Download Minimal number of periodic points for C self

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Forum
Mathematicum
Forum Math. (2008), 1–19
DOI 10.1515/FORUM.2008.
© de Gruyter 2008
Minimal number of periodic points for
compact simply-connected manifolds
C1
self-maps of
Grzegorz Graff, Jerzy Jezierski
(Communicated by Michael Brin)
Abstract. Let f be a self-map of a smooth compact connected and simply-connected manifold of
dimension m ≥ 3, r a fixed natural number. In this paper we define a topological invariant Drm[f ] which
is the best lower bound for the number of r-periodic points for all C 1 maps homotopic to f . In case
m = 3 we give the formula for Dr3 [f ] and calculate it for self-maps of S 2 × I.
2000 Mathematics Subject Classification: 37C25, 55M20; 37C05.
1 Introduction
In 1983 Albrecht Dold [D] formulated the necessary conditions that must be fulfilled by
a sequence of indices of iterations. He considered a map f : U → X , where U ⊂ X is an
open subset of an ENR, satisfying the following fixed-compactness condition. We denote
inductively U0 = U , Un+1 = f −1 (Un ) i.e. Un = {x ∈ U : x, f (x), ..., f n (x) ∈ U }. We assume
that the fixed point set Fix(f n ) = {x ∈ Un : f n (x) = x} is compact for each n ∈ N. In
such situation the fixed point index ind(f n ) = ind(f n , Un ) is well-defined. Dold proved that
the sequence of fixed point indices {ind(f n )}∞
n=1 must satisfy the following congruences,
(called Dold relations): ∑k |n μ (n/k)ind(f k ) ≡ 0 (mod n) for each n ∈ N, where μ denotes
the Möbius function.
The classical issue in periodic point theory is the question of what is the minimal number
of periodic points in the homotopy class of a given f , a self-map of a compact manifold.
It appears that this problem is strictly related to the question what are possible forms of
local indices (at an isolated fixed point) of iterations of maps homotopic to f . Namely, let
f : U → R m be a continuous map defined on a neighborhood of the point 0 ∈ R m . We define
the iterations f n as above, and assume that 0 is the only fixed point of f n for each n ∈ N
in some small neighborhood Vn ⊂ Un of 0 and consider ind(f n , 0) = ind(f n , 0, Vn). Which
n
∞
integer sequences {cn }∞
n=1 can be obtained as {ind(f , 0)}n=1? It is shown in [BaBo] and
[GNP2] that once the Dold congruences are satisfied, there is a continuous map f realizing
an arbitrary sequence {cn }∞
n=1 , for m ≥ 2, as a sequence of indices at a single fixed point.
This fact has a strong impact on the possibility of cancelling periodic points in the
given homotopy class. If f : M → M is a self-map of a compact connected and simplyResearch supported by KBN grant No 1 P03A 03929.
Manuscript nr.:
08/1-B30
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G, Graff, J. Jezierski
connected manifold of dimension ≥ 3, then it is homotopic to a continuous map g with
Fix(g r ) consisting of a single point, where r is a prescribed natural number ([Je2] Thm.
5.1). In the smooth category however, the situation is much different. (We say that f is
smooth in the meaning that it is C 1 .) In 1974 Shub and Sullivan [SS] proved that fixed point
indices of iterations of a smooth map, at an isolated fixed point, are periodic. This is a very
strong restriction making the theory quite different than in the continuous case. Later, in
1981, Chow, Mallet-Paret and Yorke [CMPY] showed that in the smooth case a sequence of
indices may be represented as the sums of some elementary sequences whose coefficients
depend on the derivative of a map at the fixed point. This leads to a system of conditions
1
that any sequence {ind(f n , 0)}∞
n=1, for f being C map, must satisfy. It was conjectured in
[CMPY] that these conditions are also sufficient to realize given sequence as the sequence of
indices of iterations of some C 1 map. The conjecture was answered positively by Babenko
and Bogatyi [BaBo] in dimension 2 and by Graff and Nowak-Przygodzki in dimension 3
[GNP1].
Basing on the results mentioned above and methods of Nielsen fixed point theory developed in [Je2] (cf. also [HK1], [HK2]), we are able to deal with the smooth case.
First of all we show in this paper that, unlike in the continuous case, Fix(f r ) cannot be
reduced to a point by a smooth map g, homotopic to the given f , where f : M → M , M
satisfies the same assumptions as in the continuous case, r is a fixed natural number.
Next, we express in terms of local indices of iterations the least cardinality of Fix(g r )
which can be obtained by a smooth map g in the homotopy class of f (cf. Theorem 3.8).
We reduce this problem to the decomposition of the sequence of Lefschetz numbers L(f n ),
where n|r, into the sum of sequences each of which can be locally realized as a sequence
of indices at an isolated periodic orbit for some C 1 map. As a result we get the topological
invariant Drm [f ] which is the best lower bound for the number of r-periodic points for all
C 1 maps homotopic to f . Let us notice that Drm [f ] may be interpreted purely in the terms of
smooth category, namely we may assume that f is smooth and approximate the homotopy
which joins f with g by a smooth one. Then Drm [f ] gives the minimal number of periodic
points in the smooth homotopy class of the given f .
In the second part of the paper, we use the classification of sequences of indices of
iterations of smooth maps in dimension 3, described in [GNP1], to give the explicit formulae
for Dr3 [f ]. We also calculate Dr3 [f ] for a 3-manifold with simple homology groups, namely
S2 × I .
Let us remark that in the non simply-connected case the problem of finding minimal
number of periodic points in smooth category is much more complicated. In the possible
version of the non simply-connected counterpart of Theorem 3.8, which we intend to prove in
the forthcoming paper, the restrictions for local indices for a smooth map must be combined
with topological properties of a space expressed in terms of orbits of Reidemeister classes
of the iterated map.
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Minimal number of periodic points for C 1 maps
3
2 Theorem of Chow, Mallet-Paret and Yorke and DD m sequences
Definition 2.1. Let f : U → R m , where U is an open subset of R m , be a map such that 0
is an isolated fixed point for each iteration of f . For each n we define integers (called local
Dold coefficients) in (f , 0) by the following equality:
in (f , 0) = ∑ μ (n/k)ind(f k , 0),
k |n
where μ is the Möbius function, i.e. μ : N → Z is defined by three properties: μ (1) = 1,
μ (k) = (−1)r if k is a product of r different primes, μ (k) = 0 otherwise.
2
We may write down Dold relations, mentioned in the introduction, as: for each natural n
(1)
in (f , 0) ≡ 0 (mod n).
Dold relations make it possible to decompose a sequence of indices of iterations into a sum
of elementary sequences. This decomposition is called a periodic expansion.
Definition 2.2. For a given k we define the basic sequence (basic expansion):
k if k|n,
regk (n) =
0 if k |n.
2
Notice that each basic sequence regk is a periodic sequence of the form:
(0, . . ., 0, k, 0, . . ., 0, k, . . ., . . .), where the non-zero entries appear only for indices of the
sequence which are multiples of k.
Theorem 2.3 (cf. [JM]). Any sequence ψ : N → C can be written uniquely in the following
form of a periodic expansion:
∞
ψ (n) = ∑ ak (ψ )regk (n),
k =1
where an (ψ ) = in (ψ )/n = 1n ∑k |n μ (k)ψ (n/k).
Moreover, ψ takes integer values and satisfies Dold relations iff an (ψ ) ∈ Z for every
n ∈ N.
Definition 2.4. Assume that a periodic expansion for a sequence {ψ (n)}∞
n=1 is given. Let
B(ψ ) = {n ∈ N : an = 0}. The set B(ψ ) will be called the basic set for a periodic expansion
of ψ .
2
Now consider f : U → R m , where U is an open subset of R m containing 0, a C 1 map with
0 an isolated fixed point for each iterations. Let us denote by Δ the set of degrees of all
primitive roots of unity which are contained in σ (Df (0)), the spectrum of derivative at 0.
1
Chow, Mallet-Paret and Yorke showed that the basic set B({ind(f n , 0)}∞
n=1), for a C map f ,
is finite and depends on the set Δ.
We denote by σ+ the number of real eigenvalues of Df (0) greater than 1, σ− the number
of real eigenvalues of Df (0) less than −1, in both cases counting with multiplicity. Let
O = {LCM(K ) : K ⊂ Δ},
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G, Graff, J. Jezierski
where by LCM(K ) we mean the least common multiple of all elements in K with the
convention that LCM(0)
/ = 1. Let Oodd = O ∩ {n ∈ N : n = 2k − 1, k = 1, 2, . . .}.
We present below the result of Chow, Mallet-Paret and Yorke ([CMPY]) expressed in the
language of periodic expansions (cf. [GNP1], [MP], [JM]).
Theorem 2.5. Let U ⊂ R m be an open neighborhood of 0, f : U → R m be a C 1 map having
0 as an isolated fixed point for each iteration. Then:
(∗)
ind(f n , 0) = ∑ ak regk (n), where
k ∈O
O=
O
if σ− is even,
O ∪ 2Oodd if σ− is odd.
(∗∗)
If σ− is odd and k ∈ 2Oodd \ O, then ak = −ak /2 .
(∗ ∗ ∗) The bounds for the coefficients a1 and a2 are the following:
(1) a1 = (−1)σ+ if 1 ∈ σ (Df (0)).
(2) a1 ∈ {−1, 0, 1} if 1 is an eigenvalue of Df (0) with multiplicity 1.
(3) a2 ∈ {0, (−1)σ++1 } if 1 ∈ σ (Df (0)) and −1 is the eigenvalue of Df (0) with multiplicity 1.
2
Remark 2.6. Let us notice that there are three different types of restrictions described in the
theorem. Namely: bounds for the set O (which contains the basic set B), relations between
coefficients ak and ak /2 and conditions for a1 and a2 .
Theorem 2.5 gives the description of indices for a map f for which 0 is an isolated fixed
point for all iterations. In the further considerations we take some fixed r and consider
more general situation, in which 0 is an isolated fixed point only for n|r. Nevertheless, for a
finite sequence {ind(f n , 0)}n|r the formula and conditions for indices of iterations given in
Theorem 2.5 remain true.
Remark 2.7. Let r be a fixed natural number, U ⊂ R m be an open neighborhood of 0,
f : U → R m be a C 1 map having 0 as an isolated fixed point for f n , where n|r. Then, for
n|r, ind(f n , 0) = ∑k |r,k ∈O ak regk (n) and the coefficients ak (if they appear, so if k|r) satisfy
the conditions (∗) and (∗∗) of Theorem 2.5.
Notice that by the assumption {ind(f n , 0)}n|r is well-defined. Remark 2.7 may be relatively
easily obtained by analyzing the original proof of Theorem 2.5. Namely, it results from two
facts. First, so-called Virtual Period Proposition (Prop. 3.2 [CMPY]) remains true without
the assumption that 0 is an isolated fixed point. Second, if we choose as in [CMPY] the
approximation of f by a map with simple fixed points, the contribution to the sequence
{ind(f n , 0)}n|r comes only from such n-orbits, for which n|r (by an n-orbit we understand
an orbit whose elements are periodic points with the minimal period equal to n).
Definition 2.8. In the rest of the paper we will use the following notation (cf. Definition 2.4):
B(f , x0 ) = B({ind(f n , x0 )}∞
n=1 ),
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Minimal number of periodic points for C 1 maps
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Br (f , x0 ) = B({ind(f n , x0 )}n|r ),
B(f ) = B({L(f n )}∞
n=1 ),
Br (f ) = B({L(f n )}n|r .
m
Definition 2.9. A sequence of integers {cn }∞
n=1 is called DD (p) sequence if there is an
m
1
isolated p-orbit P, a neighborhood U ⊂ R of P and a C map φ : U → R m such that
cn = ind(φ n , P) (notice that cn = 0 if n is not a multiple of p). The finite sequence {cn }n|r
will be called DDm (p|r) sequence if this equality holds for n|r, where r is fixed. In other
words, DDm (p) sequence is a sequence that can be realized as a sequence of indices of
iterations on an isolated p-orbit for some smooth map φ .
The following lemma gives the description of DDm (p) sequences.
m
Lemma 2.10. A sequence {cn }∞
n=1 is a DD (1) sequence if and only if
0
for p |n,
c̃n =
p cn/p for p|n
m
is a DDm(p) sequence. We will say that a DDm (p) sequence {c̃n }∞
n=1 comes from a DD (1)
∞
sequence {cn }n=1 .
Proof. “⇒” Let φ : R m → R m be a smooth map with the isolated fixed point x1 , such that
ind(φ k , x1 ) = ck . We fix different points x1, ..., xp ∈ R m . Let us choose small enough mutually
disjoint neighborhoods Ui xi (for i = 1, ..., p), and diffeomorphisms φi : Ui → Ui+1 (for
i = 1, ..., p − 1). At last we define φp : Up → U1 by the formula:
(2)
−1
φp (x) = φ (φ1−1(φ2−1(· · · φp−1
(x))))
for x near xp . We define Φ :
(3)
p
i=1 Ui
→ R m putting
Φ(x) = φi (x) for x ∈ Ui .
Then
(4)
Φp (x) = φp (φp−1 (· · · φ1 (x))) = φ (x)
in a neighborhood of x1 . What is more,
ind(Φkp, x1 ) = ind(φ k , x1 ) = ck .
Since the indices in all points of the orbit are the same,
ind(Φkp, {x1 , ..., xp}) = p ck .
Finally, for each natural n ind(Φn , {x1 , ..., xp}) is equal either to p · cn/p if p|n, or to zero
otherwise, which proves that c̃n is a DDm (p) sequence.
p
1
“⇐” If φ is a map from Definition 2.9 which realizes {c̃n }∞
n=1 , then φ is the C map
m
∞
which gives the needed DD (1) sequence {cn }n=1 .
2
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G, Graff, J. Jezierski
Remark 2.11. Let us notice that the same construction, as in the proof of “⇒”, can be done
for an arbitrary orbit of points in a manifold.
Remark 2.12. Lemma 2.10 gives an easy procedure which enables one to obtain all forms
of DDm (p) sequences once we know the forms of DDm (1) sequences. Namely by the relation
between respective basic sets: B(c̃n ) = pB(cn ) = {pk : k ∈ B(cn )}, we find that in order to
get any DDm (p) sequence it is enough to replace all basic sequences ak regk by ak regpk in
the periodic expansion of some DDm (1) sequence.
3 Minimal number of periodic points
Let f : X → X be a self-map of a topological space and r ∈ N a fixed number. What is the
minimal number of periodic points #Fix(g r ), where g is a map homotopic to f ? The Nielsen
periodic point theory gives an invariant NFr (f ), introduced by Boju Jiang in [Ji], which is a
homotopy invariant and the lower bound for the number of periodic points:
NFr (f ) ≤ #Fix(g r )
for each g homotopic to f . In [Je1] it is shown that, for X – a compact PL-manifold of
dimension not less than 3, NFr (f ) is the best such lower bound, i.e. for each f there is a map
g homotopic to f such that #Fix(g r ) = NFr (f ).
In this paper we confine ourselves to the simply-connected case. Then Nielsen theory is
trivial and NFr (f ) is equal to 0 or 1. In such a case Theorem 5.1 in [Je2] gives:
Theorem 3.1. Any continuous self-map f of a compact connected simply-connected PLmanifold of dimension ≥ 3 is homotopic to a map g such that
0/ if L(f n ) = 0 for all n|r,
r
Fix(g ) =
{∗}
otherwise,
where {∗} - Fix(g) denotes the set which consists of one point.
2
Now we study a similar question in the smooth category. We are given a manifold M . We
will assume through the rest of the paper that M is a smooth compact connected and simplyconnected manifold. Let f be a continuous self-map of M and r ∈ N a given number. If the
boundary ∂ M = 0,
/ then we assume that all considered maps have no periodic points on the
boundary. The question we ask is: what is the minimal number of periodic points #Fix(g r ),
where g is a smooth map homotopic to f ?
The results presented in the previous section show that here the situation is drastically
different. It is easy to notice that a self-map of a simply-connected manifold is not homotopic,
in general, to a smooth map g with Fix(g r ) = {∗}, a point. In fact:
Remark 3.2. If a self-map f of M is homotopic to a smooth map g with Fix(g r ) = {∗}, then
the sequence of Lefschetz numbers {(L(f n )}n|r is a DDm(1|r) sequence.
Proof. For every n|r we have:
L(f n ) = L(g n ) = ind(g n ) = ind(g n , ∗).
2
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In the next section we will see that the set of periodic points can be reduced to a single point
in rather exceptional cases. On the other hand the following lemma holds:
Lemma 3.3. If f , a self-map of M , is homotopic to a smooth map g with #Fix(g r ) = N
finite, then the sequence of Lefschetz numbers of f decomposes as
L(f n ) = c1 (n) + . . .+ cs (n),
for n|r, where ci is a DDm (li |r) sequence and l1 + . . .+ ls = N .
Proof. If #Fix(g r ) = N , then
Fix(g r ) = a1 ∪ . . .∪ as,
where ai is an orbit of the length li . We get for n|r:
s
s
i=1
i=1
L(f n ) = L(g n ) = ind(g n ) = ∑ ind (g n , ai ) = ∑ ci (n),
where ci (n) = ind (g n , ai ) is a DDm (li |r) sequence, since g is smooth. It remains to notice
that l1 + . . . + ls = #a1 + . . .+ #as = #Fix(g r ) = N .
2
The above lemma suggests the following definition.
Definition 3.4. Let {ξn }n|r be a sequence of integers satisfying Dold relations. Assume that
we are able to decompose it into the sum:
ξ (n) = c1 (n) + . . .+ cs (n),
where ci is a DDm (li |r) sequence for i = 1, . . ., s. Each such decomposition determines the
number l = l1 +. . .+ls . We define the number Drm [ξ ] as the smallest l which can be obtained
in this way.
2
We will denote briefly Drm [f ] = Drm [{L(f n )}n|r ]. It is easy to see that Drm [f ] is a homotopy
invariant. On the other hand, by Kupka-Smale Theorem (cf. for example [R]) every f is
homotopic to some smooth map g which has a finite number of r-periodic points (r is fixed).
Thus, Lemma 3.3 shows that Drm[f ] is a lower bound of the number of r-periodic points for
g: #Fix(g r ) ≥ Drm [f ].
The aim of this section is to prove Theorem 3.8 which states that Drm [f ] is the best such
lower bound, provided the dimension of the manifold is not less than 3. The main idea of
the proof may be described as follows. We fix a decomposition L(f n ) = c1 (n) + · · · + cs (n)
realizing Drm [f ] i.e. the sum l = l1 + · · ·+ ls in Definition 3.4 is the smallest. Then we deform
f to a map f¯ adding an isolated orbit of points Oi satisfying
ind(f¯ n ; Oi ) = ci (n)
for each i = 1, ..., s (Creating Procedure, Theorem 3.5). It turns out that the remaining
periodic points Fix(f¯ r ) \ (O1 ∪ · · · ∪ Os ) can be removed (Canceling Lemma, Lemma 3.7).
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G, Graff, J. Jezierski
The map f1 , obtained after this cancelation, satisfies
#Fix(f1r ) = #(O1 ∪ · · ·∪ Os ) = #O1 + · · · + #Os = l1 + · · · + ls = Drm[f ],
hence f1 realizes the lower bound of the number of periodic points Drm [f ].
The following two results i.e. Creating Procedure and Cancelling Lemma, proved in
[Je2] (valid not only for smooth but also PL-manifold M ) will be crucial in the proof of
Theorem 3.8.
The Creating Procedure enables one to create an additional orbit in the homotopy class
of f , by a use of a homotopy ft which is constant near periodic points of f (up to the given
period r) and such that f1r near the created orbit may be given by an arbitrarily prescribed
formula.
Theorem 3.5. (Creating Procedure, [Je2] Theorem 3.3) Given numbers p, r ∈ N, p|r and
a map f : M → M , where dimM ≥ 3, such that Fix(f r ) is finite and a point x0 ∈ Fix(f r ).
Then there is a homotopy {ft }0≤t≤1 satisfying:
1. f0 = f .
2. {ft } is constant in a neighborhood of Fix(f r ).
p
3. f1 (x0 ) = x0 and f1i (x0 ) = x0 for i = 1, . . ., p − 1.
p
4. The map f1 is given near x0 by an arbitrarily prescribed formula φ with the property
r
φ p (x) = x ⇐⇒ x = x0 .
p−1
2
5. The orbit x0 , f1(x0 ), . . ., f1 (x0 ) is isolated in Fix(f1r ).
Proposition 3.6. The map f1 in Theorem 3.5 can realize an arbitrarily prescribed DDm (p|r)
sequence c̃n on the created p-orbit O, i.e f1 can be smooth in a neighborhood of O and
c̃n = ind(f1n , O).
Proof. The statement may be obtained by a slight modification of the proof of Creating
Procedure from [Je2]. Namely, it results from the fact that f1 may be chosen in such a way
that f1 (x) = Φ(x) near O, where Φ has the form from the formula (3) in Lemma 2.10, i.e. φi
are some fixed diffeomorphisms and φ is an arbitrary smooth map that realizes any DDm(1)
sequence (for more details cf. [Je2]).
Now we take as φ a map that realizes DDm (1| pr ) sequence cn = 1p c̃np and by Lemma 2.10
obtain that:
ind(f1n , O) = ind(Φn , O) = c̃n .
2
The next lemma makes possible to cancel subsets of periodic points which have indices of
iterations equal to zero.
Lemma 3.7. (Cancelling Lemma, [Je2] Lemma 5.2) Let f be a continuous self-map of M .
Suppose that S ⊂ Fix(f r ) satisfies:
1. S is finite and f -invariant i.e. f (S) = S.
2. Fix(f r ) \ S is compact.
3. ind(f n , Fix(f n ) \ S) = 0 for all n|r.
Then there is a homotopy ft , starting from f0 = f , constant near S and such that Fix(f1r ) = S.
2
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Now we are in a position to prove the main result of this section.
Theorem 3.8. (Main Theorem). Let M be a smooth compact connected and simply-connected manifold of dimension m ≥ 3 and r ∈ N a fixed number, then
min{#Fix(g r ) : g is C1 and is homotopic to f } = Drm [f ].
In other words Drm [f ] is the best lower bound for #Fix(g r ) for smooth g in the homotopy
class of f .
2
Proof. The fact that the minimum is greater or equal to Drm [f ] follows from Lemma 3.3. Let
us decompose
s
(5)
L(f n ) = ∑ ci (n),
i=1
where n|r, ci is a DDm (li |r) sequence and the sum ∑si=1 li is minimal i.e. Drm [f ] = ∑si=1 li .
We have to show that Drm [f ] is reachable by some map in the homotopy class of f i.e. find a
smooth map f1 homotopic to f with Fix(f1r ) = ∑si=1 li .
First we use Creating Procedure and Proposition 3.6, deforming f to f¯ in such a way
that we subordinate to every i ∈ {1, . . ., s} an isolated orbit Oi of the length li for which
ind(f¯ n , Oi ) = ci (n). Each ci is a DDm(li |r) sequence, so by Proposition 3.6 f¯ is smooth near
the created orbits.
If the boundary of M is nonempty we take Oi inside M . We denote S = si=1 Oi . We will
show that there is a deformation of f¯ , rel. a neighborhood of S, to a smooth map f1 satisfying
Fix(f1r ) = S. Then f1 is the desired map, since
s
#Fix(f1r ) = #S = ∑ li = Drm [f ].
i=1
It remains to perform the above deformation of f¯ . We will apply Lemma 3.7. We check that
the assumptions of this lemma are satisfied. S, as the sum of orbits, is f¯ -invariant. Since S
is isolated, Fix(f¯ r ) \ S is compact. Now we notice that for each n|r:
ind(f¯ n ) = ind (f¯ n , Fix(f¯ n ) \ S) + ind(f¯ n , S)
s
= ind (f¯ n , Fix(f¯ n ) \ S) + ∑ ind (f¯ n , Oi )
i=1
s
= ind (f¯ n , Fix(f¯ n ) \ S) + ∑ ci (n) = ind (f¯ n , Fix(f¯ n ) \ S) + ind(f n ),
i=1
where the last equality results from the equation (5) and from the fact that Lefschetz number
and global index coincide.
Thus
ind (f¯ n ) = ind (f¯ n , Fix(f¯ n ) \ S) + ind(f n ).
Since ind(f¯ n ) = ind(f n ) = L(f n ), ind (f¯ n , Fix(f¯ n ) \ S) = 0, so the third assumption is also
satisfied. The map f1 is smooth in some neighborhood W of S and f1r has no fixed points
outside W . Thus, if f1 is not smooth as the global map, we may easily approximate it by a
smooth map constantly equal to f1 on W without adding any new r-periodic points in the
compact set M \ W .
2
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G, Graff, J. Jezierski
Remark 3.9. Let us consider the set
MPr (f ) = min{#Pr (g) : g is homotopic to f },
where Pr (g) denotes the set of periodic points of g with the minimal period r. In the simplyconnected case we have, by Remark 3.1, that MPr (f ) = 0 for r > 1. If we define its smooth
counterpart:
MPsr (f ) = min{#Pr (g) : g is smooth and homotopic to f },
then it follows from the proof of Theorem 3.8 that the same statement holds: MPsr (f ) = 0
for r > 1. Namely, we may find smooth g homotopic to f such that all its r-periodic points
are fixed points (which however, does not imply that Fix(g r ) = Drm [f ]).
The problem of determining the least number of Fix(f r ), i.e. Drm [f ], reduces to the decomposition of the sequence of Lefschetz numbers {L(f n )}n|r into the sum of some elementary
sequences i.e. DDm (p|r) sequences. This becomes a combinatorial task, once we know what
they are. In the next section we will use the complete description of DD3 (1) sequences from
[GNP1] to give explicit formulae for the least number of periodic points in dimension 3.
4 3-dimensional case
First of all we remind the reader of the result from [GNP1] (cf. Theorem 4.1 below) which
describes all DD3 (1) sequences (and thus, by Lemma 2.10, all DD3 (p) sequences).
Theorem 4.1. There are seven kinds of DD3 (1) sequences:
(A) cA (n) = a1 reg1 (n) + a2 reg2 (n),
(B) cB (n) = reg1 (n) + ad regd (n),
(C) cC (n) = −reg1 (n) + ad regd (n),
(D) cD (n) = ad regd (n),
(E) cE (n) = reg1 (n) − reg2 (n) + ad regd (n),
(F) cF (n) = reg1 (n) + ad regd (n) + a2d reg2d (n), where d is odd,
(G) cG (n) = reg1 (n) − reg2 (n) + ad regd (n) + a2d reg2d (n), where d is odd.
In all cases d ≥ 3 and ai ∈ Z.
Proposition 4.2. Each DD3 (p|r) sequence is also a DDm (p|r) sequence (for m ≥ 3), as a
consequence Dr3 [ξ ] ≥ Drm [ξ ].
Proof. Let {cn }n|r be a DD3 (p|r) sequence, and let f be a map whose existence is guaranteed
by Definition 2.9. Then {cn }n|r may be realized as the sequence of indices of the map f × g,
where g : R m−3 → R m−3 is a sink type map for which ind(g n , 0) = 1, for example g(x) = 12 x.
2
Lemma 4.3. ([BaBo] Corollary 3.6) Let f be a C 1 self-map of a smooth m-dimensional
manifold and let x0 be a fixed point of f . Then the basic set B(f , x0 ) for the periodic expansion
m+1
of indices of iterations at x0 satisfies: #B(f , x0 ) ≤ 2[ 2 ] .
2
The same inequality holds for each orbit of periodic points.
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Minimal number of periodic points for C 1 maps
11
Corollary 4.4. Let O(p) = {x1 , . . ., xp } be a p-orbit of a smooth map f : M → M (dim M = m).
By B(f , O(p) ) we denote B({ind(f n , O(p) )}∞
n=1 ) (cf. Definitions 2.4 and 2.8). Let us consider
the map f p . Notice that
ind((f p )n , x1 ) =
∑
ak regk (n)
k ∈B(f p ,x1 )
implies
ind(f n , O(p) ) =
∑
as/p regs (n)
s∈B(f ,O(p) )
and B(f , O(p) ) = pB(f p , x1 ) = {pk : k ∈ B(f p , x1 )} (cf. [BaBo]).Thus #B(f , O(p) ) = #B(f p , x1 ).
As a consequence by Lemma 4.3 applied to f p we get:
#B(f , O(p) ) ≤ 2[
m+1
2
].
Theorem 4.5. Let f be a self-map of a smooth compact simply-connected m-dimensional
manifold. Then:
Drm [f ] ≥
#Br (f )
2[
.
m+1 ]
2
Proof. Each basic sequence regk which appears with a non-zero coefficient in the periodic
expansion of Lefschetz numbers of f must also appear in expansions of some periodic orbits
of f , thus:
Br (f ) ⊂
Br (f , O(p) ).
{O(p) :p|r}
As a consequence, by Corollary 4.4:
m+1
#Br (f ) ≤ ∑ #Orbp (f ) · 2[ 2 ] ,
(6)
p|r
where Orbp (f ) denotes the set of all p-orbits of f .
Assume now that g is a smooth map homotopic to f for which #Fix(g r ) is minimal. We
get: Br (f ) = Br (g), next we apply the inequality (6) for g and obtain:
#Br (f )
m+1
2[ 2 ]
=
#Br (g)
2[
m+1 ]
2
) ≤ ∑ #Orbp (g) ≤ ∑ #Orbp (g) · p = Drm [f ].
p|r
p|r
This ends the proof.
2
Corollary 4.4 leads to the following observation:
Remark 4.6. In the definition of Drm [f ] it is enough to consider only DDm (p|r) sequences
m+1
such that p < 2[ 2 ] . In order to justify this fact, consider {cn }n|r a DDm (p|r) sequence with
p ≥ 2[
m+1 ]
2
, which gives the contribution p in Definition 3.4. By Theorem 4.1 and Proposition
“fm˙08˙01” — 2008/6/24 — 14:42 — page 12 — #12
12
G, Graff, J. Jezierski
4.2 it can be replaced by #B(f , O(p) ) expressions of the type (D), so DDm (1|r) sequences,
each of which gives the contribution equal to 1. Finally, as by Corollary 4.4 these sequences
m+1
in sum give the contribution equal at most 2[ 2 ] , we can always replace {cn }n|r by them
during the calculation of Drm[f ].
Notice that, in particular for m = 3, we have p < 22 , hence p ≤ 3.
2
Definition 4.7. Let us list three DD3 (2) sequences which come from DD3 (1) sequences of
the form (E), (F) and (G):
(E’) cE (n) = reg2 (n) − reg4 (n) + a2d reg2d (n), where d ≥ 3,
(F’) cF (n) = reg2 (n) + a2d reg2d (n) + a4d reg4d (n), where d ≥ 3 is odd,
(G’) cG (n) = reg2 (n) − reg4 (n) + a2d reg2d (n) + a4d reg4d (n), where d ≥ 3 is odd.
2
Let us emphasize that the contribution to Dr3 [f ] of each sequence (A)-(G) is 1, while of each
(E’)-(G’) is 2.
Lemma 4.8. Let f : M → M be a C 1 map, dim M = 3, then in the definition of Dr3 [f ] it is
enough to consider only DD3 (1|r) sequences i.e. sequences which, for n|r, are of the forms
(A)–(G); and DD3 (2|r) sequences of the forms (E’), (F’) and (G’).
Proof. By Remark 4.6 it is enough to use only DD3 (p|r) sequences with p ≤ 3. Let us
now consider a DD3 (3|r) sequence. If it comes from one of DD3 (1|r) sequences of the
type (A) - (F), then the basic set Br (f , O) for the sequence of indices of its realization on
some 3-orbit has no more than 3 elements, thus we can replace it by at most three DD3 (1|r)
sequences of the types (D) or (A). If the DD3 (3|r) sequence comes from (G), then it has the
expansion: g(n) = reg3 (n) − reg6 (n) + ad reg3d (n) + a2d reg6d (n), where d is odd. It can be
replaced by three DD3 (1|r) sequences, namely: c1F (n) = reg1 (n) + ad reg3d (n) + a2d reg6d (n),
c2F (n) = reg1 (n) + reg3 (n) − reg6 (n), c3A (n) = −2reg1 (n).
Finally, DD3 (2|r) sequence which comes from one of the DD3 (1|r) sequences of the
types (A) - (D) may be replaced by two DD3 (1|r) sequences of the types (A) or (D).
2
4.1 Formula for D3r [f]
In this section we add the assumption that M is 3-dimensional, thus we consider f , a selfmap of M , where M is a smooth compact connected simply-connected 3-manifold. We will
give a formula for the least number of periodic points, up to r periodic, for a smooth map
homotopic to f . By Theorem 3.8 the number is equal to Dr3 [f ] = Dr3 [{L(f n )}n|r ]. We will
apply Lemma 4.8 which allows us, during the calculation of Dr3 [f ], to use only sequences
(A)–(F), (E’), (F’) and (G’).
Let r be a fixed natural number. The sequence of Lefschetz numbers {L(f n )}n|r satisfies
Dold relations, hence it can be written in the form of a periodic expansion, for n|r:
(7)
L(f n ) = b1 reg1 (n) + b2 reg2 (n) + b4reg4 (n) + ∑ bk regk (n),
k ∈G
where b1 , b2 , b4 are arbitrary integers,
G = {k ∈ N : k|r and k ∈ {1, 2, 4} and bk = 0} = Br (f ) \ {1, 2, 4}.
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Minimal number of periodic points for C 1 maps
13
Let us define H , the subset of G:
H = {k ∈ G : k is odd and bk = 0, b2k = 0}.
Definition 4.9. We will say that a sequence ψ of one of the types (A)-(G), (E’)-(G’) reduces
a basic sequence bk regk in the periodic expansion of Lefschetz numbers (7) if k ∈ Br (ψ )
and ak = bk , i.e. bk regk appears in the periodic expansion of ψ .
Theorem 4.10. If r is odd, then:
#G if |L(f )| ≤ #G,
(∗)
Dr3 [f ] =
#G + 1 otherwise.
If r is even and r > 4, then:
(∗∗)
Dr3 [f ] ∈ [#G − #H , #G − #H + 2].
Proof. Let r be odd so b2 and b4 are not needed and the sequences of Definition 4.7 are not
relevant. Then, seeking for the exact value of Dr3 [f ], it is enough to use only the sequences
ci (n) of the types (A)-(D). Each of the sequences (B), (C), (D) reduces exactly one basic
sequence bk regk in the periodic expansion of {L(f n )}n|r for k ≥ 3. Thus we need at least #G
basic expansions to get the equality (7):
(8)
L(f n ) = c1 (n) + ... + c#G (n)
for n|r, n > 1.
On the other hand notice that to get the above equality for n > 1 we can use tB , tC , tD sequences
of the types (B), (C), (D) respectively, where tB , tC , tD are prescribed non-negative integers
satisfying tB + tC + tD = #G. Since the contribution of each of these sequences to b1 = L(f )
is +1, −1, 0 respectively, we may choose them to get the equality (8) also for n = 1 if and
only if −#G ≤ L(f ) ≤ #G. Otherwise, i.e. for |L(f )| > #G, we have to use one additional
sequence of the type (A) with a2 = 0 to get the equality for n = 1. This ends the proof of
part (∗).
Now we assume that r is even. We need at least #G − #H basic expansions to get the
equality (7) for r = 2, 4. In fact, only sequences (F) and (G) reduce two regk ’s, k = 1, 2, 4,
and we may use them in such a way at most #H times ((F’) and (G’) also reduces two regk ’s
but each of them is counted twice). To reduce the remaining regk ’s we will need #G − 2#H
of them. Thus Dr3 (f ) ≥ #H + (#G − 2#H ) = #G − #H . On the other hand notice that we may
always realize {L(f n )}n|r by #H sequences (F), #G − 2#H sequences (D), one sequence (A)
with the coefficients a1 = b1 − #H , a2 = b2 and one sequence (D) with d = 4 and a4 = b4 .
Thus Dr3 [f ] ≤ #G − #H + 2.
2
Now we consider some particular cases, which are not included in Theorem 4.10
Proposition 4.11. 1. If {L(f n )}n|r is constantly equal to 0, then Dr3 [f ] = 0.
Assume that {L(f n )}n|r is not constantly equal to 0, then
“fm˙08˙01” — 2008/6/24 — 14:42 — page 14 — #14
14
2. D23 (f ) = 1.
3. D43 (f ) =
G, Graff, J. Jezierski
1 if (a) or (b) or (c) is satisfied,
2
otherwise,
where
(a) L(f ) = 1 and L(f 2 ) = −1,
(b) L(f 4 ) = L(f 2 ),
(c) L(f 2 ) = L(f ) and L(f ) ∈ {−1, 0, 1}.
Proof. 1. L(f n ) ≡ 0 iff the sequence {L(f n )}n|r is the sum of zero basic sequences iff
Dr3 [f ] = 0.
2. Each nonzero sequence of two elements b1 , b2 is of the type (A).
2
)
3. Let us find when the sequence of three elements b1 = L(f ), b2 = L(f )−L(f
, b4 =
2
L(f 4 )−L(f 2 )
4
is one of the types (A)-(G):
(A) ⇐⇒ b4 = 0, which gives the condition (b).
(B), (C) or (D) ⇐⇒ (|b1| ≤ 1 and b2 = 0) ⇐⇒ (|L(f )| ≤ 1 and L(f ) = L(f 2 )) which
gives the condition (c).
(E) ⇐⇒ (b1 = 1 and b2 = −1) ⇐⇒ (L(f ) = 1 and L(f 2 ) = −1, which leads to condition (a).
(F) and (G) are reduced to (A).
It remains to notice that each sequence b1 , b2, b4 is the sum of two sequences of the types
(A), (D) which implies D43 (f ) = 2 in the remaining cases.
2
Corollary 4.12. Let r = 1. As D13 [f ] ≤ 1, each map f is homotopic to a smooth map with (at
most) one fixed point, in other words MPs1 (f ) ≤ 1 (compare Remarks 3.9 and 3.2 for r = 1).
Let us remind the reader that by ζ (r) we denote the number of all divisors of r.
Corollary 4.13. Let r ∈ N. Each map f is homotopic to a smooth map g with
#Fix(g r ) ≤ ζ (r).
#Br (f )
m+1
2[ 2 ]
≤
Proof. We begin with the proof of the second inequality. By definition, G = Br (f ) \ {1, 2, 4},
on the other hand Br (f ) is the number of basic sequences bk regk in the expansion of Lefschetz
numbers and thus #Br (f ) ≤ ζ (r). By Proposition 4.2 and Theorem 4.10 Drm [f ] ≤ Dr3 [f ] ≤
ζ (r). Finally we apply Theorem 3.8 and get the desired inequality. The first inequality in
Corollary results from Theorem 4.5 and Theorem 3.8.
2
Corollary 4.14. Let r = q be a prime number. Then the map f is homotopic to a smooth
map g with #Fix(g r ) ≤ 1 if and only if at least one of the conditions holds:
1. q = 2,
2. |L(f )| ≤ 1,
3. L(f q ) = L(f ).
Proof. For q = 2 the equivalence results from part 2 of Proposition 4.11. Let us consider
now prime q > 2. By Corollary 4.13 Dr3 [f ] ≤ 2. We must find all conditions in Theorem
4.10 part (∗) under which Dr3 [f ] ≤ 1. We get that Dr3 [f ] ≤ 1 is equivalent to the following
“fm˙08˙01” — 2008/6/24 — 14:42 — page 15 — #15
Minimal number of periodic points for C 1 maps
15
alternative: either #G ≤ 1 and L(f ) ∈ [−#G, #G], or #G = 0. The first possibility leads to
q
)
|L(f )| ≤ 1 and the second, by the definition of G, forces bq = L(f )−L(f
= 0.
2
2
4.2 Calculation of D3r [f ] for self maps of S 2 × I
As in the previous sections we assume that M is a smooth compact connected and simplyconnected 3-dimensional manifold. Now, assume that additionally M has non-empty boundary. We remind the reader that for a manifold with boundary we consider self-maps which
have no periodic points on the boundary. We take homology with the coefficients in the field
of rational numbers Q.
Notice that connectivity of M implies that H0 (M ) = Q, furthermore H1 (M ) = 0 as M is
simply-connected. Because M is a manifold with boundary, H3 (M ) = 0. Thus the homology
spaces of M have the following form: H0 (M ) = Q, H1 (M ) = 0, H2 (M ) = Q i , H3 (M ) = 0.
Equivalently, the boundary of M consists of i + 1 pairwise disjoined 2-spheres (cf. [Gr]).
In this section we will give the complete description of the least number of periodic points
of smooth maps, up to homotopy, in case i = 1, i.e. for M = S 2 × I , the only connected and
simply-connected 3-manifold with boundary and H2 (M ) = Q.
We define global Dold coefficients in (f ) by the formula:
in (f ) = ∑ μ (n/k)L(f k ).
k |n
Recall that in (f )/n = bn (f ), where bn (f ) = bn is the nth coefficient of the periodic expansion
of the sequence {L(f n )}∞
n=1 in the formula (7) (cf. also Theorem 2.3).
Let f : M → M . Then the homomorphism f∗2 : H2 (M ; Q) → H2 (M ; Q) is a multiplication
by a number β ∈ Z (called the degree of f ). Assume r ∈ N is fixed. Then, finding the least
cardinality of Fix(g r ), where g is a smooth map homotopic to f , is equivalent to determining
the number Dr3 [f ] = Dr3 [{L(f n )}n|r ], where L(f n ) = 1 + β n . We will need the exact form of
the basic set Br (f ) (cf. Definitions 2.4 and 2.8). This knowledge is provided by the following
result proved in [LPR].
Lemma 4.15. ([LPR] Theorem 1.2)
(a) b1 (f ) = 1 + β .
(b) b2 (f ) = 0 if and only if β ∈ {0, 1}.
(c) If n > 2, then bn (f ) = 0 if and only if β ∈ {−1, 0, 1}.
2
Corollary 4.16. If |β | > 1, then B(f ) = N, so Br (f ) = {n ∈ N : n|r}, or equivalently
#Br (f ) = ζ (r), where ζ (r) is the number of all divisors of r. 2
Under the assumption that |β | > 1 Corollary 4.16 implies:
Corollary 4.17. For r odd G = Br (f ) \ {1}, thus #G = ζ (r) − 1. 2
Corollary 4.18. For r even there is:
if 4|r, then G = Br (f ) \ {1, 2, 4}, thus #G = ζ (r) − 3,
if 4 |r, then G = Br (f ) \ {1, 2}, thus #G = ζ (r) − 2. 2
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16
G, Graff, J. Jezierski
Notice that #H is the number of pairs k, 2k, where k|r, k > 1 is odd, in G. By Lemma 4.15
each n, such that n|r, belongs to G, thus we obtain for even r:
Corollary 4.19. #H= η (r) − 1, where η (r) is the number of all odd divisors of r.
Taking into account that L(f ) = 1 + β we get, by Theorem 4.10 and Corollary 4.17, the
following description of Dr3 [f ] for odd r and all f with the absolute value of the degree
greater than 1:
Theorem 4.20. If r is odd, |β | > 1 then:
ζ (r) − 1 if − ζ (r) ≤ β ≤ ζ (r) − 2,
3
Dr [f ] =
ζ (r)
otherwise.
Example 4.21. Let f : M → M have the degree β ∈ {−1, 0, 1} and r = 15. The Lefschetz
numbers L(f n ) for n|15 are: 1 + β , 1 + β 3 , 1 + β 5 , 1 + β 15 . We represent this sequence (for
n = 1, 3, 5, 15) in the form of a periodic expansion.
(9)
L(f n ) = (1 + β )reg1 (n) + (
+(
β3 −β
β5−β
)reg3 (n) + (
)reg5 (n)
3
5
β 15 − β 5 − β 3 + β
)reg15 (n).
15
Now we apply Theorem 4.20: ζ (15) = 4, r = 15 is odd, hence:
3 for β ∈ {−4, −3, −2, 2},
3
D15 [f ] =
4 for β < −4 or β > 2.
2
Proposition 4.2 allows us to derive some additional information in the case of higher dimensional manifolds.
Example 4.22. Let f be a self-map of a smooth compact connected and simply-connected
m-dimensional manifold M̃ , such that {L(f n )}n|15 is the same as in Example 4.21, i.e. L(f n )
are expressed by the right hand-side of the formula (9). By Proposition 4.2 there is a C 1
map g homotopic to f such that Fix(g 15 ) has at most 3 points if β ∈ {−4, −3, −2, 2} and at
most 4 points if β < −4 or β > 2. Moreover, if m = 3 then Fix(g 15 ) cannot have less points
than this number.
2
Now we analyze the case of r even.
Theorem 4.23. Assume that r is even and |β | > 1. Then
Dr3 [f ] = ζ (r) − η (r).
Proof. For r = 2, 4 the thesis follows from Proposition 4.11. Assume r > 4. First we consider
the case 4|r.
By Corollaries 4.18 and 4.19, we may replace in the thesis ζ (r) − η (r) by #G − #H + 2.
On the other hand, by Theorem 4.10, Dr3 [f ] ∈ [#G − #H , #G − #H + 2].
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Minimal number of periodic points for C 1 maps
17
Let
(10)
L(f n ) = b1 reg1 (n) + b2reg2 (n) + b4 reg4 (n) + ∑ bk regk (n) = ∑ci (n),
k ∈G
i
where ci
i |r) sequences which give minimal decomposition of Lefschetz numbers
of iterations, i.e. ∑i pi = Dr3 [f ]. We remind the reader that by Lemma 4.8 ci have one of the
forms (A)-(G), (E’)-(G’) (thus pi ≤ 2).
Suppose that:
are DD3 (p
(11)
Dr3 [f ] = #G − #H .
We show that this cannot happen. The equation (11) forces that the coefficients b1 , b2 , b4 ,
of the periodic expansion of {L(f n )}n|r , must be obtained as a sum of coefficients a1 , a2 ,
a4 , respectively; of #G − #H sequences (B)-(G) and (E’)-(G’), where the last are counted
twice and where each involve ad regd with d ∈ G. This is impossible because any sum of
such sequences gives always non-positive contribution to b4 ((E’) and (G’) allow to produce
4
2
−reg4 ), while b4 = β −4 β > 0. As a result we must use an additional sequence, say cα , of
the type (B), (C), (D), or (E), with the coefficient a4 = 0, which makes possible to reduce
b4 reg4 .
Thus we get:
(12)
Dr3 [f ] ∈ [#G − #H + 1, #G − #H + 2].
Suppose now that Dr3 [f ] = #G − #H + 1.
As we require minimal realization, (E’) is not valid, because it reduces only one basic
sequence bk regk , k ≥ 6, but is counted twice. By the same reason (F’), (G’) could be used
only under the condition that each of them reduces two basic sequences. Now we show
that (F’), (G’) may always be replaced in the minimal realization by sequences of the types
(A)-(G). For odd d (d > 1, 4d|r) we consider a triple:
(13)
bd regd (n) + b2d reg2d (n) + b4d reg4d (n),
which appears in the periodic expansion of Lefschetz numbers of iterations. By Lemma
4.15 each coefficient near the basic sequences in the formula (13) is nonzero. Notice that
(F’), (G’) may be used only if each of them reduces a part of such triple (two last terms).
Then we have to use one sequence (B)-(E) to reduce bd regd , which gives the contribution
2 + 1 = 3 to Dr3 [f ]. On the other hand, we may use one (F) or (G) to reduce two first terms
and one sequence (B)-(E) to reduce the last and one sequence (A) with a1 = b1 and a2 = b2 ,
getting the same contribution 1 + 1 + 1 = 3, independently of the values of b1 and b2 . Thus
in the minimal realization, we may consider only sequences (A)-(G).
We need at least #G − #H sequences (B)-(G) to reduce bk regk for k = 1, 2, 4 (cf. also
proof of Theorem 4.10), and one cα of the type (B)-(E) to reduce b4 reg4 . Finally, we need
2
also one sequence of the type (A) to reduce b2 reg2 with b2 = β 2−β > 0, because (B)-(G)
have coefficients a2 non-positive. As a result Dr3 [f ] = #G − #H + 2.
Now consider the case 4 r. By Corollary 4.17 and the second part of Corollary 4.18 we
know that ζ (r) − η (r) = #G − #H + 1. Thus, to finish the proof, it is enough to state that
Dr3 [f ] = #G − #H + 1.
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18
G, Graff, J. Jezierski
Because b4 = 0, we see that Dr3 [f ] ∈ [#G − #H , #G − #H + 1] (compare the proof of
Theorem 4.10). On the other hand, if Dr3 [f ] = #G − #H , then b2 > 0 must be a sum of
a2 -coefficients of #G − #H sequences of minimal realization. This is impossible, because
the only sequences which may be used to realize b2 in such a way, are of the types (E’), (F’)
or (G’), but in the case 4 r each of them reduces only one sequence (for k = 1, 2) in the
expansion of {L(f n )}n|r , but is counted twice. This ends the proof. 2
Proposition 4.24. In case β ∈ {−1, 0, 1} Dr3 [f ] = 1 as we can use one DD3 (1|r) sequence
of the type (A). Namely:
3
β {L(f n )}∞
n=1 Dr [f ]
0
reg1 (n)
1
1
2reg1 (n)
1
−1 reg2 (n)
1
Theorems 4.20 and 4.23 together with Proposition 4.24 give the complete description of the
least number of periodic points of smooth self-maps, up to homotopy, for S 2 × I .
References
[BaBo]
Babenko I. K., Bogatyi S. A.: The behavior of the index of periodic points under iterations
of a mapping. Math. USSR Izv. 38 (1992), 1–26
[B]
Brown R. F.: The Lefschetz Fixed Point Theorem. Glenview, New York 1971
[Ch]
Chandrasekharan K.: Introduction to Analytic Number Theory. Springer, Berlin 1968
[CMPY] Chow S. N., Mallet-Parret J., Yorke J. A.: A periodic point index which is a bifurcation
invariant. Geometric Dynamics (Rio de Janeiro, 1981), 109–131, Springer Lecture Notes
in Math. 1007, Berlin 1983
[D]
Dold A.: Fixed point indices of iterated maps. Invent. Math. 74 (1983), 419–435
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Received July 20, 2007; revised December 4, 2007; in final form December 10, 2007
Grzegorz Graff, Faculty of Applied Physics and Mathematics, Gdansk University of Technology, ul.
Narutowicza 11/12, 80-952 Gdansk, Poland
[email protected]
Jerzy Jezierski, Institute of Applications of Mathematics, Agricultural University of Warsaw (SGGW),
00-665 Warsaw, Poland
[email protected]
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