This is a preprint of: “Periods of the Morse-Smale diffeomorphisms on S n and S m ×S n and HP n ”, Clara Cufı́-Cabré, Jaume Llibre, J. Fixed Point Theory Appl., vol. 24(4), 1–12, 2022. DOI: [10.1007/s11784-021-00918-5] PERIODS OF MORSE–SMALE DIFFEOMORPHISMS ON Sn , Sm × Sn , CPn AND HPn . CLARA CUFÍ-CABRÉ AND JAUME LLIBRE Abstract. We study the set of periods of the Morse–Smale diffeomor- phisms on the n-dimensional sphere Sn , on products of two spheres of arbitrary dimension Sm × Sn with m 6= n, on the n-dimensional complex projective space CPn and on the n-dimensional quaternion projective space HPn . We classify the minimal sets of Lefschetz periods for such Morse–Smale diffeomorphisms. This characterization is done using the induced maps on the homology. The main tool used is the Lefschetz zeta function. 1. Introduction Understanding the periodic orbits and the set of periods of a map is a very important problem in dynamical systems. The Lefschetz numbers are one of the most useful tools to study the existence of fixed points and periodic orbits of self-maps on compact manifolds. In this paper we obtain information on the set of periods of certain diffeomorphisms on compact manifolds using the Lefschetz zeta function, which is a generating function of the Lefschetz numbers of the iterates of a map. Let M be a compact manifold, let f : M → M be a continuous map, and denote by f m the m−th iterate of f . A point x ∈ M such that f (x) = x is called a fixed point, or a periodic point of period 1 of f . A point x ∈ M is called periodic of period k > 1 if f k (x) = x and f m (x) 6= x for all m = 1, . . . , k − 1, and the set formed by the iterates of x, i.e. {x, f (x), . . . , f k−1 (x)}, is called the periodic orbit of the periodic point x. As usual N denotes the set of all positive integers. Then Per(f ) is the set {k ∈ N : f has a periodic orbit of period k}. A fixed point x of a C 1 map f is called hyperbolic if all the eigenvalues of Df (x) have modulus different than one. A periodic point x of f of period k is called a hyperbolic periodic point if it is a hyperbolic fixed point of f k . 2010 Mathematics Subject Classification. 37C05, 37C25, 58B05. Key words and phrases. Morse–Smale diffeomorphisms, periodic orbit, Lefschetz zeta func- tion, minimal Lefschetz period. 1 2 C. CUFÍ-CABRÉ AND J. LLIBRE We denote by Diff(M ) the space of all C 1 diffeomorphisms on a compact Riemannian manifold M . We say that two maps f, g ∈ Diff(M ) are topo- logically equivalent if there exists a homeomorphism h : M → M such that h ◦ f = g ◦ h. A map f ∈ Diff(M ) is called structurally stable if there exists a neighbourhood U ⊂ Diff(M ) of f such that every g ∈ U is topologically equivalent to f . A map f ∈ Diff(M ) is quasi unipotent on homology if all the eigenvalues of the nontrivial induced maps on the homology groups of M with rational coefficients, are roots of unity. We denote by F(M ) the set of elements of Diff(M ) being quasi unipotent on homology and having finitely many periodic points, all of them hyperbolic. Among the maps in F(M ) there are the Morse-Smale diffeomorphisms (see [7, 21]). Those are very important from the dynamical point of view because the family of the Morse–Smale diffeomorphisms is structurally stable inside the class of all diffeomorphisms (see for details [7, 19, 20, 23]). Even if we will refer to Morse–Smale diffeomorphisms along the paper because of their dynamical interest, all the results hold for any map of the class F(M ). In order to define the Morse-Smale diffeomorphisms we introduce the following concepts. We say that x ∈ M is a nonwandering point of f if for any neighborhood U of x there is a positive integer m such that f m (U ) ∩ U 6= ∅. We denote by Ω(f ) the set of nonwandering points of f . Clearly if γ is a periodic orbit of f , then γ ⊆ Ω(f ). Let d be the metric on M and suppose that x ∈ M is a hyperbolic fixed point of f . We define the stable manifold of x as the set W s (x) = {y ∈ M : d(x, f m (y)) → 0 as m → ∞}, and the unstable manifold of x as the set W u (x) = {y ∈ M : d(x, f −m (y)) → 0 as m → ∞}. In the same way we define the stable and unstable manifolds of a hyper- bolic periodic point x ∈ M of period k as the stable and unstable manifolds of the hyperbolic fixed point x under f k , respectively. We say that the sub- manifolds W s (x) and W u (x) have a transversal intersection if at every point of intersection, their separate tangent spaces at that point together generate the tangent space of the ambient manifold M at that point. A diffeomorphism f : M → M is Morse–Smale if (1) Ω(f ) is finite, (2) all the periodic points of f are hyperbolic, and (3) for every x, y ∈ Ω(f ), W s (x) and W u (y) have a transversal inter- section. PERIODS OF MORSE–SMALE DIFFEOMORPHISMS 3 Clearly, condition (1) implies that Ω(f ) is the set of all periodic points of f. In the last quarter of the 20th century there appeared some papers ded- icated to understand the connections between the dynamics of the Morse– Smale diffeomorphisms and the topology of the manifold where they are defined. Without trying to be exhaustive, see for instance [5, 18, 21, 22, 23]. This interest continues during this first part of the 21st century, see for example [2, 3, 9, 10, 15, 16, 17]. In this paper we study the set of periods of Morse–Smale diffeomorphisms on the n−dimensional sphere Sn , on products of two spheres of arbitrary dimension Sm × Sn with m 6= n, and on the projective spaces CPn and HPn . More precisely, our goal is to describe the set MPerL (f ) (see Definition 4) that those diffeomorphisms can exhibit for arbitrary values of n and m. The set of periods for Morse–Smale diffeomorphisms on a product of any number of spheres of the same dimension has been studied in [2]. For the par- ticular case of Sn , we give more detailed results considering the orientation of the diffeomorphisms. The set of periods for Morse–Smale diffeomorphims on the two-dimensional sphere has been studied with more details in [1, 11]. In [8] the authors study the Lefschetz periodic point free self-maps on Sm × Sn , CPn and HPn . Our results give extended information in the same line for the Morse–Smale diffeomorphisms. The main results of this paper are Theorems 8, 9 and 10, where we char- acterize the possible sets for MPerL (f ) in function of the action of f on the homology and on the parity of the numbers n and m. Related with these results the reader can look at the set of periods for homeomorphisms (respectively continuous maps) on Sn and on Sm × Sn which have been studied in [13] (respectively [14]). We also remark that the results obtained along this paper hold in any compact manifold with the same homology as the manifolds considered here. More precisely, they hold for any manifold homotopy equivalent to Sn , Sm × Sn , CPn and HPn , respectively. 2. Lefschetz numbers and the Lefschetz zeta function Let f : M → M be a continuous map on a compact manifold of dimension n. We denote by H0 (M, Q), . . . , Hn (M, Q) the homology groups of M with rational coefficients. A continuous map f : M → M induces n+1 morphisms on the homology groups of M , f∗i : Hi (M, Q) → Hi (M, Q), i ∈ {0, . . . , n}. For more details, see [24]. 4 C. CUFÍ-CABRÉ AND J. LLIBRE Definition 1. Let M be a compact manifold and let f : M → M be a continuous map. The Lefschetz number of f is defined as n X L(f ) = (−1)i tr(f∗i ), i=0 where tr(f∗i ) denotes the trace of f∗i . A very important result which relates the Lefschetz number of a map f and the existence of fixed points of f is the Lefschetz Fixed Point Theorem, which says that if L(f ) 6= 0, then f has a fixed point. For a proof, see [4]. In order to obtain information about the set of periods of a map f it is useful to have information of the sequence {L(f m )}∞ m=0 of the Lefschetz numbers of all the iterates of f . Definition 2. Let f : M → M be a continuous map on a compact manifold. The Lefschetz zeta function of f is defined as X  ∞ L(f m ) m Zf (t) = exp t . m=1 m This function generates the whole sequence of Lefschetz numbers of f , and it may be independently computed through n Y (1) Zf (t) = det (Ink − t f∗k )(−1) k+1 , k=0 (see for more details [6]) where n is the dimension of M , nk is the dimension of Hk (M, Q), Ink is the nk × nk identity matrix, and we take det (Ink − t f∗k ) = 1 if nk = 0. Note that the expression given in (1) is a rational function of t. For a C 1 map f having a finite number of periodic points, all of them being hyperbolic, we give another characterization of the Lefschetz zeta function introduced by Franks in [6]. Let f : M → M be a C 1 map on a compact manifold without boundary, and let γ be a hyperbolic periodic orbit of f of period p. For each x ∈ γ, let Exu denote the linear subspace of the tangent space Tx M of M at x, generated by the eigenvectors of Df p corresponding to the eigenvalues whose moduli are greater than 1, and let Exs denote the linear subspace of Tx M generated by the remaining eigenvectors. We denote by u and s the dimensions of the spaces Exu and Exs , respectively. We define the orientation type ∆ of γ to be +1 if Df p (x) : Exu → Exu preserves orientation, that is, if det Df p (x) > 0 with x ∈ γ, and to be −1 if it reverses orientation, that is, if det Df p (x) < 0. Note that the definitions of ∆ and u do not depend on the periodic point x, but only on the periodic orbit γ. PERIODS OF MORSE–SMALE DIFFEOMORPHISMS 5 For a C 1 map f : M → M having only finitely many periodic orbits, all of them hyperbolic, we define the periodic data, Σ, as the collection composed by all triples (p, u, ∆) corresponding to all the hyperbolic periodic orbits of f , where a same triple can occur more than once provided that it corresponds to different periodic orbits. The following result was proved by Franks in [6]. Theorem 3. Let f : M → M be a C 1 map on a compact manifold without boundary, having finitely many periodic points, all hyperbolic, and let Σ be the periodic data of f . Then the Lefschetz zeta function of f satisfies Y Zf (t) = (1 − ∆ tp )(−1) u+1 . (p, u, ∆)∈Σ Note that the Morse–Smale diffeomorphisms satisfy the hypotheses of Theorem 3. Then the result will be useful to obtain information on the set of periods for Morse-Smale diffeomorphisms from the comparsion of the expressions of the Lefschetz zeta functions in Theorem 3 and in (1). 3. Minimal set of Lefschetz periods of Morse–Smale diffeomorphisms Definition 4. Let f be a map satisfying the hypotheses of Theorem 3. The minimal set of Lefschetz periods of f , MPerL (f ), is the set given by the intersection of all sets of periods forced by the different representations of Zf (t) as products of the form (1 ± tp )±1 . As an example, consider the following Lefschetz zeta function of a Morse– Smale diffeomorphism f on the four-dimensional torus T4 , (1 − t3 )2 (1 + t3 ) (1 − t3 )(1 − t6 ) (1 − t3 )(1 − t6 ) Zf (t) = = = (1 − t)6 (1 + t)3 (1 − t)6 (1 + t)3 (1 − t)3 (1 − t2 )3 (2) (1 − t3 )2 (1 + t3 ) = , (1 − t)3 (1 − t2 )3 see for instance [12], which can be expressed in four ways as a product of factors of the form (1 ± tp )±1 as a quotient of polynomials of degree 9. By Theorem 3, the first expression of the Lefschetz zeta function (2) ensures the existence of periodic orbits of periods 1 and 3 for f . In the same way, the second expression of Zf (t) provides the periods {1, 3, 6} for f , the third expression of Zf (t) provides the periods {1, 2, 3, 6}, and finally the fourth expression provides the periods {1, 2, 3}. In this case we have that the minimal set of Lefschetz periods of f is MPerL (f ) = {1, 3} ∩ {1, 3, 6} ∩ {1, 2, 3, 6} ∩ {1, 2, 3} = {1, 3}. Note that if Zf (t) is constant equal to 1, then MPerL (f ) = ∅. 6 C. CUFÍ-CABRÉ AND J. LLIBRE Remark 5. Even if the minimal set of Lefschetz periods of a Morse–Smale diffeomorphism is empty, one can still obtain some information on the set of periods from Theorem 3. For example, suppose that the Lefschetz zeta function of a map f satisfying the hypotheses of Theorem 3 is Zf (t) = 1+t2 . It can be expressed as products of terms of the form (1 ± tp )±1 in infinitely many ways, 1 − t4 1 − t4 1 − t8 1 + t2 = = = , 1 − t2 (1 + t)(1 − t) (1 + t)(1 − t)(1 + t4 ) and so on. In this case we have clearly MPerL (f ) = ∅, but each of the infinitely many expressions of Zf (t) forces either the period 2 or the periods {2, 4}, or the periods {1, 4}. Then, f has either a periodic orbit of period 2, or periodic orbits of periods 2 and 4, or periodic orbits of periods 1 and 4. Remark 6. By Theorem 3 an even period n, can never be contained in the set MPerL (f ). Indeed, the following expressions 1 − t2n 1 − t2n 1 − tn = (1 + tn/2 )(1 − tn/2 ), 1 + tn = = , 1−t n (1 + tn/2 )(1 − tn/2 ) show that if the term 1 − tn or 1 + tn , with n even, appears in one of the expressions of Zf (t), one can always obtain a new expression of Zf (t) where the period n does not appear. Remark 7. Along the paper, for every possible Lefschetz zeta function of a given map f , in general we will write only one of the possible equivalent expressions of Zf (t). We will provide the expression of Zf (t) that forces a smaller set of periods, and consequently it will be sufficient to describe the set MPerL (f ). From remark 6 we note that providing an expression of Zf (f ) that forces only even periods is sufficient to ensure that MPerL (f ) = ∅. Finally note that MPerL (f ) is contained in the set of periods that are conserved under homotopy. Indeed, for a Morse–Smale diffeomorphism T f : M → M on a compact manifold consider the set MPerms (f ) := h∼f Per(h), where h runs over all the Morse–Smale diffeomorphisms of M which are ho- motopic to f . Then it is clear that MPerL (f ) ⊆ MPerms (f ), because two homotopic maps on a manifold M induce the same Lefschetz zeta functions. PERIODS OF MORSE–SMALE DIFFEOMORPHISMS 7 4. Periods of Morse–Smale diffeomorphisms on Sn Let n ∈ N and let Sn be the n−dimensional sphere. The homology groups of Sn over Q are ( Q if k ∈ {0, n}, Hk (S , Q) = n 0 otherwise. For a continuous map f : Sn → Sn the nontrivial induced maps on the homology can be written as the integer matrices f∗0 = (1) and f∗n = (d), where d is called the degree of f . Let f : Sn → Sn be a Morse–Smale diffeomorphism. As we already men- tioned before, the linear maps induced on the homology are quasi unipotent, which means that all their eigenvalues are roots of unity. Then we must have either d = 1, or d = −1. Also, the orientation of f is constant on Sn and is determined by the sign of d. Theorem 8. Let f : Sn → Sn be a Morse–Smale diffeomorphism. Then, (a) If n is even and f preserves the orientation, then MPerL (f ) = {1}. (b) If n is even and f reverses the orientation, then MPerL (f ) = ∅ but {1, 2} ∩ Per(f ) 6= ∅. (c) If n is odd and f preserves the orientation, then MPerL (f ) = ∅. (d) If n is odd and f reverses the orientation, then MPerL (f ) = {1}. Proof. Computing the Lefschetz zeta function for f using equation (1), we get n Y Zf (t) = det (Ink − t f∗k )(−1) = (1 − t)−1 (1 − td)(−1) k+1 n+1 , k=0 and so for n even we have 1 1 1 Zf (t) = if d = 1, Zf (t) = = if d = −1, (1 − t)2 (1 − t)(1 + t) 1 − t2 and for n odd we have 1+t Zf (t) ≡ 1 if d = 1, Zf (t) = if d = −1. 1−t It is clear that f satisfies the hypotheses of Theorem 3. Then the state- ments follow directly applying Theorem 3 to each of the expressions obtained for the Lefschetz zeta function Zf (t). Indeed, for 1/(1−t)2 and (1+t)/(1−t) we have MPerL (f ) = {1}, because any other expression of the same Lefschetz zeta function as a product of terms the form (1 ± tp )±1 would provide at least the period 1. 8 C. CUFÍ-CABRÉ AND J. LLIBRE For the function 1/(1−t2 ) = 1/((1−t)(1+t)) it is clear that MPerL (f ) = ∅, but by Remarks 5 and 6, we can ensure that {1, 2} ∩ Per(f ) 6= ∅. Finally we have MPerL (f ) = ∅ when Zf (t) ≡ 1.  5. Periods of Morse–Smale diffeomorphisms on Sm × Sn Let m, n ∈ N, m 6= n, and consider the product of spheres Sm × Sn , with m 6= n. Applying Künneth’s formula, the homology groups over Q of Sm × Sn can be easily computed and are given by ( Q if k ∈ {0, m, n, m + n}, Hk (S × S , Q) = m n 0 otherwise. Then for a continuous map f : Sm ×Sn → Sm ×Sn the linear maps induced on the homology are given as the integer matrices f∗0 = (1), f∗m = (a), f∗n = (b), f∗m+n = (d), where d is the degree of f . The rest of the induced maps are the zero map. Let f : Sm × Sn → Sm × Sn be a Morse–Smale diffeomorphism. Since the linear maps induced by f on the homology are quasi unipotent, in this case we have a, b, d ∈ {−1, 1}. Moreover, by the structure of the cohomology ring of Sm × Sn , one has always that ab = d (see [8]) and hence the possibil- ities for these values are restricted to a = b = d = 1, {a, b} = {−1, 1} and d = −1, and a = b = −1, d = 1. Theorem 9. Let M = Sm × Sn be with m 6= n, and let f : M → M be a Morse–Smale diffeomorphism. (i) If m and n are even, then (a) If a = b = 1, then MPerL (f ) = {1}. (b) Otherwise, MPerL (f ) = ∅ but {1, 2} ∩ Per(f ) 6= ∅. (ii) If m and n are odd, then (a) If a = b = −1, then MPerL (f ) = {1}. (b) Otherwise, MPerL (f ) = ∅. (iii) If m is even and n is odd, then (a) If b = −1 and a = 1, then MPerL (f ) = {1}. (b) Otherwise, MPerL (f ) = ∅. (iv) If m is odd and n is even, then (a) If a = −1 and b = 1, then MPerL (f ) = {1}. (b) Otherwise, MPerL (f ) = ∅. Proof. Computing the Lefschetz zeta function for f using equation (1) we obtain m+n Y Zf (t) = det (Ink − t f∗k )(−1) k+1 k=0 = (1 − t)−1 (1 − a t)(−1) (1 − b t)(−1) (1 − d t)(−1) m+1 n+1 m+n+1 . PERIODS OF MORSE–SMALE DIFFEOMORPHISMS 9 Depending on whether m and n are even or odd, and for each allowed value of a, b, d ∈ {−1, 1} with ab = d, the expressions obtained for the Lefschetz zeta functions in each case are displayed in Tables 1, 2, 3 and 4. The proof follows directly from the information obtained from the men- tioned tables and applying Theorem 3, taking into account the considera- tions of Remarks 5, 6 and 7.  Values for a, b Zf (t) 1 a=b=1 (1−t)4 1 1 {a, b} = {−1, 1} (1−t)2 (1+t)2 = (1−t1 2 )2 Table 1. Zf (t) for a quasi-unipotent diffeomorphism f on Sm ×Sn , with m 6= n and m, n even. Values for a, b Zf (t) a=b=1 1 (1+t)2 a = b = −1 (1−t)2 {a, b} = {−1, 1} 1 Table 2. Zf (t) for a quasi-unipotent diffeomorphism f on Sm ×Sn , with m 6= n and m, n odd. Values for a, b, d Zf (t) a=b=1 1 (1+t)2 b = −1, a = 1 (1−t)2 {b, d} = {−1, 1} 1 Table 3. Zf (t) for a quasi-unipotent diffeomorphism f on Sm ×Sn , with m 6= n and m even, n odd. Values for a, b, d Zf (t) a=b=1 1 (1+t)2 a = −1, b = 1 (1−t)2 {a, d} = {−1, 1} 1 Table 4. Zf (t) for a quasi-unipotent diffeomorphism on Sm × Sn , with m 6= n and m odd, n even. 10 C. CUFÍ-CABRÉ AND J. LLIBRE 6. Periods of Morse–Smale diffeomorphisms on CPn and HPn Let n ∈ N, n ≥ 1 and let CPn be the n-dimensional complex projective space and let HPn be the n-dimensional quaternion projective space. Along this section we consider these manifolds as real manifolds of dimension 2n and 4n, respectively. The homology groups of CPn over Q can be easily computed applying Künneth’s formula and are given by ( Q if k ∈ {0, 2, . . . , 2n}, Hk (CP , Q) = n 0 otherwise. For a continuous map f : CPn → CPn , the induced linear maps on the homology can be written as integer matrices as f∗k = (dk/2 ) for k ∈ {0, 2, 4, . . . , 2n}, with d ∈ Z, and f∗k = (0) otherwise (see [24]). Similarly the homology groups of HPn over Q are given by ( Q if k ∈ {0, 4, . . . , 4n}, Hk (HP , Q) = n 0 otherwise, and for a continuous map f : HPn → HPn , the induced linear maps on the homology can be written as f∗k = (dk/4 ) for k ∈ {0, 4, 8, . . . , 4n}, with d ∈ Z, and f∗k = (0) otherwise (see [24]). Let f : CPn → CPn (respectively, f : HPn → HPn ) be a Morse-Smale diffeomorphism. Since the linear maps induced on the homology are quasi unipotent, we will have either d = 1 or d = −1, which determines also whether f preserves or reverses the orientation. Theorem 10. Let f : CPn → CPn (respectively, f : HPn → HPn ) be a Morse–Smale diffeomorphism. Then, (a) If n is odd and f reverses the orientation, then MPerL (f ) = ∅, but {1, 2} ∩ Per(f ) 6= ∅. (b) Otherwise, MPerL (f ) = 1. Proof. We develop the proof for CPn , being the proof for HPn completely analoguous. We start with the case d = 1. Computing the Lefschetz zeta function for f using (1), we get 2n Y 1 Zf (t) = det (Ink − t f∗k )(−1) = k+1 , k=0 (1 − t)n+1 since we have f∗k = (1) for each k ∈ {0, 2, . . . , 2n}. We consider next the case d = −1. As before, we have n + 1 nontrivial induced maps, with f∗k = (1) if k/2 is even and f∗k = (−1) if k/2 is odd. PERIODS OF MORSE–SMALE DIFFEOMORPHISMS 11 For n even, computing the Lefschetz zeta function for f using equation (1), we get 2n Y 1 1 1 1 Zf (t) = det (Ink −t f∗k )(−1) = n = k+1 , (n (1 − t) 2 +1) (1 + t) 2 (1 − t ) 2 (1 − t) 2 n k=0 and for n odd we have 2n Y 1 1 1 Zf (t) = det (Ink − t f∗k )(−1) = = k+1 . (1 − t) (1 + t) (1 − t2 ) n+1 n+1 n+1 2 2 2 k=0 It is clear that f satisfies the hypotheses of Theorem 3. Then the re- sults follow directly applying Theorem 3 to each of the expressions ob- tained for the Lefschetz zeta function Zf (t). For Zf (t) = (1−t)1 n+1 and 1 1 Zf (t) = we have MPerL (f ) = {1}, since any other expression (1−t2 ) 2 (1−t) n of the same Lefschetz zeta function as a product of terms the form (1 ± tp )±1 would provide at least the period 1. 1 1 1 For Zf (t) = n+1 n+1 = n+1 it is clear that MPerL (f ) = ∅, (1−t) 2 (1+t) 2 (1−t2 ) 2 but taking into account the considerations of Remarks 5 and 6, one has {1, 2} ∩ Per(f ) 6= ∅.  Remark 11. A map f is called Lefschetz periodic point free if L(f m ) = 0, for all m ∈ N. In [8] is claimed that there are no Lefschetz periodic point free maps on CPn and HPn , that is, all self maps of CPn and HPn have periodic points. Here we see that in particular Morse–Smale diffeomorphisms in CPn and HPn always have fixed points, unless when n is odd and d = −1, where in this case there are always fixed points or periodic points of period 2. 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An introduction to algebraic topology, Second edition, Graduate Texts in Mathematics 145, Springer-Verlag, New York, 1994. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: clara@mat.uab.cat, jllibre@mat.uab.cat