Combinatorial scheme of ﬁnding minimal number of periodic points for smooth self-maps of simply connected manifolds

. Let M be a closed smooth connected and simply connected manifold of dimension m at least 3, and let r be a ﬁxed natural number. The topological invariant D mr [ f ], deﬁned by the authors in [ Forum Math. 21 (2009), 491–509], is equal to the minimal number of r -periodic points in the smooth homotopy class of f , a given self-map of M . In this paper, we present a general combinatorial scheme of computing D mr [ f ] for arbitrary dimension m ≥ 4. Using this approach we calculate the invariant in case r is a product of diﬀerent odd primes. We also obtain an estimate for D mr [ f ] from below and above for some other natural numbers r .


Introduction
Let f be a self-map of a compact manifold M . The problem of minimizing the number of fixed or periodic points in a homotopy class of f is one of the important challenges in modern periodic point theory. In this paper, we consider the smooth version of this question, asking about minimal number of r-periodic points in the smooth homotopy class of f , i.e., for min # Fix(g r ) : g where s 64 G. Graff and J. Jezierski JFPTA introduced by Jiang [18] in 1983 as a lower bound for the number of r-periodic points in the homotopy class, and it was proved in 2006 that NF r (f ) is the best such lower bound, i.e., it is equal to the minimum in (1.1); see [16]. During the last decade, NF r (f ) was computed in many particular cases; see [13,14,15,19,20,21]. Recent investigations of the authors showed that the smooth and continuous theories do not coincide. In [6,9] two counterparts of NF r (f ) were defined for smooth category: D m r [f ] for simply connected manifolds and its generalization NJD m r [f ] for non-simply connected ones. The difference between continuous and smooth categories is clearly noticeable for the simply connected case. In such situation NF r (f ) ∈ {0, 1} but D m r [f ] is usually greater than 1. It turned out that then the only obstacle (as the fundamental group is trivial) to minimize the number of periodic points comes from their fixed point indices. By the classical Poincaré-Lefschetz theorem, for each n the Lefschetz number L(f n ) is equal to the sum of fixed point indices of f n at points that are fixed by f n . On the other hand, the sequence of fixed point indices at an isolated fixed or periodic point for a smooth map has a very special form. As a result, to obtain the sequence {L(f n )} n|r as a sum of indices, one usually needs many periodic points (unlike in a continuous case, where the forms of sequences of indices are more arbitrary, and thus Lefschetz numbers can be realized by one such sequence [16]).
The invariant D m r [f ] is equal to the minimal number of sequences in the decomposition of Lefschetz numbers of iterations {L(f n )} n|r into sequences, each of which can be realized as fixed point indices at a periodic orbit of a smooth local map. As a consequence, to find the value of D m r [f ], one needs to know all possible forms of local fixed point indices of a smooth map in the given dimension m. All such forms were described for three-dimensional maps in [12] which allowed us to find D 3 r [f ] for S 2 × I [6], S 3 [7], two-holed three-dimensional closed ball [5] and also NJD 3 r [f ] for RP 3 [10]. Recently, the complete list of all sequences of local indices of iterations in arbitrary dimension has been found [11], which enabled us to calculate D m r [f ] in dimension 4 [8].
The main goal of this paper is to provide the effective methods of computing D m r [f ] for arbitrary higher-dimensional manifolds. In order to do that, at first we show that finding the value of the invariant may be simplified in higher-dimensional case (cf. Theorem 4.2). This observation is also an answer to the question (asked during a discussion in the conference Nielsen Theory and Related Topics, St. John's, Newfoundland, Canada, 2009) about the differences between three-and higher-dimensional cases in smooth category. Namely, we prove that for m > 3 one may find smooth g homotopic to f such that Fix(g r ) = D m r [f ] and all r-periodic points of g are fixed points, while for m = 3 in addition to fixed points some 2-periodic orbits for g may remain irreducible (

Sketch of the construction
At first we sketch the definition of D m r [f ] to provide the general topological background of our idea; for further details the reader may consult [6]. Problem 2.1. We are given a smooth self-map f : M → M of a smooth closed connected and simply connected manifold of dimension m ≥ 4 and a number r ∈ N. We seek the minimal number of r-periodic points in the smooth homotopy class of f : where s ∼ means that the maps g and f are C 1 -homotopic.
We will briefly describe in the items below how this question reduces to a calculation of our combinatorial-type invariant denoted as D m r [f ]. (1) Let us consider an isolated periodic point x ∈ Fix(f p ). Then the integer sequence {c k } k = {ind(f k , x)} k must satisfy strong restrictions found by Chow, Mallet-Paret and Yorke [3]. We will call each integer sequence that satisfies such conditions DD m (p) sequence.
(2) Assume now for simplicity that the minimal number of r-periodic points can be realized by fixed points: there is a smooth map g smoothly homotopic to f satisfying # Fix(g r ) = MF diff r (f ) and Fix(g r ) = Fix(g).
(In fact, one of our results, i.e., Theorem 4.2, states that this is true for m ≥ 4.) (3) Consider the above map g. Now where u = MF diff r (f ). This implies that Thus the (finite) sequence {L(f k )} k|r is the sum of u DD m (1) sequences.

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G. Graff and J. Jezierski JFPTA (4) In [6] we proved, using advanced Nielsen techniques, that the inverse is also true. If {L(f k )} k|r is the sum of u DD m (1) sequences, then f is homotopic to a smooth map g with # Fix(g r ) = u.

Definitions and theorems
Now we give more information concerning the invariant D m r [f ].
is open; and P , an isolated p-orbit of φ, such that c n = ind(φ n , P ) (notice that c n = 0 if n is not a multiple of p). The finite sequence {c n } n|r will be called DD m (p|r) sequence if this equality holds only for n | r, where r is fixed.
Let us fix an integer r ≥ 1. The value of the invariant D m r [f ] is given as the minimal decomposition of the sequence of Lefchetz numbers of iterations into DD m (p|r) sequences. where c i is a DD m (l i |r) sequence for i = 1, . . . , s. Each such decomposition determines the number l = l 1 + · · · + l s . We define the number D m r [f ] as the smallest l which can be obtained in this way.
The invariant D m r [f ] was defined in [6] and it is equal to the minimal number of r-periodic points in the smooth homotopy class of f . [6]). Let M be a closed smooth connected and simply connected manifold of dimension m ≥ 3 and r ∈ N a fixed number. Then,

Theorem 2.4 (see
The convenient way of writing down sequences of indices of iterations is to represent each of them as an integral combination of some basic periodic sequences {reg k (n)} n . Definition 2.5. For a given k we define the basic sequence as Vol. 13 (2013) Combinatorial scheme 67 Any sequence of indices of iterations (and also Lefchetz numbers of iterations) may be represented in the form of periodic expansion (cf. [17]), namely where a n = 1 n k|n μ(k) ind(f (n/k) , x 0 ), μ is the classical Möbius function, i.e., μ : N → Z is defined by the following three properties: Moreover, all coefficients a k in (2.2) are integers, which was proved by Dold [4].
The invariant D m r [f ] is defined by a use of DD m (p) sequences, but it turns out that it is enough to know only the forms of DD m (1) sequences, because the complete list of all DD m (p) sequences can be obtained from the list of DD m (1) ones, by replacing each reg k by reg pk (see Definition 2.6 and Theorem 2.7 below for the formal explanation of this statement). As a consequence, the forms of DD m (1) sequences that are given in Theorem 3.2 in Section 3 allow one to easily determine all forms of DD m (p) sequences. Definition 2.6. We will say that the DD m (p) sequence {c n } n comes from the given DD m (1) sequence {c n } n with the periodic expansion if the periodic expansion of {c n } n has the form Theorem 2.7 (see [6]). Every DD m (p) sequence comes from some DD m (1) sequence.

Indices of iterations in R m
In this section, we give the complete list of all forms of indices of iterations of smooth maps in a given dimension m ≥ 3.
Let us remark here that the problem of finding the forms of indices of iterations of particular class of maps is difficult in general. Nevertheless, last years brought some important results concerning planar homeomorphism [23], R 3 -homeomorphisms [2,24] and holomorphic maps [25,26,27]. For natural s we denote by L(s) any set of natural numbers of the form L with #L = s and 1, 2 ∈ L.
By L 2 (s) we denote any set of natural numbers of the form L with #L = s + 1 and 1 ∈ L, 2 ∈ L. where where a 1 = 1 and (II) For m even, in the case (E e ); where a 1 = 1.

Minimal number of periodic points can be realized at fixed points
Let M be a closed smooth connected and simply connected manifold of dimension m ≥ 4 and r ∈ N a fixed number. In this section, based on the knowledge of the forms of indices of iterations, we will prove that it is always possible to find in a given smooth homotopy class a map g with the minimal number of r-periodic points such that g has only fixed points (up to the rth iteration).
Proof. By the relation p LCM(K) = LCM(pK) which holds for K = ∅, we get Proof. We will show that every DD m (p) sequence with p ≥ 2 is a sum of at most two DD m (1) sequences, which proves our theorem.
By Theorem 2.7 every DD m (p) sequence can be represented in the form k∈p·G a k reg k (n), (4.1) where the forms of G are described in Theorem 3.2, with perhaps some additional restrictions on coefficients. We will prove that the sequence (4.1) with arbitrary coefficients a k is always a sum of at most two DD m (1) sequences. The dimension m is fixed, we will consider two cases in dependence on the parity of m.
Case I (m is odd). Here we consider two subcases: 2 ) (remind that we ignored the influence of the restrictions for a 1 , a 2 ).
We will consider each of the above subcases separately. Notice that the set {pd 1 , . . . , pd s , 2p} consists of s + 1 = m 2 elements and 2 does not belong to the set. Thus we can realize all a k reg k with k ∈ {pd 1 , . . . , pd s , 2p} by one sequence of the type (F e ), which gives the contribution to a 1 equal to 1. The remaining expression has the form − reg 1 + a p reg p , (4.3)

Vol. 13 (2013)
Combinatorial scheme 71 and can be realized either by one sequence of the type (D e ) for p > 2, or by one sequence of the type (A e ) for p = 2. Finally, in each subcase we are able to realize the sum (4.1) by no more than two sequences. This completes the proof for m even and the proof of the whole theorem.
Assume we have a given decomposition of Lefschetz numbers of iterations into DD m (p|r) sequences. Then, by the construction described in [9], one can find in the smooth homotopy class of f a map g for which p-periodic orbits are in the one-to-one correspondence with DD m (p|r) sequences. The above fact and Theorems 2.4 and 4.2 imply the following result.

Combinatorial scheme of finding D m r [f ] for maps with nonvanishing coefficients of periodic expansion
We fix the natural number r. For the divisors of r we represent the sequence of Lefschetz numbers of iterations in the form of periodic expansion: In the rest of the paper we will work under the following assumptions. First, it is convenient to find the minimal decomposition of the sum L(f n ) = k|r b k reg k (n) into DD m (p|r) sequences modulo reg 1 ; i.e., we require that equality (2.1) holds only for all divisors i|r different from 1 (thus we temporarily ignore the coefficient at reg 1 ). JFPTA

5.1.
Finding D m r [f ] modulo reg 1 Let Div(r) denote the set of all divisors of r different from 1. We will show that finding the minimal decomposition is equivalent to finding a minimal family of subsets of Div(r) satisfying some simple conditions.
Let us consider a decomposition of Lefschetz numbers into DD m (1) sequences for k|r.
As we consider the case of odd r and ignore the coefficient b 1 , the only sequences {c i } i that may appear in (5.3)  Notice that condition (ii) is equivalent to (ii) for each k|r, k = 1, there exist an i = 1, . . . , h and a subset K ⊂ A i such that k = LCM(K). As a consequence, we get the following lemma. Then, for even m there is for some i, v 0 + 1 otherwise.  .7) is satisfied, we have to use one sequence more of the type (A o ) with the coefficient a 1 = b 1 . If m is even, the proof is analogous, with the difference that we can use only sequences of the type (F e ).
Remark 5.6. In the first part of our Standing Assumption (II) we restrict ourselves to the simpler case of odd r. Our aim is to describe the essence of the introduced method rather than use it to find the exact formulas in every case. For even r it could be complicated, however also possible, for example for any self-map f of S 3 the value of D 3 r [f ] was found also for even r in [7]. Remark 5.7. Notice that in case the second part of Standing Assumption (II) is not satisfied, i.e., there are some b k = 0 in the periodic expansion of Lefschetz numbers in (5.1), then the right-hand sides of equalities (5.6) and (5.7) give the upper bound for the number of DD m (1) sequences in the decomposition of {L(f n )} n|r . As a consequence, we always get (independently of the map) the estimates from above for the minimal number of r-periodic points in the smooth homotopy class of a given map.

D
Proof. We will show that conditions (1) and (2) of Lemma 6.1 are equivalent to conditions (5.4) and (5.5). As, by our assumption, r = p 1 · · · p v is a product of v different odd primes, there is a natural bijection D : Div(r) → 2 Iv \ {∅} given by Thus we obtained exactly condition (2) of Lemma 6.1.
By the equalityD(A i ) = B i , condition (5.4) is obviously transformed into condition (1).
The inverse map D −1 gives the inverse transformation of the conditions, which shows that they are equivalent. Find the explicit formula for h s (v).
The next theorem gives a formula for the number h s (v). To make this formula uniform we will use the following convention. We will uniquely represent each natural number v as v = k · s + R, where k ∈ N ∪ {0} and R = 1, . . . , s. In particular, if s divides v, then v = k · s + s.
In other words, Before we give the proof of Theorem 6.3, we will prove some helpful lemmas.
is the least integer greater than or equal to 2 sk+R −1 2 s −1 . Proof. Let us notice that is an integer. On the other hand, To complete the proof, it remains to notice that 0 ≤ 2 s −2 R 2 s −1 < 1 for R = 1, . . . , s. The next lemma shows that the sequence expressed by the right-hand side of (6.3) can be given inductively.
Proof. Since in our convention s = s · 0 + s, which proves the first inductive step. Now, we assume that the formula holds for sk + R and we will prove it for sk + R + 1. We will consider two cases in the dependance on the value of R.