One of the main results connecting the algebraic topology with the fixed point theory is the Lefschetz Fixed Point Theorem which establishes the existence of a fixed point if Lð f Þ -0, see for instance [4]. If we consider the Lefschetz number of f m , i.e. Lðf m Þ, then in general it is not true that Lðf m Þ -0 implies that f has a periodic point of period m; it only implies the existence of a periodic point with a periodic divisor of m. The Lefschetz numbers have been used frequently for studying the set of periods of different kinds of maps, see for instance [10,20,22,23] and the references cited therein. We say that a continuous map f : X ! X is Lefschetz periodic point free if Lðf m Þ ¼ 0 for all m $ 1. We say that the map f is Lefschetz partially periodic point free up to period n if Lðf m Þ ¼ 0 for all 1 # m # n. If n ¼ 1, we say that f is Lefschetz fixed point free. These are weaker notions of periodic point free and partially periodic point free up to period n, since Lefschetz periodic point free is a necessary condition to be a periodic point free, but not sufficient as it is shown by considering the identity map on the circle. The Lefschetz periodic free maps on some connected compact manifolds have been studied in [11,21].
One of the goals of this paper is to study the self-continuous maps on graphs and closed surfaces and other manifolds which are (Lefschetz) partially periodic point free up to period n.
A graph is a union of vertices and edges, which are homeomorphic to the closed interval and have mutually disjoint interiors. The endpoints of the edges are vertices (not necessarily different), and the interiors of the edges are disjoint from the vertices. Some graphs are homotopic to particular cases of wedge sums of spheres, which we shall define later on. However not all graphs can be obtained as particular cases of wedge sums of circles, e.g. the interval or the topological space with the shape of the capital letter sigma.
Here a closed surface means a connected compact surface without boundary, orientable or not. More precisely, an orientable connected compact surface without boundary of genus g $ 0, M g , is homeomorphic to the sphere if g ¼ 0, to the torus if g ¼ 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 closure. In what follows M g;0 ¼ 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 ¼ 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 closure. In what follows N g;0 ¼ N g .
Theorem 1. Let f : X ! X be a continuous map partially periodic point free up to period n. Assume that the induced map in the first homology space f *1 : H 1 ðX; QÞ ! H 1 ðX; QÞ is invertible. Then the following statements hold.
(a) Let X be a connected compact graph such that dim H 1 ðX; QÞ ¼ r. If n $ 2 then n , r.
All the results stated in this introduction are proved in Section 2.
One of the first results that show the relation between the topology of a compact topological space M and the existence of periodic points of a homeomorphism f : M ! M is due to Fuller [8]. In particular, from Fuller's result it follows that if g $ 1 and f : M g ! M g is a homeomorphism then Perð f Þ > {1; 2; . . . ; 2g} -B; for more detail see [7].
For homeomorphisms there are results improving Theorem 1 without boundary components by Wang [28], and with boundary components by Chas [5]. In fact, the proof of Theorem 1 uses ideas of [28].
The technique of using Lefschetz numbers to obtain information about the periods of a map is also used in many other papers, see for instance the book of Jezierski and Marzantowicz [17], the article of Gierzkiewicz and Wójcik [9] and the references cited in both.
Given topological spaces X and Y with chosen points x 0 [ X and y 0 [ Y, then the wedge sum X _ Y is the quotient of the disjoint union X and Y obtained by identifying x 0 and y 0 to a single point (for more detail, see, page 10 of [15]). The wedge sum is also known as 'one point union'. For example, S 1 _ S 1 is homeomorphic to the figure '8', two circles touching at a point. Some graphs can be obtained as particular cases of wedge sums of S 1 , and a compact connected graph X such that dim ðH 1 ðX; QÞÞ ¼ r is homotopic to S 1 _ · · · r2times _ S 1 .
£ S n and f : X ! X be a continuous map. If X ¼ S 1 £ · · · £ S 1 the previous result is well known, see [3,13]. The proof of Theorem 3 in the case n odd is similar to the case n ¼ 1.
In [11] the case k ¼ 2 is considered. Necessary and sufficient conditions are given for a self-continuous map to be Lefschetz periodic free point. However those conditions are stated in a different manner; they are special cases of the present theorem.
In Proposition 4 we consider continuous self-maps on S n £ S m , with nm. We provide conditions in order that they are Lefschetz (partially) periodic point free. The results presented here are generalizations of results contained in [11]. We also consider continuous self-maps on CP n , HP n and OP n , the n-dimensional projective plane over the complex numbers, the quaternions and the octonions. We also provide conditions in order that they are Lefschetz (partially) periodic point free. We present them as examples of the techniques expounded in this article. Proposition 4. Let X ¼ S n £ S m and f : X ! X be a continuous map with induced maps on homology f *n ¼ a, f *m ¼ b and f *nþm ¼ d. For more information on the complex projective spaces CP n or the quaternion projective spaces HP n see for instance [27], and for the octonionic projective spaces OP n , see [1].

Proof of theorems and propositions
We separate the proof for the different statements of Theorem 1.
Proof of statement (a) of Theorem 1. Let X be a connected compact graph. Since the continuous map f : X ! X does not have periodic points up to period n, by the Lefschetz fixed point theorem we have Due to the fact that X is connected we know that f *0 ¼ ð1Þ, i.e. f *0 is the identity of H 0 ðX; QÞ ¼ Q, for more detail see [24,27]. From the definition of the Lefschetz number and if a j ¼ traceðf j *1 Þ, we have Therefore, a j ¼ 1 for 1 # j # n. Let r ¼ dim H 1 ðX; QÞ and l 1 ; . . . ; l r be the eigenvalues of f *1 , so The characteristic polynomial of f *1 is Due to Newton's formulae for symmetric polynomials (see for instance [26]), we have ð21Þ i a i a r2i þ ð21Þ r ra r ¼ 0: ð3Þ So we get a 1 ¼ a 1 ¼ 1, a 2 ¼ 0, and by induction a j ¼ 0 for 2 # j # n. Since a r ¼ detðf *1 Þ -0, if n $ 2 then n , r, and statement (a) is proved.  [24,27]). In the next computations we must take From the definition of the Lefschetz number and if a j ¼ traceðf j *1 Þ, we have Using the Newton's formulae (3) we get a 1 ¼ 1 þ d, a 2 ¼ d, a 3 ¼ 0, and by induction a j ¼ 0 for 3 # j # n. Note that if b . 0 then a j ¼ 0 for 2 # j # n. Since a n 1 ¼ detðf *1 Þ -0, if n $ 3 then n , n 1 , and statement (b) is proved. where n 0 ¼ 1, n 1 ¼ g þ b 2 1 and n 2 ¼ 0; and the induced linear map f *0 ¼ ð1Þ (see again for additional details [24,27]). From the definition of the Lefschetz number and if a j ¼ traceðf j *1 Þ, we have Therefore, a j ¼ 1 for 1 # j # n. Now the characteristic polynomial of f *1 is (2) with r ¼ n 1 . Using the Newton's formulae (3) we get a 1 ¼ 1, a 2 ¼ 0, and by induction a j ¼ 0 for 2 # j # n. Since a n 1 ¼ detðf *1 Þ -0, if n $ 2 then n , n 1 , and statement (c) is proved. A Proof of Theorem 2. Let X ¼ S 2m _ S m _ · · · s2times _ S m . Using the properties of the wedge sum (see page 160 of [15]), the homology spaces of X with coefficients in Q are as follows: where n 0 ¼ n 2m ¼ 1, n m ¼ s and 0 otherwise. So the non-trivial induced linear maps are f *0 , f *m and f *2m , where f *0 ¼ 1 and f *2m ¼ ðdÞ, the degree of f. We adapt here the argument used in the proof of Theorem 1. Using the fact that m is odd, the Lefschetz numbers of the iterates of f are as follows: where a j ¼ traceðf j *m Þ. If f is partially periodic point free up to period n. Then Lðf j Þ ¼ 0 for 1 # j # n. Now the characteristic polynomial of f *m is (2) with r ¼ n m ¼ s. Using the Newton's formulae (3) we get a 1 ¼ 1 þ d, a 2 ¼ d and a j ¼ 0 for 3 # j # n. Since a s ¼ detðf *m Þ -0 then n , n m ¼ s. This completes the proof of statement (a).
If m is even, then Lðf j Þ ¼ 1 þ a j þ d j . Assuming Lðf j Þ ¼ 0 for 0 # j # n, using the Newton's formulae (3) and induction we get for 1 # j # n. If d ¼ 21 and j odd, then a j ¼ 0. So, if s is odd and s # n, then a s ¼ 0. This contradicts the hypothesis of detðf *m Þ -0. Therefore n , s. This completes the proof of statement (b). A Proof of Theorem 3. Let X ¼ S n £ · · · k2times £ S n . According to the Künneth theorem, the homology groups of X over Q are is an exterior algebra over C with k generators of dimension 1, for more detail see [27] (p. 203). Since H jn ðX; CÞ is torsion free for 1 # j # k, we compute the Lefschetz numbers over {H jn ðX; CÞ} k j¼0 . Let A ¼ ða i;j Þ 1#i;j#k be a complex matrix that represents f *n , the induced map on the cohomology group H n ðX; CÞ, since C is algebraically closed we can suppose that A is an upper (or lower) triangular matrix. Since f *n is the dual of f *n , the trace of A, or f *n , is P k i¼1 a ii . By the exterior algebra structure the trace of f *2n is P i,j a ii a jj and the trace f *jn is X i 1 ,· · ·,i j a i 1 i 1 · · ·a i j i j : So, using the fact that n is odd, the Lefschetz number of f is a ii a jj þ · · · þ ð21Þ nk a 11 a 22 · · ·a kk ¼ ð1 2 a 11 Þð1 2 a 22 Þ· · ·ð1 2 a kk Þ ¼ detðId 2 AÞ: Similarly the trace of f l *jn is X i 1 ,· · ·,i j a l i 1 i 1 · · ·a l i j i j : Hence f is Lefschetz periodic point free if and only if 1 is an eigenvalue of f *n . This completes the proof of statement (a). If n is even Lð f Þ ¼ traceðf *0 Þ þ ð21Þ n traceðf *n Þ þ · · · þ ð21Þ nk traceðf *kn Þ a ii a jj þ · · · þ a 11 a 22 · · ·a kk ¼ ð1 þ a 11 Þð1 þ a 22 Þ· · ·ð1 þ a kk Þ ¼ detðId þ AÞ: is an eigenvalue of A l for 0 # l # k; then ð21Þ 1=l is an eigenvalue of A. We consider A s , with s ¼ 2k!, then its eigenvalues are of the form ð21Þ 2k!=l ¼ ð21Þ 2kðk21Þ· · ·ðlþ1Þðl21Þ· · ·2 ¼ 1: Hence 1 is the only eigenvalue of A s . So Lðf s Þ -0, therefore f is not Lefschetz periodic point free.
If f is Lefschetz periodic point free up to order m, with m $ 2 then 21 and ffiffiffiffiffiffi ffi 21 p are roots of the characteristic polynomial of A, so its degree should be greater than or equal to 3. Since the degree of the characteristic polynomial of the matrix A is k, then k $ 3, if m $ 2. for all l $ 1. If n and m are even we get that Lðf 2 Þ ¼ 1 þ a 2 þ b 2 þ d 2 , so Lðf 2 Þ -0. So f is not Lefschetz periodic point free up to period 2 and it is Lefschetz fixed point free if 1 þ a þ b þ d ¼ 0. This completes the proof of statement (a).
If n is even and m odd then the solution of the linear system Lð f Þ ¼ Lðf 2 Þ ¼ 0 is a ¼ b and d ¼ 1, or a ¼ d and b ¼ 1, so this implies that Lðf l Þ ¼ 0 for l . 2. Similarly for the other cases when n and m are not simultaneously even. Therefore f is Lefschetz periodic point free. This completes the proof of statement (b). A Example 1. The homology groups of CP n over Q are H 2l ðCP n ; QÞ ¼ Q, if 0 # l # n and are trivial otherwise. So the induced maps on homology are f *2l ¼ a l if 0 # l # n, with a [ Z, and f *l ¼ 0 if l is odd. Therefore f m *2l ¼ ða l Þ m for 0 # l # n. Hence the Lefschetz numbers of the iterates of f are: Lð f Þ ¼ 1 þ a þ · · · þ a n ; . . .
Lðf m Þ ¼ 1 þ a m þ · · · þ a mn : If jaj -1 then Lðf m Þ ¼ ð1 2 a mðnþ1Þ Þ=ð1 2 a m Þ -0 for all m, so f m has fixed points. If a ¼ 1 is clear that Lðf m Þ -0, for all m. If a ¼ 21 the result depends on n being odd or even, if n is even then Lðf m Þ -0, for all m. If a ¼ 21 and n odd then Lð f Þ ¼ 0 and Lðf 2 Þ -0, so f is Lefschetz fixed point free.
Example 2. The homology groups of HP n over Q are H 4l ðHP n ; QÞ ¼ Q, if 0 # l # n and trivial otherwise. So the induced maps on homology are f *4l ¼ a l if 0 # l # n, with a [ Z, and f *l ¼ 0 if l is not a multiple of 4. On the other hand the homology groups of OP n over Q are H 8l ðOP n ; QÞ ¼ Q, if 0 # l # n and otherwise it is trivial. So the induced maps on homology are f *8l ¼ a l if 0 # l # n, with a [ Z, and f *l ¼ 0 otherwise. Now the computation follows in a similar manner as in the case of CP n .