A sufficient condition for the realizability of the least number of periodic points of a smooth map

There are two algebraic lower bounds of the number of n-periodic points of a self-map f:M→M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f : M \rightarrow M}$$\end{document} of a compact smooth manifold of dimension at least 3: NFn(f)=min{#Fix(gn);g~f;gcontinuous}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$NF_{n}(f) = min\{\#Fix(g^{n}); g ~ f; g continuous\}$$\end{document}and NJDn(f)=min{#Fix(gn);g~f;gsmooth}.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$NJD_{n}(f) = min\{\#Fix(g^{n}); g ~ f; g smooth\}.$$\end{document}In general NJDn(f) may be much greater than NFn(f). In the simply connected case, the equality of the two numbers is equivalent to the sequence of Lefschetz numbers satisfying restrictions introduced by Chow, Mallet-Parret and Yorke (1983). The last condition is not sufficient in the non-simply connected case. Here we give some conditions which guarantee the equality when π1M=Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\pi_{1}M = \mathbb{Z}_{2}}$$\end{document}.

imply the equality NF n (f ) = NJD n (f ) (Theorem 6.1). In particular, the equality holds for each compact Lie group with π 1 M = Z 2 and smoothly realizable (L(f k )) k|n .

Indices of iterations of a smooth map
This paper is based on [14] and we refer the reader to it for the details.
In this section we study the following problem. We are given a self-map of a smooth compact connected simply connected manifold f : M → M and an integer n. We ask if f is homotopic to a smooth map g with Fix(g n ) a point. We recall a necessary and sufficient condition for the existence of such a smooth map g.
Let us start with the remark that the same question in the class of continuous maps always has a positive answer (in dimensions greater than or equal to 3); see [15,Theorem 5.3.2]. However, the smooth case turned out to be quite different. The crucial difference is the sequence of Lefschetz numbers L(f k ).
In 1983, Dold [5] noticed that a sequence of fixed point indices A k = ind(f k ; x 0 ), where f is a continuous self-map of a Euclidean space R m and x 0 is an isolated fixed point for each k, must satisfy some congruences. Namely, Vol. 18 (2016) Least number of periodic points 611 Least number of periodic points 3 for each n ∈ N, ∑ k|n µ ( n k where µ denotes the Möbius function. It was shown in [1] that each sequence of integers (A k ) satisfying Dold congruences can be realized as A k = ind(f k ; x 0 ), for a continuous self-map of R m , m ≥ 3. In other words, Dold congruences are the only restrictions for the sequence of fixed point indices of a continuous map.
Surprisingly it turned out that there are much more restrictions on the sequences A k = ind(f k ; x 0 ) when f is smooth [4,18].
Definition 2.1. A sequence of integers (A n ) is called smoothly realizable in R m (or in dimension m) if there exist a smooth self-map f : R m → R m and an isolated fixed point x 0 ∈ Fix(f ), which is also an isolated fixed point of each iteration f n , so that A k = ind(f k ; x 0 ).
In Theorem 2.4 we recall all possible sequences which can be obtained as fixed point indices of a smooth self-map of R m (for m ≥ 3). Then, in Theorem 2.5, we give an equivalent but shorter list of smoothly realizable sequences.
It is convenient to present the sequences of integers as the sum of elementary periodic sequences as follows.
Definition 2.2. For a given k ∈ N, we define In other words, reg k is the periodic sequence where the nonzero entries appear for indices divisible by k. It turns out that each sequence of integers (A n ) can be uniquely written as a periodic expansion: Moreover, all coefficients a k are integers if and only if the sequence (A n ) satisfies Dold congruences.
Remark 2.3. Later we will be interested in iterations f k , for k dividing a prescribed n, so we will consider only finite sequences labeled by {k ∈ N; k|n}. We will say that a finite sequence (B k ) k|n is smoothly realizable in R m if it is the restriction of a sequence smoothly realizable in R m . For a finite subset A ⊂ Z, we denote by lcm(A) the least common multiplicity of its elements and we denote In the next theorem, L(s) denotes where L ⊂ {3, 4, 5, . . . } is any subset of cardinality s. Moreover, we denote where L is as above. where where a 1 = 1 and (II) For m even, where Least number of periodic points 5 where a 1 = 1.
The next theorem is a more concise version of the above list of smoothly realizable sequences. Proof. In the first two columns of Table 1 we list the cases of the theorem and in the last column we give the corresponding cases in Theorem 2.4.
We notice that ∑ L(s) and ∑ L2(s) in Theorem 2.4 mean LCM and LCM 2 , respectively. It remains to check that the data in the last column coincide with the conditions of the theorem and to notice that the two missing cases (A o ), (A e ) correspond to m ≥ 2s + 3, hence they need no restrictions.
Corollary 2.6. If the above sequence (D n ) is smoothly realizable in dimension m, then (1) |α 1 | ≥ 2 implies m ≥ 2s + 3 or (m = 2s + 2 and LCM); (2) α 1 = 0 implies m ≥ 2s + 2 or (m ≥ 2s + 1 and LCM). J. Jezierski It is easy to notice that if f : M → M is a self-map of a compact manifold and Fix(f n ) is a point, then the sequence (L(f k )) k|n is smoothly realizable in R m , where m = dim M . It turns out that if M is simply connected and dim(M ) ≥ 3, then the inverse implication is also true.

Nielsen fixed point theory
Now we consider spaces with nontrivial fundamental groups. It will be impossible to reduce the set of fixed (or periodic) points to a single point when the Nielsen-type invariants introduced below are greater than 1.
We consider a self-map of a compact connected polyhedron f : X → X and its fixed point set Fix(f ). We define the Nielsen relation on this set by: x ∼ y if and only if there is a path ω joining x and y such that fω and ω are fixed-endpoint homotopic.
This relation splits Fix(f ) into Nielsen classes. We denote the set of Nielsen classes by N (f ). We say that a Nielsen class A is essential if its fixed point index is nonzero: ind(f ; A) ̸ = 0. The number of essential Nielsen classes is called Nielsen number and is denoted by N (f ). This is a homotopy invariant and, moreover, it is the lower bound of the number of fixed points in the (continuous) homotopy class: [3,15,16].
On the other hand, we define the set of Reidemeister classes of the map f as the quotient space of the action of the fundamental group π 1 M on itself given by Here we take as the basepoint a fixed point of f . We denote the quotient space by R(f ). There is a natural injection from the set of the Nielsen classes to the set of Reidemeister classes N (f ) ⊂ R(f ) defined as follows. We choose a point x in the given Nielsen class A and a path ω from the basepoint x 0 to x. Then the loop ω * (fω) −1 represents the corresponding Reidemeister class. Now we consider iterations of the map f . For fixed natural numbers l|k, there is a natural inclusion which induces the map N (f l ) → N (f k ) (which may be not injective). This map extends to the map i kl : Vol. 18 (2016) Least number of periodic points 615 Least number of periodic points 7 The functorial equalities i kl i lm = i km and i kk = id are satisfied and, moreover, the diagram The group Z k acts on Fix(f k ) by and on R(f k ) by commutes. We denote by OR(f k ) the set of orbits of the above action (orbits of Reidemeister classes).

Consider the Reidemeister classes
Let ER(f k ) and IR(f k ) denote the sets of essential and irreducible classes, respectively.
A map f is called essentially reducible if A ≼ B and B is essential implies A is essential.
We denote by IEOR(f ), or simply IEOR, the set of irreducible essential orbits of the Reidemeister classes of f .

The Reidemeister graph
We fix a map f : M → M . It is very convenient to consider a directed graph whose vertices are orbits of Reidemeister classes and the fixed point indices are their weights. Then the problem of minimizing the number of n-periodic points is reduced to finding some minimal subsets in the graph. We denote the graph of orbits of Reidemeister classes by GOR(f ) and we define it as follows: • vertices are elements of the disjoint sum

Vert
( where µ denotes the Mobius function.
The fixed point index defines the map ind : for A ∈ OR(f a ). We will decompose this function into summands coming from smoothly realizable sequences.
Then we say that the sequence c is attached to the orbit H ∈ OR(f h ). We also say that C H comes from c.
We denote by The sequence attached to the orbit H, then gives where the summation runs over the set Vert(GOR(f )).
On the other hand, any Dold function D : Vert(GOR(f )) → Z may be uniquely represented as where the summation runs over the set Vert(GOR(f )) and a B are integers Vol. 18 (2016) Least number of periodic points 617 Least number of periodic points 9 (see [8]). Now we may reformulate the problem of realizing the least number of periodic points of a smooth map as an algebraic question.
for each B ∈ OR(f k ), k|n.
Now we illustrate the above theorem in the case π 1 M = Z 2 (the simplest nontrivial non-simply connected case). Compare [8,Section 6].
We consider a connected manifold M with π 1 M = Z 2 . Then the induced homotopy homomorphism f # : Z 2 → Z 2 may be either the zero map or the identity map.
First, we consider the case when f # = 0. Then the set of Reidemeister classes R(f ) = Z 2 /(im(id −0)) = 0 and the same holds for each iteration: R(f n ) = 0. Now the Reidemeister graph is reduced to the graph of natural numbers. Let Then and similarly R(f n ) = Z 2 for any iteration n. On the other hand, since f # = id Z2 , the action of Z n on R(f n ) = Z 2 is trivial for each n. This means that each orbit reduces to a point and OR(f n ) = R(f n ) = Z 2 for each n. We recall that the map i kl : OR(f l ) → OR(f k ) is given by id when k/l is odd, 0 when k/l is even.
We introduce the following notation. We denote The element 1 ′ corresponds to the choice of a point in the definition of the homotopy group π 1 M = Z 2 . Then we denote Let us recall that the orbit A ∈ OR(f l ) precedes B ∈ OR(f k ) if l|k and i kl (A) = B. An irreducible orbit is the orbit which is preceded only by itself.
Formula (4.4) gives irreducible orbits of Reidemeister classes for f # = id Z2 . We notice that i kl (l ′ ) = (k ′ ) for all l|k and i kl (l ′′ ) =

Self-maps of odd-dimensional projective spaces
Each odd self-mapf : S m → S m (i.e.,f (−x) = −f (x)) defines a self-map f of the projective space RP m which induces the identity homomorphism of the fundamental group π 1 (RP n ) = Z 2 . We assume, moreover, that m is also odd and | deg(f ) | ≥ 3. We show that then the sequence of Lefschetz numbers is constant L(f k ) = 1, hence it is smoothly realizable in each dimension, but there exist natural numbers l such that the least number of l-periodic points in the homotopy class of f cannot be realized by a smooth map (in fact it is enough to take an odd l that is a product of sufficiently many different primes, where the number of the primes depends only on the dimension m).
Since f # = id, OR(f k ) = Z 2 , Section 4 implies that i kl (l ′ ) = k ′ and i kl (l ′′ ) = k ′′ for odd l|k. On the other hand, since the dimension m is odd, RP n is not a Jiang space, it turns out that gives the splitting of Fix(f ) into Nielsen classes (the same holds for all iterations). Now We assume that for each r, the least number of r-periodic points in the homotopy class of f can be realized by a smooth map. We get a contradiction.
We recall that and we notice that the only elements in IEOR preceding orbits l * (for odd l) may be only 1 ′ or 1 ′′ . Now by Theorem 4.2 there exist expressions, coming from sequences smoothly realizable in R m , so that Vol. 18 (2016) Least number of periodic points 619 Least number of periodic points 11 for all l * . But tends to infinity. On the the hand, the smoothly realizable expressions C ′ 1 and C ′′ 1 in dimension m can take at most 2 [(m+1)/2] values (see [1]). Now the number of values on the right-hand side is bounded by a number which depends only on m.
We have obtained a contradiction, since the indices of Reidemeister orbits were growing exponentially but their sum was always equal to 1. This explains why in the main theorem of the paper (Theorem 6.1) some extra assumptions for the equality will be necessary.

The main result
The main result of the paper is the following. (1) f is essentially reducible; The proof is based on Theorem 4.2: we must find for each essential irreducible orbit The following lemma will be useful. Proof. Let us assume that l 0 is the smallest number satisfying a l0 ̸ = 0. We will inductively prove the formula a (2 r 0 l0) ′ = −a l0 2 r0+1 (6.2) J. Jezierski for all r 0 ≥ 1. Since the number a (2 r 0 l0) ′ must be integer, a l0 = 0 and we get a contradiction. We start with some general remarks. Let us notice that a l ′ = a l ′′ = 0 for all l < l 0 , since then a l = 0 by the assumption and the Jiang property. Moreover, for all r 0 ≥ 0. The following equalities follow from the Reidemeister graph: since l|l 0 and l ̸ = l 0 imply 2l < l 0 .
On the other hand, by the Jiang property, the above indices are equal to 1 2 L(f 2l0 ) and hence to Moreover, a (2l0) ′ + a (2l0) ′′ = 0, since a 2l0 = 0 by the assumption. Now which implies that a (2l0) ′ = a l0 −2 2 , hence we get the desired formula for r 0 = 1.
In a similar way we perform the inductive step. We assume that formula (6.2) holds for all r < r 0 .
The following equalities follow from the Reidemeister graph and the inductive assumption: since a k ′ + a k ′′ = 0 for even k and a (2 r 0 +1 l) ′ = 0 for l < l 0 .
Vol. 18 (2016) Least number of periodic points 621 Least number of periodic points 13 On the other hand, by the Jiang property, the above indices are equal, hence Since L(f 2 r 0 l0 ) = L(f l0 ) = a l0 · l 0 , we get a (2 r 0 +1 l0) ′ = −a l0 2 r0+2 which ends the inductive step.

Dold decomposition
We give the Dold decomposition of the index function of the map satisfying the assumption of Theorem 6.1; i.e., for each orbit k * we exhibit the number a k * such that the equality ) holds for each orbit B.
We present the natural numbers as k 0 = 2 r0 l 0 , where l 0 is an odd number. We define (for . Now we consider k * 0 = k ′ 0 . Then we put Now we show that equality (6.1) is satisfied for each n * . Let us recall that f is a Jiang map, so for n * = n ′ or n ′′ , and it remains to show that ) . (7.1) We may rewrite the right-hand side of equality (7.1) as (n * ) = S 1 + S 2 + S 3 = ( * ).
Now we notice that under the assumption m ≥ 2s + 3 in Theorem 2.5, the only restriction is that the summation in ∑ k α k reg k (n) runs over the set LCM ({2; d 1 , . . . , d s }). Now, by Theorem 4.2, it remains to show that one can attach to each essential irreducible Reidemeister class an expression coming from an expression of the type as above so that their sum realizes the fixed point index of each orbit (Theorem 4.2).
We may rewrite (7.2) as We will show that the expressions and come from expressions smoothly realizable in R m attached to the orbits 1 ′ and (2 r ) ′′ , respectively. Consider the sequence C 1 ′ . We notice that it comes from We recall that the sum ∑ k a k reg k satisfies (a) the summation runs over LMC(2; d 1 , . . . , d s ), (b) 2s + 3 ≥ m. Now the sum ∑ k 1 2 a k reg k also satisfies (a) and (b), but (a) and (b) form a sufficient condition for the sum to be smoothly reducible in dimension m (Theorem 2.5). J. Jezierski Now we consider C (2 r 0 ) ′′ , for a fixed number r 0 ≥ 0. We notice that the sum (8.2) comes from ∑ l odd ( a (2 r l) ′′ · reg l − 1 2 a (2 r l) ′′ , a 2 r l0 ̸ = 0 for an r ≤ r 0 . This implies that 2 r0 l 0 = lcm(d i1 , . . . , d is ) for some 1 ≤ i 1 < · · · < is ≤ s. Now we eliminate the powers of 2 and we get which implies that l 0 ∈ LCM(d ′ 1 , . . . , d ′ s ).

Corollary 8.3. The assumptions
• f is essentially reducible, • each iteration f k is Jiang, i.e., the indices of all the Reidemeister classes in R(f k ) are the same hold for all self-maps of the so-called DJ spaces (introduced in [14], see also [6]), so to get the equality NF n (f ) = NJD n (f ), it is enough to assume then that π 1 M = Z 2 and the smooth realizability of the sequence of Lefschetz numbers (L(f k )) k∈N . Since the class of DJ spaces contains compact Lie groups, we get the similar implication for compact Lie groups with π 1 M = Z 2 .
Vol. 18 (2016) Least number of periodic points 625 Least number of periodic points 17