Periodic expansion in determining minimal sets of Lefschetz periods for Morse–Smale diffeomorphisms

We apply the representation of Lefschetz numbers of iterates in the form of periodic expansion to determine the minimal sets of Lefschetz periods of Morse–Smale diffeomorphisms. Applying this approach we present an algorithmic method of finding the family of minimal sets of Lefschetz periods for Ng\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_g$$\end{document}, a non-orientable compact surfaces without boundary of genus g. We also partially confirm the conjecture of Llibre and Sirvent (J Diff Equ Appl 19(3):402–417, 2013) proving that there are no algebraic obstacles in realizing any set of odd natural numbers as the minimal set of Lefschetz periods on Ng\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_g$$\end{document} for any g.


Introduction
Let f : M → M be a Morse-Smale diffeomorphism, where M is a compact manifold without boundary. Morse-Smale diffeomorphisms, structurally stable and having relatively simple dynamics, constitute an important subclass of diffeomorphisms that were carefully studied during past decades (cf. [17] and the references therein).
One of the problems studied for Morse-Smale diffeomorphisms is the structure of the set of its minimal periods. The promising results in this direction may be obtained by the comparison (via Lefschetz-Hopf theorem) of the global behavior of f expressed by Lefschetz numbers of iterates (L(f n ) n ) and local properties of f near periodic points represented by local fixed point indices of iterates. Basing on this relation MPer L (f ) called minimal set of Lefschetz periods is considered (cf. Definition 3.3). MPer L (f ) provides the information about the set of periodic points of f as it is the subset of minimal

Lefschetz numbers of iterates and Lefschetz zeta function
First we remind the definition of Lefschetz numbers of iterates; for simplicity we will consider homology with rational coefficients.
Let K be a CW-complex of dimension m with the homology groups H i (K; Q), where i = 0, 1, . . . , m. In case the homology coefficients are equal to Q the groups H i (K; Q) are finite dimensional linear spaces over Q. For a self-map f of K we denote by f * i the linear map induced by f on H i (K; Q) and by f * the self-map where tr(f n ) * i is the trace of the integer matrix representing (f n ) * i :H i (K; Q) → H i (K; Q). Notice that if A is a matrix of f * i , then A n is a matrix of (f n ) * i , representing the homomorphism induced on H i (K; Q) by f n (cf. [25]). Lefschetz zeta function, Z f , which is a useful tool in periodic points theory, codes information on the whole sequence of Lefschetz numbers of iterates: We will also consider zeta function associated with a given integer sequence (a n ) n , which will be defined in an analogous way: Z an (t) = exp ∞ n=1 a n n t n . (3.3) Remark 3.1. An alternative formula for the Lefschetz zeta function may be given by a use of eigenvalues of f * , namely if λ i is an eigenvalue of f * , taken with algebraic multiplicities k i , then where m i denotes the index of the homology group associated with λ i [25].
It turns out that the Lefschetz zeta function for Morse-Smale diffeomorphisms has very special form, first described by Franks in [9].
Let M be a smooth manifold and x be a hyperbolic p-periodic point of a map f : M → M . Denote by E un x the subspace of the tangent T M x spanned by eigenvectors of Df p (x) which correspond to eigenvalues which are greater than one in absolute value. Let γ be an orbit of x, we define u = dim E un x and Δ, the orientation type of γ, as +1 if Df p (x) preserves the orientation and −1 if it reverses the orientation.
By Σ we denote periodic data, i.e. a collection of all the triples (p, u, Δ) which corresponds to periodic orbits of f .

Theorem 3.2. [9]
Let M be a closed manifold, and f : M → M be a C 1 map with finitely many periodic points, all of them hyperbolic; then where (p, u, Δ) belongs to periodic data of f . Definition 3.3. Let Z f (t) = 1, the minimal set of Lefschetz periods of f is defined as where the intersection is considered over all the possible expressions of Z f (t) in the following form: The importance of the minimal sets of Lefschetz periods for Morse-Smale diffeomorphisms results from the following fact, which is a straightforward consequence of Theorem 3.2 MPer L (f ) ⊂ MPer(f ). (3.7) Now, we introduce the notion of quasi-unipotent maps. The following fact makes it possible to determine Lefschetz zeta function for a Morse-Smale diffeomorphism in a relatively easy way. Proposition 3.5 [32]. Let f be a Morse-Smale diffeomorphism of a compact manifold, then f is quasi-unipotent.
Due to Proposition 3.5 and the formula (3.4) Lefschetz zeta function for a Morse-Smale diffeomorphism may be expressed as a rational function with the nominator and denominator being a product of cyclotomic polynomials, whose degrees are bounded by the dimensions of homology spaces. As for a given n there is a finite number of cyclotomic polynomials of degree ≤ n, for a given manifold M there is a finite number of different forms of zeta functions on M . This observation was a base for a strategy of finding minimal set of Lefschetz periods for all Morse-Smale diffeomorphisms on a given manifold M , which may be described in the following steps. For a given Morse-Smale diffeomorphism f find zeta functions Z f (t) (expressed in terms of cyclotomic polynomials), next determine all their decompositions into the products of elements of the form (3.6), and finally take the common part of coefficients r i (which are related to minimal periods of f ) over all such products. We proposed below the alternative strategy which is based on representing (L(f n )) n as the sum of basic periodic sequences and decomposing it into sum of sequences related to periodic orbits of the considered map.

Periodic expansion of Lefschetz numbers of iterates
Definition 4.1. A sequence of integer numbers (a n ) ∞ n=1 will be called a Dold sequence if the following congruences (called Dold congruences or Dold relations) are fulfilled: k|n μ(k)a n k ≡ 0 (mod n) for each n ≥ 1, (4.1) where μ : N → Z is the classical Möbius function, given by the following formula: Dold sequences play important role not only in dynamics but also in number theory (cf. [2,16]). There is a convenient way of writing down a Dold sequence by using so-called periodic expansion, i.e. representing the sequence as a combination of some basic periodic sequences. Let k be a fixed natural number. We define Thus, reg k is the periodic sequence: where the non-zero entries appear for indices divisible by k. can be written uniquely in the following form of a periodic expansion: Moreover, (a n ) n is integral valued and satisfies Dold congruences iff b k ∈ Z for every k ∈ N. There is a deep relation between sequences (a n ) n and (b n ) n that appear in Proposition 4.3, which may be expressed in the language of formal power series and formal products.  [6]). Let (a n ) n , (b n ) n be some complex valued sequences. Then (a n ) n is a Dold sequence with Dold coefficients A sequence of fixed point indices of iterates turned out to be a Dold sequence.
Theorem 4.6 [5]. The sequence of fixed point indices of iterates (ind(f n )) n is a Dold sequence (provided it is well-defined). As a consequence it has a periodic expansion of the form (4.2) with integral coefficients.
The language of periodic expansion is a convenient tool that has been recently used in different contexts such as: study of the fixed point indices of an iterated map [4,24,34] and minimization of the number of periodic points in homotopy class [13,14,30].
In particular the sequence (L(f n )) n is also Dold sequence (cf. [33] for different proofs of this fact). As a consequence, by Remark 4.4, we get where b k are integers. Remark 4.7. The coefficients b k in the periodic expansion of Lefschetz numbers in the formula (4.4) are called Lefschetz numbers for periodic points (denoted also as l(f k )) and play important role in the periodic points theory (cf. [26]). In particular, the determination of the set L = {k ∈ N : b k = 0} provides valuable knowledge about the structure of periodic points obtained by confronting the information carried by L with the local information about periodic points (expressed by fixed point indices at orbits). This comparison could be done either straighforwardly (Lefschetz-Hopf theorem) or by zeta function or equivalently by periodic expansion. During the past 30 years the program of determination of the set L for different types of manifolds (in terms of action of induced maps on homology groups) and its application to various classes of maps has been realized. Among other cases this program was accomplished for: • transversal maps of a compact manifold M such that its rational ho- where J is a subset of N with cardinality 1, 2 or 3 [27], • transversal maps of a compact manifold M such that its rational ho- • C 1 maps of rational exterior spaces and simple rational Hopf spaces [10,12] (see also [11]), and transversal maps of some simple rational Hopf spaces [20], • holomorphic maps of some complex compact manifolds [8].

Periodic expansion of Lefschetz numbers of Morse-Smale diffeomorphisms
For the considered class of maps, i.e. Morse-Smale diffeomorphisms periodic expansion of Lefschetz numbers may be expressed by a use of roots of cyclotomic polynomials.
Definition 4.8. The dth cyclotomic polynomial ω d (z) is defined by the following formula: where U d denotes the set of all primitive dth roots of unity.
Let us remark that ω d (z) is an irreducible polynomial with integer coefficients of degree ϕ(d), where ϕ is the Euler function (i.e. ϕ(d) is the number of positive integers less than or equal to d that are co-prime to d). For example . Now, our aim is to establish the coefficients b k for the periodic expansion of Lefschetz numbers of iterated quasi-unipotent maps.
Let ε 1 , . . . , ε ϕ(d) be the all dth primitive roots of unity. For a given d we define The cyclotomic polynomial We will call an eigenvalue λ = 0 essential provided e(λ) = 0. It is obvious that only essential eigenvalues give the contribution to {L(f n )} ∞ n=1 . Denote by σ es (f ) the set of essential eigenvalues of f . We define Notice that the essential dth primitive roots of unity appear in groups of ϕ(d) elements, contributing e(d) ϕ(d) L d (n) to L(f n ). As a result we get: As a consequence, to find the periodic expansion of as an integral combination of basic sequences reg k : where b d k are integers, d is fixed. The following theorem gives the value of b d k , and thus allows us to determine the periodic expansion of {L d (n)} ∞ n=1 . It was proved in [15] in an elementary but rather long way. Below we will give much simpler proof that is based on Theorem 4.5.
Proof. If a n = r i=1 m i λ n i , where m i are integers and λ i are some complex numbers, then (cf. [25] (3.1.26)): On the other hand, in our case each λ i ∈ U (d), where U (d) denotes the set of all primitive roots of unity of degree d and r = ϕ(d) . As ϕ(d) On the other hand, the following well-known fact holds (cf. for example [7]): Applying the formula (4.3) for a n = L d (n) we get that Comparing the formulas (4.10) and (4.11) we get the equality (4.9).

Periodic expansion of indices of iterates at periodic points for Morse-Smale diffeomorphisms
Let us denote for short the derivative of f at x 0 ∈ Fix(f ) by D = Df (x 0 ) and by σ(D) its spectrum. By σ + we denote the number of real eigenvalues of D greater than 1 and by σ − the number of real eigenvalues of D less than −1, in both cases counting with multiplicity. We consider hyperbolic maps (i.e. maps having only hyperbolic periodic points). For a fixed point x 0 we get [4]: In dependence of parity of the values σ + , σ − and n, we obtain four possibilities:  (A) for a n = l reg k (n), Z an (t) = 1 (1 − t k ) l .
Proof. We will consider each case separately: (A) Let a n = l reg k (n); by Theorem 4.5 we have the following relation between sequences a n and b n : where b n = 1 n s|n μ(s)a n s and we get by the definition of (reg k ) n that b n is equal to 1 for n = k, and zero otherwise, which gives us the desired form of Z an (t). (B) Let a n = l(reg k (n) − reg 2k (n)); using the result from the previous case we have

The minimal set of Lefschetz periods expressed by periodic expansions
By Lefschetz-Hopf formula we may represent the sequence of Lefschetz numbers in the following form: where Orb k (f ) denotes the set of k-orbits of f and each O has the form (5.3).
For j ∈ {1, 2, 3, 4} let us denote by c j ri an integer sequence that has one of the forms of fixed point indices of an r i -orbit in (5.3), i.e. where the intersection is taken over all possible decompositions of (L(f n )) n given by the formula (6.1) into the sum of sequences of the form (6.2). Proof. Each representation of Z f (t) in the form (3.6) (with the factors (1 + Δ i t ri ) mi ) is equivalent by Theorem 5.1 to the unique representation of (L(f n )) n as the sum of the sequences of the form (6.2) (with k = r i representing the minimal period of an orbit).
Theorem 6.2. The following formula holds: where Odd denotes the set of odd natural numbers.
Proof. It is obvious that for odd k for which b k = 0 in (4.4) we get that k ∈ MPer(f ). We show that there are no even numbers in MPer(f ). Assume, contrary to our claim, that there is an even r i = 2u in MPer L (f ). Then for every decomposition of L(f n ) into (6.2) there must be the term c j 2u (n) for some j. However, for j = 1 we have Analogously, for j = 2, 4 we obtain another decompositions of c j 2u (n), which contradicts our assumption. Remark 6.3. Theorem 6.2 holds in fact for a larger class of maps, namely for maps having finitely many periodic points, all of them hyperbolic. For a map f in this class the sequence (L(f n )) n is bounded (cf. [4]) and thus the only non-zero eigenvalues of f * which give the contribution to (L(f n )) n (i.e. have different multiplicity in odd and even homology) are roots of unity [1]. As a consequence, the analysis of such maps reduces to quasi-unipotent case.

Applications: the minimal sets of Lefschetz periods on N g , a non-orientable compact surface without boundary of genus g
Let us remind that N g is homeomorphic with the connected sum of g real projective planes and its homology groups are the following: H 0 (M, Q) = Q, H 2 (M, Q) = 0 and

Realization of the minimal set of Lefschetz periods on N g
Conjecture 7.1. Llibre and Sirvent [28] formulated the following conjecture: can any finite set of odd positive integers be the minimal set of Lefschetz periods for a C 1 Morse-Smale diffeomorphism on some non-orientable compact surface without boundary with a convenient genus?
We will show that there are no algebraic obstacles on the homology which would prevent the validity of Conjecture 7.1.
Let d be a degree of some primitive root of unity; we take maximal m such that 2m − 1 ≤ d and denote by c d the vector Observe that for odd k there is b d k = −b 2d k and as a consequence, . Consider the matrix A composed of vectors c i in rows: As a consequence, the matrix A is lower triangular and det A = 1, so the set A m is a basis of Z m .
Proof. (a) Remind that by the formula (4.9) Lefschetz numbers of self-map of N g have the form We are searching for a homomorphism f * satisfying for each odd i the equality: .
Consider the vector [a 1 + 1, a 3 where in the last equality we used the relation (7.1). Now we define the map f * 1 represented by a diagonal matrix of dimension and by Theorem 4.11 we get the formula (7.3).
Remark 7.6. There is a "topological" part of Conjecture 7.1 which is still unsolved. Namely, it remains an open question whether the quasi-unipotent map f * found in Theorem 7.3 could be realized as a map that is induced by some Morse-Smale diffeomorphism on N g .

Algorithmic approach to determine the minimal sets of Lefschetz periods
for self-maps of N g In this section we describe a simple algorithm that enables us to determine the minimal sets of Lefschetz periods for self-maps of N g . As an application we use a computer program based on this algorithm to verify MPer L (f ) for g < 10 found in [28] and to compute MPer L (f ) for g ≥ 10.
We will sketch below the main scheme of the algorithm, while the detailed description is placed in Appendix. The program for computation of MPer L (f ) based on the algorithm (in Mathematica) is available on the webpage of Myszkowski. 1 We assume that g > 1; otherwise the matrix f * 1 has the dimension 0 and the only possibility is MPer L (f ) = {1}. The dimension of the matrix f * 1 1 http://www.mif.pg.gda.pl/homepages/amyszkowski/. We define a finite family P of partitions P j of the number g − 1 into possible values of ϕ, i.e.
The next lemma allows us to find the bound for the number of elements in the family P . By application of the inequality (7.4) we get that: We can associate with each partition P j (i.e. each set of degrees) the corresponding sequence of Lefschetz numbers in the form of a periodic expansion: where b di k are defined by the formula (4.9) of Theorem 4.9. By Theorem 6.2 the set of minimal periods M Pj for a map corresponding to a group of degrees P j is Finally, the family of the minimal sets of Lefschetz periods MPer L (f ) over all f being Morse-Smale diffeomorphisms of N g is given as follows:  Example 7.9. Let g = 4; then dim f * 1 = 3. There are 9 possible families P j of degrees of primitive roots of unity (each corresponding to some induced map f * ). In Table 1 the set M Pj = MPer L (f ) is described for each P j and MPer L is determined.
Thus. the zeta Lefschetz function for f has the possible forms . Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix: The algorithm for the determination of MPer L
For the sake of clarity we divided our algorithm into some steps (Algorithms 1-4 below). In Algorithm 1 we find the set ϕ(ϕ −1 [1, g − 1]) by using the function P hiV alues.