Science in China Series A: Mathematics
Mar., 2008, Vol. 51, No. 3, 351–360
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Nielsen type numbers and homotopy minimal
periods for maps on the 3-nilmanifolds
Jong Bum LEE1 & ZHAO XueZhi2†
1
2
Department of Mathematics, Sogang University, Seoul 121-742, Korea;
Department of Mathematics, Capital Normal University, Beijing 100037, China
(email: [email protected], [email protected])
Abstract
For all continuous maps on the 3-nilmanifolds, we give explicit formulae for a complete
computation of the Nielsen type numbers N Pn (f ) and N Φn (f ). The most general cases were explored
by Heath and Keppelmann and the complementary part is studied in this paper. While studying the
homotopy minimal periods of all maps on the 3-nilmanifolds, we also give a complete description of
the homotopy minimal periods for all maps on the 3-nilmanifolds, including a correction to Jezierski
and Marzantowicz’s result.
Keywords:
MSC(2000):
1
homotopy minimal period, Nielsen number, Nielsen type number, nilmanifold
55M20, 57S30
Introduction
Given a self-map f : X → X, there are well-known invariants in fixed point theory which are
the Lefschetz number L(f ) and the Nielsen number N (f ). The map f must have a fixed point
if L(f ) = 0, while N (f ) serves as a lower bound of the number of the fixed points for all maps
in the homotopy class of f , hence providing more information than L(f ). For the periodic
points, two Nielsen type numbers N Pn (f ) and N Φn (f ) were introduced by Jiang, which are
lower bounds for the number of periodic points of least period exactly n and the set of periodic
points of period n, respectively.
It is obvious that these Nielsen numbers are much more powerful than the Lefschetz number
in describing the periodic point sets of self-maps, but the computation of those homotopy
invariants is difficult in general. However using fiber techniques on nilmanifolds and some
solvmanifolds, Heath and Keppelmann[1,2] succeeded in showing that the Nielsen numbers and
the two Nielsen type numbers are related to each other under certain condition.
In this paper we will give a complete computation (see Theorem 4.7) of the Nielsen type
numbers N Pn (f ) and N Φn (f ) of all iterates of f when f is a self-map on the 3-nilmanifolds.
To this end, we will classify all self-maps on each 3-nilmanifold up to homotopy (Theorem 2.2).
The most general cases were explored by Heath and Keppelmann[1,2] and the complementary
Received January 29, 2007; accepted September 14, 2007
DOI: 10.1007/s11425-008-0003-5
† Corresponding author
This work was supported in part the Sogang University (KOREA) Research Grant in 2006 and in part by a
grant from ABRL by Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (Grant
No. H00021), and by a BMEC grant (Grant No. KZ200710025012)
Jong Bum LEE & ZHAO XueZhi
352
part is studied in this paper.
One of the natural problems in dynamical system is the study of the existence of periodic
points of least period exactly n. Homotopically, a new concept, namely the homotopy minimal
periods,
HPer(f ) =:
{m | g m (x) = x, g q (x) = x, q < m for some x ∈ X}
gf
was introduced by Alsedà, Baldwin, Llibre, Swanson and Szlenk in [3]. Since the homotopy
minimal period is preserved under a small perturbation of a self-map f on a manifold X, we can
say that the set HPer(f ) of homotopy minimal periods of f describes the rigid part of dynamics
of f . Jezierski and Marzantowicz in [4] gave a description of the sets of homotopy minimal
periods of all self-maps on the 3-nilmanifolds. They used the facts that every 3-nilmanifold
M admits a fibration with S 1 as the fiber and T 2 as the base and that every map on M is
homotopic to a fiber preserving map of this fibration, so that they could use the previous results
on S 1 and T 2 .
However we find one error in their work. The second purpose of this paper is to correct
this error and give a complete description (see Theorem 5.5) of the sets of homotopy minimal
periods of all self-maps on the 3-nilmanifolds, which are not homeomorphic to 3-dimensional
tori. Our method of computing the sets of homotopy minimal periods is much simpler and
direct when compared with the method employed in [4].
In Section 2, we describe the general form of homomorphisms of the lattice Γk in the Heisenberg group G. This can be found also in literature, for example in [4]. Section 3 deals with
the linearization of a self-map f on the 3-nilmanifold to understand the Nielsen numbers of the
iterates of f by using Anosov’s theorem. Linearizations for self-maps of arbitrary nilmanifolds
and solvmanifolds of type N R are explored and surveyed in for example [5, 6]. With these preliminary sections, we determine the Nielsen type numbers in Section 4 and the sets of homotopy
minimal periods in Section 5 for maps on the 3-nilmanifolds.
2
Maps on the 3-nilmanifolds
Let G be the 3-dimensional Heisenberg group. That is,
⎫
⎧⎡
⎤
⎬
⎨ 1 y z G = ⎣ 0 1 x ⎦ x, y, z ∈ R .
⎭
⎩
0 0 1
Then G is the unique 3-dimensional non-abelian connected simply connected nilpotent Lie
group, (cf [7]). For all integers k > 0, we consider the subgroups Γk of G:
⎫
⎧⎡
l ⎤
⎪
⎪
⎬
⎨ 1 n k ⎥
⎢
Γk = ⎣ 0 1 m ⎦ m, n, l ∈ Z .
⎪
⎪
⎭
⎩
0 0 1
These are uniform lattices of
Letting
⎡
1 0
⎣
x= 0 1
0 0
G, and every uniform lattice of G is isomorphic to some Γk .
⎤
0
1⎦,
1
⎡
1
⎣
y= 0
0
⎤
1 0
1 0⎦,
0 1
⎡
1 0
⎢
z = ⎣0 1
0 0
1⎤
−
k⎥
0 ⎦,
1
Nielsen type numbers and homotopy minimal periods for maps on the 3-nilmanifolds
353
we obtain a presentation of Γk : Γk = x, y, z | [x, z] = [y, z] = 1, [x, y] = z k . Every element of
Γk can be written uniquely as the form
⎡
l ⎤
1 n −
k⎥
⎢
xm y n z l = ⎣ 0 1 m ⎦ .
0 0
1
Lemma 2.1 (cf. [4, Proposition 2.12]).
is given as follows:
ϕ(x) = xα y γ z μ ,
Any homomorphism ϕ : Γk → Γk on the lattice of G
ϕ(y) = xβ y δ z ν ,
where α, β, γ, δ, μ, ν ∈ Z. Equivalently, ϕ maps
⎡
r⎤
⎡
1 q −
1 γp + δq
k
⎢
⎣0 1
p ⎦ = xp y q z r −→ ⎣ 0
1
0 0
1
0
0
ϕ(z) = z αδ−βγ ,
(2.1)
r ⎤
−
k ⎥
αp+βq γp+δq r y
z ,
αp + βq ⎦ = x
1
where r = (αδ − βγ)r − αγ p(p−1)
k − βδ q(q−1)
k − βγpqk + μp + νq.
2
2
Proof. Note that the center of Γk is z, we must have ϕ(z) = z ζ for some ζ ∈ Z. Write
ϕ(x) = xα y γ z μ and ϕ(y) = xβ y δ z ν , where α, β, γ, δ, μ, ν ∈ Z. Since ϕ preserves the relation
[x, y] = z k , we must have [ϕ(x), ϕ(y)] = ϕ(z)k , i.e. (xα y γ )(xβ y δ )(xα y γ )−1 (xβ y δ )−1 = z kζ . It
follows that z k(αδ−βγ) = z kζ .
Let Γ\G be a nilmanifold. Since Γ is isomorphic to some Γk , by Mal’cev theorem[8] , this
isomorphism extends uniquely to an automorphism on G, and this automorphism restricts to
a diffeomorphism from Γ\G onto Γk \G. Hence we may assume that we are given a nilmanifold
Γk \G.
By Lemma 2.1, any homomorphism ϕ : Γk → Γk is determined by the set of integers
k
{α, β, γ, δ, μ, ν}. We denote such a homomorphism in (2.1) by ϕΓα,β,γ,δ,μ,ν
. By Mal’cev theorem
Γk
again, any homomorphism ϕα,β,γ,δ,μ,ν : Γk → Γk extends uniquely to a Lie group homomork
on the nilmanifold Γk \G,
phism on G and then this endomorphism restricts to a map ϕΓα,β,γ,δ,μ,ν
Γk
which is still written as ϕα,β,γ,δ,μ,ν .
We can state our classification theorem for self maps in 3-nilmanifolds.
k
Theorem 2.2. Any self map on Γk \G is homotopic to a map of the form ϕΓα,β,γ,δ,μ,ν
: Γk \G →
Γk
Γk
Γk \G. Two maps ϕα,β,γ,δ,μ,ν and ϕα ,β ,γ ,δ ,μ ,ν on Γk \G are homotopic to each other if and
only if α = α , β = β , γ = γ , δ = δ , and
α β
[ μ ν ] − [ μ ν ] ∈ k · Im
.
γ δ
Proof. Let f : Γk \G → Γk \G be a self map on the nilmanifold Γk \G. Identify Γk as the
fundamental group of Γk \G, f induces a homomorphism fπ on Γk . By Lemma 2.1, fπ =
k
ϕΓα,β,γ,δ,μ,ν
for some integers α, β, γ, δ, μ, ν. Since the nilmanifold Γk \G is a K(Γk , 1)-manifold,
k
.
we have that f ϕΓα,β,γ,δ,μ,ν
If [ μ ν ] − [ μ ν ] ∈ k · Im αγ βδ , then there are integers m and n such that
α β
.
[ μ ν ] − [ μ ν ] = k [ n −m ]
γ δ
Jong Bum LEE & ZHAO XueZhi
354
k
k
A simple calculation shows that wϕΓα,β,γ,δ,μ,ν
(u)w−1 = ϕΓα,β,γ,δ,μ
,ν (u) for all u ∈ Γk , where
⎡
1
⎢
w = ⎣0
0
l ⎤
−
k⎥
m ⎦ ∈ Γk .
1
n
1
0
k
k
Since the nilmanifold Γk \G is a K(Γk , 1)-manifold, we obtain that ϕΓα,β,γ,δ,μ,ν
and ϕΓα,β,γ,δ,μ
,ν on Γk \G are homotopic.
The proof of the other direction of the second conclusion is a simple computation.
3
Nielsen numbers N (f n )
Let G denote the Lie algebra of the unique 3-dimensional non-abelian connected simply connected nilpotent Lie group G. That is,
⎫
⎧⎡
⎤
⎬
⎨ 0 b c G = ⎣ 0 0 a ⎦ a, b, c ∈ R .
⎭
⎩
0 0 0
It is known that the exponential map exp : G → G is a diffeomorphism. Denoting its inverse
by log, we have the following properties:
• For any homomorphism φ : G → G on the Lie group G, there exists a unique homomorphism dφ : G → G (differential of φ) on its Lie algebra G, making the following diagram
commuting:
φ
G
⏐
⏐
log −→
G
−→
dφ
G
⏐
⏐
log
G
• Conversely, for any homomorphism dφ : G → G on the Lie algebra, there exists a unique
homomorphism φ : G → G on the Lie group, making the above diagram commuting.
Note that the exponential map is given as follows:
⎡
⎤
⎡
ab ⎤
0 b c
1 b c+
2 ⎥
⎢
exp : ⎣ 0 0 a ⎦ ∈ G −→ ⎣ 0 1
a ⎦ ∈ G.
0 0 0
0 0
1
Let
⎡
0 0
⎣
e1 = 0 0
0 0
⎤
0
1⎦,
0
⎡
0
⎣
e2 = 0
0
⎤
1 0
0 0⎦,
0 0
⎡
0 0
⎢
e3 = ⎣ 0 0
0 0
1⎤
−
k⎥
0 ⎦.
0
Then exp(e1 ) = x, exp(e2 ) = y, exp(e3 ) = z. Consider the homomorphism ϕ on Γk and hence
on G which is given in Lemma 2.1. We see that its differential dϕ on G is given as follows:
αγ
dϕ(e1 ) = αe1 + γe2 + μ +
k e3 ,
2
βδ
dϕ(e2 ) = βe1 + δe2 + ν +
k e3 ,
2
dϕ(e3 ) = (αδ − βγ)e3 .
Nielsen type numbers and homotopy minimal periods for maps on the 3-nilmanifolds
355
Hence with respect to the ordered basis {e1 , e2 , e3 },
⎤
⎡
α
β
0
⎥
⎢
γ
δ
0
dϕ = ⎣
⎦.
βδ
αγ
k ν+
k αδ − βγ
μ+
2
2
Its characteristic polynomial is χϕ (s) = (s − det Aϕ )(s2 − trAϕ s + det Aϕ ), where Aϕ = αγ βδ .
Remark 3.1. The above observation is not only true for maps on the 3-nilmanifold but also
for maps on a nilmanifold of any dimension. For example, see the proof of [9, Theorem 2.4] for
self-maps on an arbitrary nilmanifold.
Definition 3.2. Let f : Γk \G → Γk \G be a continuous map, which is determined by a
homomorphism ϕ in (2.1). The eigenvalues λ1 and λ2 of Aϕ = αγ βδ , i.e.,
(α + δ) ± (α − δ)2 + 4βγ
λi =
2
are said to be the basic eigenvalues of f .
Corollary 3.3. Let f : Γk \G → Γk \G be a continuous map with basic eigenvalues λ1 and
λ2 . Then the eigenvalues of the characteristic polynomial of f∗ : G → G are λ1 , λ2 and λ1 λ2 .
Remark 3.4.
Note that the nilmanifold Γk \G has a principal fiber bundle structure
Z(Γk )\Z(G) → Γk \G −→ (Γ/Z(Γk ))\(G/Z(G)),
where the fiber Z(Γk )\Z(G) is the circle and the base (Γ/Z(Γk ))\(G/Z(G)) is the 2-torus. The
k
map f = ϕΓα,β,γ,δ,μ,ν
is a fiber-map, which induces the base map f¯ and the typical fiber map fˆ.
Note further that f¯ is restricted from the linear map Aϕ on G/Z(G) = R2 and fˆ is restricted
from the linear map αδ − βγ on Z(Γk )\Z(G) = R. Hence deg(f¯) = det(Aϕ ) = λ1 λ2 = deg(fˆ).
See also [4].
By Anosov theorem[10] , the Lefschetz number and the Nielsen number of the n-th iterates of
f are given as follows:
L(f n ) = det(I − dϕn ) = (1 − det Anϕ )(1 − trAnϕ + det Anϕ ),
N (f n ) = | det(I − dϕn )| = |1 − det Anϕ ||1 − trAnϕ + det Anϕ |.
Thus, we have
Proposition 3.5. Let f : Γk \G → Γk \G be a continuous map with basic eigenvalues λ1 and
λ2 . Then
L(f n ) = (1 − λn1 λn2 )(1 − λn1 )(1 − λn2 ), N (f n ) = |1 − λn1 λn2 ||1 − λn1 ||1 − λn2 |.
This implies that for the continuous maps on 3-nilmanifolds, the Nielsen numbers and Lefschetz numbers are totally determined by the two basic eigenvalues of the given maps. In the
following sections, we shall show that these two basic eigenvalues also determine all the Nielsen
type numbers, such as N Pn (f ), N Φn (f ) and HPer(f ).
4
Nielsen type numbers N Pn (f ) and N Φn (f )
Heath and Keppelmann[1,2] proved that the two Nielsen type numbers N Pn (f ) and N Φn (f )
can be computed according to the Nielsen numbers for most maps on nilmanifolds.
356
Jong Bum LEE & ZHAO XueZhi
Proposition 4.1 ([1, Theorem 1], [2, Theorem 1.2]). Let f be a continuous map on a nilmanifold. Suppose that N (f n ) = 0. Then for all m | n, N Φm (f ) = N (f m ), N Pm (f ) =
m
q
q|m μ(q)N (f ), where μ is the Möbius function.
Since there are already formulae for the Nielsen numbers of continuous maps on the 3nilmanifolds (Proposition 3.5), N Pn (f ) and N Φn (f ) are known provided N (f n ) = 0. Here, we
shall focus on the case where N (f n ) = 0.
The following lemma gives us a necessary and sufficient condition for N (f n ) = 0.
Lemma 4.2. Let f : Γk \G →Γk \G be a continuous map on the 3-nilmanifold Γk \G with
basic eigenvalues λ1 and λ2 . Then N (f n ) = 0 if and only if one of the following is satisfied:
(1) (λ1 λ2 , λ1 + λ2 ) = (1, l);
(2) (λ1 λ2 , λ1 + λ2 ) = (−1, l) and n is even;
(3) (λ1 λ2 , λ1 + λ2 ) = (l, l + 1);
(4) (λ1 λ2 , λ1 + λ2 ) = (l, −l − 1) and n is even.
Proof. Recall that N (f n ) = |1−λn1 λn2 ||1−λn1 ||1−λn2 |. If N (f n ) = 0, then either |1−λn1 λn2 | = 0
or |1 − λn1 ||1 − λn2 | = 0.
Suppose that |1−λn1 λn2 | = 0. Because λ1 λ2 is an integer, there are two possibilities: (i) λ1 λ2 =
1, (ii) λ1 λ2 = −1 and n is even. They are just the first two cases in the statement of our lemma,
where l is an integer.
Suppose that |1 − λn1 ||1 − λn2 | = 0. If λ1 and λ2 are not real numbers, they are conjugate.
The condition |1 − λn1 ||1 − λn2 | = 0 implies that |λ1 | = |λ2 | = 1. Note that their real part
1
+λ2 ) is a half integer.
The√ two basic eigenvalues must be one of the pairs:
Re λ1 = Re λ√
2 = 2 (λ1 √
√
1
3 1
3
1
3
1
(i, −i), ( 2 + 2 i, 2 − 2 i) or (− 2 + 2 i, − 2 − 23 i). Correspondingly, (λ1 λ2 , λ1 + λ2 ) = (1, 0),
(1, 1) or (1, −1), which are included in (1).
If λ1 and λ2 are real numbers, the condition |1 − λn1 ||1 − λn2 | = 0 implies that either λn1 = 1 or
λn2 = 1. These are included in (3) and (4), where l and −l are respectively the other eigenvalue,
which is automatically an integer.
Conversely, it is easy to check that each of the four conditions ensures that N (f n ) = 0.
In fact, all of the four cases may happen. For example, Aϕ can be chosen respectively as
l 1
l 1
l 0
−l
0
,
,
,
.
−1 0
1 0
0 1
0 −1
Immediately, we have
Corollary 4.3. Let f : Γk \G → Γk \G be a continuous map. We have
(1) If N (f n ) = 0, then N (f m ) = 0 for all m | n;
(2) If N (f n ) = 0 for some odd n, then N (f m ) = 0 for all m;
(3) If N (f n ) = 0 for some even n, then N (f m ) = 0 for all even m.
By definition of basic eigenvalues, we have
Corollary 4.4. Let f : Γk \G → Γk \G be a continuous map, which is determined by a
homomorphism ϕ in (2.1). Then N (f n ) = 0 if and only if the integers α, β, γ, δ satisfy one of
the following:
(1) αδ − βγ = 1;
Nielsen type numbers and homotopy minimal periods for maps on the 3-nilmanifolds
357
(2) αδ − βγ = −1 and n is even;
(3) βγ = (α − 1)(δ − 1);
(4) βγ = (α + 1)(δ + 1) and n is even.
Combining Lemma 4.2 with Proposition 3.5, we obtain
Corollary 4.5. Let f : Γk \G → Γk \G be a continuous map with basic eigenvalues λ1 and
λ2 . Then the following statements are equivalent:
(1) L(f ) = 0;
(2) N (f ) = 0;
(3) (λ1 λ2 , λ1 + λ2 ) = (1, l) or (l, l + 1);
(4) N (f k ) = 0 for all k;
(5) N Φk (f ) = 0 for all k.
Now let us consider the Nielsen type number N Φn (f ) when N (f n ) = 0.
Proposition 4.6. Let f : Γk \G → Γk \G be a continuous map with basic eigenvalues λ1 and
λ2 . Suppose that N (f n ) = 0 and n = 2r n0 with non-negative integer r and odd n0 .
⎧
2|1 − λn1 0 ||1 − λn2 0 |,
⎪
⎨
N Φn (f ) = 2|1 − l2n0 |,
⎪
⎩
0,
Proof.
if (λ1 λ2 , λ1 + λ2 ) = (−1, l), r > 0,
if (λ1 λ2 , λ1 + λ2 ) = (l, −l − 1), r > 0,
otherwise.
Since N (f n ) = 0, the pair (λ1 λ2 , λ1 + λ2 ) has four possibilities listed in Lemma 4.2.
If either (λ1 λ2 , λ1 + λ2 ) = (1, l) or (λ1 λ2 , λ1 + λ2 ) = (l, l + 1), N (f m ) = 0 for all m. The
map has no essential periodic orbit classes of any period. It follows that N Φn (f ) = 0.
Assume that (λ1 λ2 , λ1 + λ2 ) = (−1, l) and n is even. If l = ±1, i.e. (λ1 λ2 , λ1 + λ2 ) = (1, −2)
or (−1, 0), then N (f m ) = 0 for all m. Our proposition holds. On the other hand, if l = ±1,
then N (f m ) = 0 for all odd numbers. Especially, N (f n0 ) = 0. The essential periodic orbit
classes of period q for all q | n are the same as all those of period q for all q | n0 . It follows
from definition that N Φn (f ) = N Φn0 (f ). Note that λ1 λ2 = −1. By Propositions 3.5 and 4.1,
we have N Φn (f ) = N Φn0 (f ) = 2|1 − λn1 0 ||1 − λn2 0 |.
In the case of (λ1 λ2 , λ1 + λ2 ) = (l, −l − 1) and even n, we have similarly that N Φn (f ) =
N Φn0 (f ) = 2|1 − l2n0 |. Here, basic eigenvalues are −1 and −l.
If N (f n ) = 0, then by definition N Pn (f ) = 0. But, N Φn (f ) may be non-zero. From the
above proposition, this phenomenon does happen to some continuous maps on 3-nilmanifolds,
where n is even, and either (λ1 λ2 , λ1 + λ2 ) = (−1, l) with l = 0 or (λ1 λ2 , λ1 + λ2 ) = (l, −l − 1)
with l = ±1. By Lemma 4.2, these are equivalent to that N (f m ) = 0 for even m and N (f m ) = 0
for odd m.
Now we summarize how to compute the Nielsen type numbers for maps on the 3-nilmanifold
Γk \G as follows.
Theorem 4.7.
Let f : Γk \G → Γk \G be a continuous map. We have
(1) if N (f n ) = 0, then N Pn (f ) = 0, N Φn (f ) = N (f n0 );
n
(2) if N (f n ) = 0, then N Pn (f ) = q|n μ(q)N (f q ), N Φn (f ) = N (f n ),
where n = 2r n0 with r 0 and n0 odd, and μ is the Möbius function.
Jong Bum LEE & ZHAO XueZhi
358
5
Homotopy minimal periods HPer(f )
The following lemma is originally due to Jiang and Llibre in [11, Theorem 3.4].
Lemma 5.1. Let f : M → M be a continuous map on a smooth manifold with dimension
greater or equal to 3. Suppose that for any integer n, either N (f n ) = m|n N Pm (f ) or N (f ) =
0. If there is an integer matrix A such that N (f n ) = | det(I − An )| for all n. Then
HPer(f ) = {n | N Pn (f ) = 0}
n
= {n | N (f n ) = 0, N (f n ) = N (f q ) for all prime q|n, q n}.
Proof.
Since N Pn (f ) is a lower bound for the number of periodic points with least period n,
by definition, we have HPer(f ) ⊃ {n | N Pn (f ) = 0}. If N Pn (f ) = 0, by [12], there is a map g
homotopic to f without period point with leat period n. we arrive at the first equality.
The proof of the other equality is the same as that of [11, Theorem 3.4] (cf. [13, (4.6)
Remark]).
Immediately, we have 1 ∈
/ HPer(f ) ⇐⇒ N (f ) = 0. By Corollary 4.5, this implies that all
n
N (f ) = 0 and hence all n ∈
/ HPer(f ), which means that HPer(f ) = ∅. Consequently,
Lemma 5.2. Let f : Γk \G → Γk \G be a continuous map with basic eigenvalues λ1 , λ2 . Then
the following are equivalent:
(1) 1 ∈
/ HPer(f );
(2) (λ1 λ2 , λ1 + λ2 ) is of the form (1, l) or (l, l + 1) for some integer l;
(3) N (f ) = 0;
(4) HPer(f ) = ∅.
The next lemma characterizes all the maps f for which 1 ∈ HPer(f ) and 2 ∈
/ HPer(f ).
Lemma 5.3. Let f : Γk \G → Γk \G be a continuous map with basic eigenvalues λ1 and λ2 .
Suppose that 1 ∈ HPer(f ). Then 2 ∈
/ HPer(f ) if and only if (λ1 λ2 , λ1 + λ2 ) = (0, 0), (0, −2),
(−2, 0), (−2, 2), (−1, l) with l = 0, or (l, −l − 1) with l = ±1.
Proof. By Lemma 5.1, we see that 2 ∈
/ HPer(f ) if and only if either N (f 2 ) = 0 or N (f 2 ) =
N (f ) = 0. In the last case, we have the equations (1 + λ1 )(1 + λ2 ) = ±1, and (1 + λ1 λ2 ) = ±1.
These imply that
(λ1 λ2 , λ1 + λ2 ) = (0, 0), (0, −2), (−2, 0), or (−2, 2).
The condition N (f 2 ) = |1 − λ21 λ22 ||1 − λ21 ||1 − λ22 | = 0 induces that λ1 = ±1, λ2 = ±1 or
λ1 λ2 = ±1. Since N (f ) = 0, we have λ1 = −1, λ2 = −1 or λ1 λ2 = −1. If λ1 λ2 = −1, then
λ2 = −1/λ1 and λ1 + λ2 = (λ21 − 1)/λ1 = l for some integer l. But l = 0, as λ1 = 1 and λ2 = 1.
In this case we obtain that (λ1 λ2 , λ1 + λ2 ) = (−1, l) with l = 0. If one λi = −1 then the other
λj is an integer = ±1. Hence we obtain that (λ1 λ2 , λ1 + λ2 ) = (l, −l − 1) with l = ±1. This
proves our assertion.
From the proof, we know that N (f 2 ) = 0 in the first four distinct cases, but N (f 2 ) = 0 in
the other cases.
Lemma 5.4. Let f : Γk \G → Γk \G be a continuous map with basic eigenvalues λ1 and λ2 .
Let m be an integer with m 3 such that N (f m ) = 0. If m ∈
/ HPer(f ), then (λ1 λ2 , λ1 + λ2 ) =
(0, 0) or (0, −1).
Nielsen type numbers and homotopy minimal periods for maps on the 3-nilmanifolds
359
Proof. Since N (f m ) = 0 and m ∈
/ HPer(f ), by Lemma 5.1, there would be a prime factor p
m
m
p
of m with N (f ) = N (f ). Thus,
m
m m m
p
p m
m
m
1 − λ1p 1 − λ2p .
|1 − λm
1 λ2 ||1 − λ1 ||1 − λ2 | = 1 − λ1 λ2
Then
(p−1)m (p−1)m m
m
(p−1)m 1 + (λ1 λ2 ) mp + · · · + (λ1 λ2 ) p 1 + λ p + · · · + λ p 1 + λ p + · · · + λ p = 1.
1
1
2
2
Since λ1 λ2 and λ1 + λ2 are integers, the above equations is equivalent to
m
(p−1)m
1 + (λ1 λ2 ) p + · · · + (λ1 λ2 ) p = ±1,
(p−1)m (p−1)m m
m
1 + λ2p + · · · + λ2 p
= ±1.
1 + λ1p + · · · + λ1 p
The first equality yields that
1 − (λ1 λ2 )m
m
1 − (λ1 λ2 ) p
= ±1.
Since N (f m ) = 0, (λ1 λ2 )m = 1. We must have λ1 λ2 = 0. It follows from the second equality
that (λ1 λ2 , λ1 + λ2 ) = (0, 0) or (0, −1).
Theorem 5.5. Let f : Γk \G → Γk \G be a map with basic eigenvalues λ1 and λ2 . Then
(1) (λ1 λ2 , λ1 + λ2 ) = (1, l) or (l, l + 1) if and only if HPer(f ) = ∅;
(2) (λ1 λ2 , λ1 + λ2 ) = (0, 0) or (0, −1) if and only if HPer(f ) = {1};
(3) (λ1 λ2 , λ1 + λ2 ) = (0, −2), (−2, 0) or (−2, 2) if and only if HPer(f ) = N \ {2};
(4) (λ1 λ2 , λ1 + λ2 ) = (−1, l) with l = 0 or (l, −l − 1) with l = 0, ±1 if and only if HPer(f ) =
N \ 2N;
(5) Except the above cases, HPer(f ) = N.
Proof.
Case 1.
As the sets HPer(f ) in these five cases are distinct, it suffices to show the “If” part.
Follows from Lemma 5.2.
Case 2. If (λ1 λ2 , λ1 + λ2 ) = (0, 0), the two basic eigenvalues are both 0. Thus, N (f n ) = 1
for all n. By Lemma 5.1, HPer(f ) = {1}. If (λ1 λ2 , λ1 + λ2 ) = (0, −1), the two basic eigenvalues
are 0 and −1. Thus, N (f n ) = 1 − (−1)n for all n. By Lemma 5.1 again, HPer(f ) = {1}.
Case 3.
By a computation,
n
n
⎧
|1 − (−2)n | |1 − 2 2 | |1 − (−1)n 2 2 |,
⎪
⎨
N (f n ) = |1 − (−2)n |,
⎪
√
√
⎩
|1 − (−2)n | |1 − (1 + 3)n | |1 − (1 − 3)n |,
if (λ1 λ2 , λ1 + λ2 ) = (−2, 0),
if (λ1 λ2 , λ1 + λ2 ) = (0, −2),
if (λ1 λ2 , λ1 + λ2 ) = (−2, 2).
/ HPer(f ). Note that N (f n ) = 0
Since N (f 2 ) = N (f ) = 0, we have that 1 ∈ HPer(f ) and 2 ∈
for all n 3. By Lemma 5.4, n ∈ HPer(f ) for all n 3. Thus, HPer(f ) = N \ {2}.
Case 4. Since N (f n ) = 0 for all even n, HPer(f ) ⊂ N \ 2N. As N (f ) = 0, 1 ∈ HPer(f ). Let
m be an odd number greater than 1. Note that N (f m ) = 0. Since (λ1 λ2 , λ1 + λ2 ) is neither
(0, 0) nor (0, −1), by Lemma 5.4, we have m ∈ HPer(f ). Hence, HPer(f ) = N \ 2N.
In the remaining cases, by Lemma 5.2, we have N (f ) = 0. Thus, 1 ∈ HPer(f ). Lemma 5.3
ensures that 2 ∈ HPer(f ). Since N (f m ) = 0 for each m, by Lemma 5.4, we obtain that
Jong Bum LEE & ZHAO XueZhi
360
Tabele 1
(λ1 λ2 , λ1 + λ2 )
HPer(f )
(1, l)
∅
(l, l + 1)
∅
(0, 0), (0, −1)
{1}
(0, −2), (−2, 0), (−2, 2)
N \ {2}
(−1, l) with l = 0
N \ 2N
(l, −l − 1) with l = 0, ±1
N \ 2N
Otherwise
N
m ∈ HPer(f ) for m 3. Therefore HPer(f ) =
N.
Now we tabulate what we have found about
HPer(f ) in Table 1.
Remark 5.6. The first four cases and the
sixth case coincide with the cases of [4]. However when for example (λ1 λ2 , λ1 +λ2 ) = (−1, 2),
the set of homotopy minimal periods of the corresponding map is claimed to be N in [4]; but
it should be N \ 2N.
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