Minimal number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds

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Introduction
Finding minimal number of r -periodic points in the homotopy class (for a fixed r ) is an important challenge in modern homotopy periodic point theory, with an increasing number of valuable results obtained in the last decade in many particular cases [1][2][3][4][5][6][7].
On the other hand, the smooth case in which smooth homotopies are considered, turned out to be essentially different from the continuous one (cf. [8]). First authors who observed a difference between smooth and continuous category in Nielsen theory were Brown, Greene and Schirmer for r = 1 i.e. for fixed points and in the relative case [9] (see also [10][11][12][13] B Grzegorz Graff graff@mif.pg.gda.pl Jerzy Jezierski jezierski@acn.waw.pl 1 Faculty of Applied Physics and Mathematics, Gdansk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland for related results). These authors considered a smooth manifold M with boundary ∂ M and φ, a smooth self-map of ∂ M. They asked a question whether it is possible to extend φ from ∂ M to M without introducing any more fixed points. It turned out that the answer depends whether one considers smooth or continuous extensions: there are smooth maps φ which admit continuous extensions to M with no fixed points on M\∂ M, but such that every smooth extension has a fixed point in M\∂ M. This rather unexpected result showed that there is a differences between smooth and continuous case in fixed point theory, even if there was no such difference in classical Nielsen theory i.e. for the non-relative case (cf. [14]).
In this paper we raise more general question related to minimization the number of periodic points instead of the number of fixed points in the smooth category. We consider maps of pairs f : (M, ∂ M) → (M, ∂ M) and ask for minimal number of periodic points in the smooth (i.e. C 1 ) homotopy class (preserving the boundary) of f . Let us point out that we consider all C 1 maps, generalizing the approach presented in [9] were the class of transversal maps was examined. On the other hand, we confine ourselves to the case in which both M and ∂ M are simply-connected. The reason for that is the following: there are two obstacles to minimize the number of periodic points for C 1 maps. One of them comes from the Reidemeister relations and the second results from the restrictions on the sequence of local indices of iterations of a C 1 map. It would be difficult (also from the computational point of view) to follow the restrictions that come from the both conditions simultaneously, and thus we analyze the situation in which the fundamental group is trivial, so the Reidemeister relations disappear. Then the only obstacle that we have to control is related to the forms of local indices of iterations and we can apply the topological methods developed in [15] that were used to minimize the number of periodic points for C 1 self-maps of simply-connected manifold without boundary. As these technics work only in dimension at least 3 we have to assume that dim ∂ M ≥ 3. We consider only odd r : the case of even r is conceptually analogous, but the forms of indices of iterations are much more complicated and thus analyzing the even case would overshadowed our main idea.
The paper is organized in the following way. In Sect. 2 we define the invariant D r ( f ; M, ∂ M) and prove that it is equal to the minimal number of periodic points in the C 1 -homotopy class of f (for homotopies preserving the boundary).
In Sect. 3 we consider ( f,f ) : (M, ∂ M) → (M, ∂ M), x 0 ∈ ∂ M a fixed point of f and find all possible forms of pairs of sequences (ind( f n , x 0 ), ind(f , x 0 )) n (Theorem 3.15), which is necessary for effective computation of D r ( f ; M, ∂ M). In the last section we show our invariant in action, finding necessary and sufficient conditions for a reduction of a set of r -periodic points to one point and illustrating this case in the example of self-maps of a 6-dimensional closed ball.

Definition of the invariant D r ( f ; M, ∂ M)
Let r be a fixed natural number, M be a smooth compact and simply-connected manifold with dim M ≥ 4, such that its boundary ∂ M is also simply-connected. We consider a Let us emphasize that the pair ( f,f ) is determined by f , so we will sometimes overuse the notation identifying f and ( f,f ).
In this section we define an invariant D r ( f ; M, ∂ M) of a pair and we will show that Remark 2. 1 We will assume for the rest of the paper that Fix( f r ) is a finite set, because we could always approximate our map by a map with a finite set of r -periodic points (cf. [16]).

Remark 2.2
The minimum given in (2.1) does not change if we consider only such pairs (g,ḡ) that the only r -periodic points of g (and soḡ) are fixed points (which results from the similar considerations as those given in Sect. 4 in [17], see also Lemma 4.8 in [15]). This statement is true for each r in dimension at least 4 and for odd r also in dimension 3. In other words, we could seek for the minimum over the maps having only fixed points up to the r th iteration (but we have to assume that r is odd and dim ∂ M ≥ 3).
is open; and p, an isolated fixed point of ψ, such that c(n) = ind(ψ n , p). A finite sequence {c(n)} n|r will be called a D D m (1|r ) sequence if this equality holds for all n with n|r , where r is fixed.
We introduce the following notations: We will consider C 1 maps of a pair by a C 1 map we understand a map which has continuous partial derivatives of degree 1, where for a point x 0 ∈ R m+1 0 we consider right derivatives ∂ + f i (x 0 ) ∂t , i = 1, . . . , m + 1.

Definition 2.4 A pair of sequences of integers
. A pair of finite sequences ({c(n)} n|r , {c(n)} n|r ) will be called a ∂ D D m (1|r ) pair of sequences if this equality holds for all n with n|r , where r is fixed.

Construction of the invariant
Let The above yields the following decomposition of the sequences of Lefschetz numbers of iterations of f andf : .
where d j (n) = ind(g n , x j ) and c i (n) = ind(g n , y i ).

Main theorem
In this section we will prove one of the main results of the paper: Theorem 2.10. First, we will introduce two procedures, which make it possible to create and remove fixed or periodic points in a homotopy class.
The Creating Procedure enables one to create an additional fixed point in the C 1 -homotopy class of f , by a use of a homotopy f t which is constant near periodic points of f (up to the given period r ) and such that f r 1 near the new fixed point is given by any prescribed formula. ∈ Fix( f r ). Then there is a homotopy { f t } 0≤t≤1 satisfying: (4) f 1 realizes given D D m (1|r ) sequence {c n } n|r on x 0 , i.e. f 1 could be any C 1 map in a neighborhood of x 0 such that c n = ind( f n 1 , x 0 ) for n|r.
Due to the next lemma it is possible to cancel, in the homotopy class, invariant subsets of periodic points which have indices of iterations equal to zero. (1) S is finite and f -invariant i.e. f (S) = S.
Then there is a homotopy f t , starting from f 0 = f , constant in a neighborhood of S and such that Fix( f r 1 ) = S. The next lemma enables us to construct homotopies preserving the boundary, which we will use in the proof of Theorem 2.10.
Proof We take as V 0 any open set satisfying 0 ∈ V 0 ⊂ cl(V 0 ) ⊂ B 0 and as ε any number 0 < ε < ε. For two disjoint closed subsets in R m+1 Notice that if f, g and η are C 1 maps then so is H (·, 1).

Theorem 2.10 Let M be a smooth compact and simply-connected manifold of dimension m ≥ 4 with simply-connected boundary ∂ M and r ∈ N a fixed odd number. Let
Proof First of all, let us notice that the minimum is greater or equal to D r ( f ; M, ∂ M) by the equalities (2.3) and (2.4).
To show the inverse inequality we decompose sequence. We will prove that for any sum of (α + β) sequences given in (2.5) and (2.6) we are able to find a C 1 map f 1 C 1 -homotopic to f , by a homotopy preserving the boundary, with #Fix( f r 1 ) = α + β. The proof will be done in three steps: first we minimize the number of r -periodic points in the boundary ∂ M, next we realize given indices near the boundary fixed points in the whole M and in the last step we minimize the number of r -periodic points in IntM by a homotopy which is constant near ∂ M.
Consider ( f,f ):(M, ∂ M) → (M, ∂ M). Abusing the notation we will use below the same letters for the homotopic maps obtained after consecutive applications of the Creating and the Cancelling Procedures.
Step 1. Minimization of the number of r -periodic points in the boundary ∂ M.
We may replace each B i by a smaller ball neighborhood of y i so that y i is a unique r -periodic point off in B i . Then, for n|r Thus, we get that ind(f n , ∂ M \{y 1 , . . . , y β }) = 0 for n|r , so we may apply the Cancelling Procedure (Lemma 2.7) tof to get a homotopy that removes all other r -periodic points in ∂ M and such that it is constant in cl(B 1 ∪ · · · ∪ B β ).
Step 2. We realize c i (n) in the homotopy class of the initial map as indices of g n i near the boundary fixed points y i in the whole M.
First, we extend the homotopy used in the Cancelling Procedure (in STEP 1) from ∂ M onto M. Since f (x, t) = (f (x), t) near the boundary, the above deformation off extends to a homotopy G s of the pair (M, ∂ M).
Now, we take f := G 1 and use Corollary 2.9 to f and g i (for i = 1, . . . , β). This gives us a homotopy H τ with the support cl(B 1 ∪ · · · ∪ B β ) × [0, ε], preserving ∂ M and joining f with a map which is equal to g i in a neighborhood of each y i in M. Let us notice (see the proof of Lemma 2.8) that H τ for each (x, t), is either constant or is a convex combination of f (x, t) and g i (x, t). Since for τ = 0 the mapsf andḡ i are equal in each B i , the deformation is constant in ∂ M. Thus we have obtained in the homotopy class (preserving the boundary) of the initial map, a map f : Step 3. Minimization of the number of r -periodic points in IntM. Now, we fix points {x 1 , . . . , x α } ∈ IntM. We will show that there is a homotopy constant near ∂ M joining f and h and such that Fix Since each d j is a D D m+1 (1|r ) sequence, we may assume (applying the Creating Procedure) that ind( f n , x j ) = d j (n) for n|r , j = 1, . . . , α. Now, analogously to (2.7) we will use the Cancelling Procedure for the whole M, the map f and the invariant set As a consequence, we get that ind( f n , M \ {x 1 , . . . , x α , y 1 , . . . , y β }) = 0 for n|r . Now we apply the Cancelling Procedure. It allows us to remove all other r -periodic points in IntM. Furthermore, we can do this by a homotopy constant near the boundary. This results from the fact that the homotopy used in the Cancelling Procedure changes its values only in some small neighborhoods of some arcs joining r -periodic points (cf. [19] for details).
Finally, we get a C 1 map of a pair (h,h) : The map h is a C 1 map in some neighborhood W of S and h r has no fixed points outside W . Thus, if h is not C 1 as the global map, we may approximate it by a C 1 map preserving the boundary, which is equal to h on W without adding any new r -periodic points in the compact set M \ W .

Periodicity of the indices for a boundary fixed point
The classical result of Shub and Sullivan from [21] states that for an isolated fixed point x 0 of a C 1 self-map f of R m the sequence {ind( f n , x 0 )} n is periodic. In this subsection we extend this result to a boundary fixed point.
Let us recall (1) If 0 is an isolated fixed point of f then (2) If 0 is an isolated fixed point for some f k , where k ≥ 1 and a−1 j=0 D jk is nonsingular for an a ≥ 1, then 0 is an isolated fixed point of f ak and moreover

1)
where σ = sign det( a−1 j=0 D jk ). We will show that the formula (3.1) in the above lemma is valid for a boundary fixed point.

Lemma 3.3 Let f : R m+1
and let k|n be two numbers such that (0, 0) is an isolated fixed point of f k and n/k−1 j=0 D jk is nonsingular. Then (0, 0) is an isolated fixed point of f n and ind( f n , (0, 0)) = σ · ind( f k , (0, 0)), (3.2) where σ = sign det( Proof It is enough to prove the lemma for k = 1. Since f maps R m+1 ∂t (0, 0) ≥ 0 and we will consider three cases.
• Case ∂ f m+1 ∂t < 1. Then for any fixed n and τ the homotopy H τ (x, t) = f n (x, t · τ ) has no fixed point different from (0, 0) in a neighborhood of (0, 0). We get: where the middle equality comes from part (2) of Lemma 3.2 for s = m.
• Case ∂ f m+1 ∂t > 1. Then for any fixed n the homotopy H τ (x, t) = f n (x, t + τ ) removes all n-periodic points of f near (0, 0) and both sides of the equality (3.2) are equal to zero.
• Case ∂ f m+1 ∂t = 1. This case needs more comment. We recall that ind ( f n , (0, 0) + . Let us define E(x, t) by the equality: We claim that the homotopy has no zeroes on the boundary of a sufficiently small ball K centered at (0, 0) ∈ R m+1 . This implies our lemma, because by the homotopy invariance and multiplicativity property of the degree we get: , (0, 0)).
where ε > 0 is the least norm of eigenvalues of n−1 j=0 D j . Now the homotopy H τ has no zeroes on the boundary of a ball K (for t ≥ 0) on which ||E(x, t)|| < ε||(x, t) −f (x, t)||. Now let t < 0. Then (3.3) takes the form We will show that the homotopy H τ has no zeroes different from (0, 0) on K .
Notice that the last coordinate of (3.4) has the form where π t : R m+1 → R is the projection on the last coordinate. This implies π t (E(x, t)) = t (1 − n). As a consequence, the last coordinate of the homotopy (1 − n)). It remains to notice that this expression is negative for t < 0, 0 ≤ τ ≤ 1. In particular, the homotopy has no zeroes for t < 0.  , (0, 0)).

Remark 3.5 We noticed in the proof of Lemma 3.3 that the derivative of f has the form
where A is m × m matrix equal to D 0f the derivative off at 0 ∈ R m+1 0 ≈ R m .

Periodic expansion of indices for a boundary fixed point
This subsection is devoted to finding more detailed description of the forms of fixed point indices for a boundary fixed point in terms of so-called periodic expansion, i.e. by representing {ind( f n , (0, 0))} n as a combination of some simple periodic sequences.

fixed point isolated for each iteration. Assume that n is odd natural number. Then, for some integers a k , there is the equality:
ind( f n , (0, 0)) = k∈O a k reg k (n), (3.7) where O = {LCM(Q) : Q ⊂ }.
Proof We fix an n / ∈ O and show that a n = 0. Let us denote q k = lcm{s ∈ O : s|k}. We notice that q k = lcm{s ∈ : s|k}.

Remark 3.11
For a C 1 mapf : R m → R m , with 0 as an isolated fixed point for each iteration, the formula for {ind(f n , 0)} n (n odd) is (except for some restriction on a 1 ) exactly the same as (3.7) with replaced by 0 which is equal to the set of primitive roots of unity contained in σ (D 0f ) (cf. [20,22,23]).

The form of indices for an interior fixed point
In this section we recall the description of possible forms of indices for a fixed point in an open subset of R m given in [20,22]. We only consider the case of odd iterations, and thus the forms of indices are simpler. We use the following notation: for natural s we denote by L (  (II) For m even:

(I) For m odd
where a 1 = 1.

The form of indices for a map of a pair (f,f)
In this section we will solve Problem 3.1. First, we formulate a statement which shows a relation between the index of a boundary mapf and its extension f , which we will use in the proof of Theorem 3.15.
Lemma 3.13 (Theorem 5.1 in [9]) an isolated fixed point of f . Assume that +1 is not an eigenvalue of D 0f i.e. I − D 0f is an isomorphism. Then We will also make use of the following lemma.
where by ind G (F, x 0 ) we denote ind(F |G , x 0 ).  We will realize the pairs of maps given in (F e ∂ 0 ) and (F e ∂ 1 ).

Definition 3.17
We define some subspaces of R m+1 + : We will use methods similar to those used in Section 9 in [20]. We will define a map on each of the spaces listed in Definition 3.17 as a discretization of some smooth flow. It is possible to extend this maps to a C 1 self-map of the whole R m+1 + with 0 as the only periodic point, in such a way that the fixed point indices of all iterations for the extension are the same as for the restriction to W ∪ V 0 ∪ V (cf. [20] for details).
By the formula (3.12) applied to (W ∪ V 0 ) ∪ V , taking into account that the index on the singleton {(0, 0)} is equal to 1, we get ind( f n , (0, 0)) = ind W ( f n , (0, 0)) + ind V 0 ( f n , (0, 0)) + ind V ( f n , (0, 0)) − 2reg 1 (n). (3.13) We define the self-maps on the spaces W, V 0 , V in the following way. Consider V = R m+1 0 ≈ R m . By Remark 3.11 and by Theorem 3.12 we can find a self-map of R m+1 0 (of the type F e ) such that its indices are equal to k∈L( m 2 ) a k reg k (n), where a 1 = 1. On W ≈ R m+1 0 ≈ R m we repeat the same construction with the coefficients c k = b k − a k , so that ind W ( f n , (0, 0)) = k∈L( m 2 ) c k reg k (n), where c 1 = 1. On V 0 , which is a half-line, we define (in an obvious way) f so that either ind V 0 ( f n , (0, 0)) = 0 or ind V 0 ( f n , (0, 0)) = reg 1 (n). Thus we obtain two maps, which indices may be expressed in dependence on a parameter α as ind V 0 ( f n , (0, 0) Finally, substituting obtained values of indices to (3.13), we get where α ∈ {0, 1}. In this way we constructed two kinds of maps, one of the type (F e ∂ 0 ) for α = 0 and the second of the type (F e ∂ 1 ) for α = 1, which completes the construction in the case (A).
We will realize the pairs of maps given in B o , C o and D o . We represent R m+1 in the same way as in the case (A), we can find a map of a pair ( f,f ) whose indices have basic sequences reg k with arbitrary coefficients a k , b k for k ≥ 3, k ∈ L( m−1 2 ). We have still some room, namely V × R + = R × R + , where we can realize reg 1 with the coefficient a 1 ∈ {−1, 0, 1} and arbitrary b 1 , which will complete the proof of part (B). The last realization can be done, due to the following lemma. Proof Let us define three types of smooth flows defined on R 2 0 = R with 0 as the only stationary point, and consider indices of its discretizationf .
(3) The flow such that 0 is a stationary point removable by any small perturbation, then ind(f n , 0) = 0. Now, let us consider planar flows h p : R × R + × R → R × R + with phase portraits consisting of 2| p| hyperbolic regions for p < 0, 2 p elliptic regions for p > 0 (an example of such flow is given in Fig. 1). The flow may be taken as smooth as we like, in particular with the discretization being C 1 map (see [25]).
We take its time-one map and denote it by h p (1). Next, we consider f p , the self-map of R 2 equal to h p (1) composed with retraction ρ : R 2 → R 2 + . We calculate ind( f p , (0, 0)) as the number of revolutions of the vector connecting z with f (z) for z ∈ C running along a small circle C centered at (0, 0). Notice that the fact that  , (0, 0)). Now in each of the three cases (1)-(3) we will define an extension off with arbitrary integer index.
(1) We take the extension equal to h p for p = 0 and a 2-dimensional sink on R × R + in case p = 0. Then we get (2) We take the extension equal to h p for p = 0 and a 2-dimensional source on R × R + in case p = 0. Then we get (3) Consider the situation presented in Fig. 2. We take the extension equal to h p (for p = 0) in the area denoted by G and one hyperbolic sector in the area denoted by H . For p = 0 we take 2-dimensional sink on R × R + . Then we get (C) m ≥ 2s + 2. We will realize the pairs of maps given in (A o ) and (B e ).
We have a similar decomposition R m+1 Again, we may find a map of a pair ( f,f ) such that on the subspace V × R + we can realize indices having basic sequences reg k with arbitrary coefficients a k , b k for k ≥ 3 and k ∈ L( m−2 2 ) for m even or k ∈ L( m−3 2 ) for k odd. Furthermore, on V × R + = R 2 × R + we can realize sequences a 1 reg 1 forf and b 1 reg 1 for f with arbitrary a 1 , b 1 (cf. [25]).

Reducing r-periodic points to one point
In this section we illustrate the whole theory, finding necessary and sufficient conditions to reduce r -periodic points in the C 1 -homotopy class just to one point (Theorem 4.1) and applying the result to self-maps of 6-dimensional closed ball (Theorem 4.2). In the case (A) β = 0 is equivalent to L(f n ) = 0 for all r |n. On the other hand, α = 1 is equivalent to the fact that (L( f n )) n is itself a D D m+1 (1|r ) sequence, which gives the condition (1).
The condition (B) may be expressed in the following form: for n|r L(f n ) =c 1 (n) and L( f n ) = c 1 (n), (4.1) where (c 1 ,c 1 ) is a ∂ D D m (1|r ) pair of sequences, so (B) is equivalent to (2). Proof As m = 5 is odd here, we get that L(f n ) = 1 − D n . Furthermore L( f n ) = 1, since B 6 is contractible. First, notice that (1) of Theorem 4.1 is equivalent to D = 1, because L( f n ) = reg 1 (n) is a D D m+1 (1|r ) sequence for any m and r .
Next, we show that for the rest of cases (a), (b), (c) listed in the thesis of Theorem 4.2 (i.e. for D = 1) the condition (2) of Theorem 4.1 is satisfied i.e. that (reg 1 (n), 1 − D n ) n|r is a ∂ D D 5 (1|r ) pair of sequences.

Case (b).
If r is a prime number, then (L( f n ), L(f n )) n|r = (reg 1 (n), a 1 reg 1 (n) + a p reg p (n)) n|r (similarly for r = 1, with a p = 0), so it is again a ∂ D D 5 (1|r ) pair of sequences of the type (α).
Case (c). Assume D = 2, then a 1 = L( f ) = 1 − D = −1. Thus (L( f n ), L(f n )) n|r = (reg 1 (n), −reg 1 (n) + a p reg p (n) + a q reg q (n) + a pq reg pq (n)) n|r if r = pq, (reg 1 (n), −reg 1 (n) + a p reg p (n) + a p 2 reg p 2 (n)) n|r if r = p 2 . Take |D| ≥ 2 and assume that r has at least two nontrivial divisors, s and t. It is known ( [26], Theorem 1.2) that for |D| ≥ 2 each a n , the nth coefficient of the periodic expansion of {L(f n )} ∞ n=1 , is non-zero. Taking into account that a 1 = 1 − D we observe that in the periodic expansion of {L(f n )} n|r must appear the following non-zero terms: (1 − D)reg 1 (n) + a s reg s (n) + a t reg t (n) + a [s,t] reg [s,t] (n).
Thus (L( f n ), L(f n )) n|r could be a ∂ D D 5 (1|r ) pair of sequences only if it is of the type (β) with δ = 1 − D ∈ {−1, 0, 1}, thus for D = 2 and only if there are no more then two different non-trivial divisors.