- Research Article
- Open access
- Published:
Eventually Periodic Points of Infra-Nil Endomorphisms
Fixed Point Theory and Applications volume 2010, Article number: 721736 (2010)
Abstract
Hyperbolic toral automorphisms provide important examples of chaotic dynamical systems. Generalizing automorphisms on tori, we study (infra-)nil endomorphisms defined on (infra-)nilmanifolds. In particular, we show that every infra-nil endomorphism has dense eventually periodic points.
1. Introduction
Let be an
nonsingular integer matrix. Then
induces a map
on the
-torus
. If
is hyperbolic, we say that
is a hyperbolic toral endomorphism. If, in addition,
, then
is called a hyperbolic toral automorphism.
A hyperbolic toral automorphism provides an important example of a chaotic dynamical system. We review the most fundamental property about hyperbolic toral automorphisms, together with some definitions which are necessary to describe this property. See [1] for details.
A continuous surjection of a topological space
is said to be topologically transitive if, for any pair of nonempty open sets
and
in
, there exists
such that
. Intuitively, a topologically transitive map has points which eventually move under iteration from one arbitrary small neighborhood to any other. The continuous map
of the metric space
is said to have sensitive dependence on initial conditions if there exists
such that, for any
and any neighborhood
of
, there exist
and
such that
. Intuitively, a map possesses sensitive dependence on initial conditions if there exist points arbitrarily close to
which eventually separate from
by at least
under iteration of
.
The following proposition shows that a hyperbolic toral automorphism is dynamically quite different from its linear counterpart.
Proposition 1.1 (see [1, Theorem ]).
A hyperbolic toral automorphism is chaotic on
. That is,
()the set of periodic points of is dense in
;
() is topologically transitive;
() has sensitive dependence on initial conditions.
Anosov diffeomorphisms play an important role in dynamics. In [2], Smale raised the problem of classifying the closed manifolds (up to homeomorphism) which admit an Anosov diffeomorphism. Franks [3] and Manning [4] proved that every Anosov diffeomorphism on an infra-nilmanifold is topologically conjugate to a hyperbolic infra-nil automorphism. In [5], Gromov proved that every expanding map on a closed manifold is topologically conjugated to an expanding map on an infra-nilmanifold.
We will consider infra-nil endomorphisms in this paper. These include Anosov diffeomorphisms and expanding maps on infra-nilmanifolds up to topological conjugacy. The purpose of this paper is to show that the infra-nil endomorphisms have dense eventually periodic points. In the case of infra-nil automorphisms, this is already known (cf. [4, Lemma ]).
2. Toral Endomorphisms
Now we show that every toral endomorphism has dense periodic points. This generalizes [1, Proposition ] in which it is shown that every toral automorphism has dense periodic points.
Definition 2.1.
For a self-map , a point
of
is called an eventually periodic point of
if
for some
. If
, then it becomes a periodic point of
with period
.
Note that if is a nonempty set of prime numbers, then the set
is a multiplicative subset of
. Let
be the ring of quotients of
by
. We denote
by
. Clearly,
and
.
Lemma 2.2.
Let be a toral endomorphism of the torus
induced by the automorphism
and let
be a nonempty set of prime numbers. Then every point with coordinates in
is an eventually periodic point of
. Moreover, if
for all
, then every point with coordinates in
is a periodic point of
.
Proof.
Let be a point of
with coordinates in
. Finding a common denominator, we may assume that
is of the form
where
and
are integers. Write
. Then there are exactly
points in
of the form
with
.
The image of any such point under may also be written in this form, since the entries of
are integers. Thus
is an eventually periodic point of
. Moreover, if
for all
,
is injective on these points and hence
is a permutation of
such points. In fact, if
, then we see that
, or
. Since
and
, we must have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ1_HTML.gif)
Hence . Therefore,
is a periodic point of
.
Corollary 2.3.
Every toral endomorphism of the torus
has dense periodic points.
Proof.
Let be a prime number with
and let
. Then by Lemma 2.2, the points with coordinates in
are periodic. Moreover,
, the set of points in
with coordinates in
, is a dense subset of the torus
.
3. Nil Endomorphisms
In this section, we first recall from [6–10] some definitions about nilpotent Lie groups and give some basic properties which are necessary for our discussion.
Let be a connected, simply connected nilpotent Lie group. A discrete cocompact subgroup
of
is said to be a lattice of
, and in this case, the quotient space
is said to be a nilmanifold.
Let be a lattice of
. Then
is a finitely generated torsion-free nilpotent group. Recall that the lower central series of
is defined inductively by
and
. Suppose that
is
-step nilpotent, that is,
, but
. The isolator of a subgroup
of
, denoted by
, is the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ2_HTML.gif)
It is well known ([6], [9, page 473] or [10]) that the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ3_HTML.gif)
forms a central series with . It follows that it is possible to choose a generating set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ4_HTML.gif)
of in such a way that
is the group generated by
and
for each
. We refer to
as a preferred basis of
.
We use to indicate the Lie algebra of
. This Lie algebra
has the same dimension and nilpotency class as
. Moreover, in the case of connected, simply connected nilpotent Lie groups it is known that the exponential map
is a diffeomorphism. We denote its inverse by
. If
is another connected, simply connected nilpotent Lie group, with Lie algebra
, then we have the following properties.
(i)For any homomorphism of Lie groups, there exists a unique homomorphism
(differential of
) of Lie algebras, making the following diagram commuting:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ5_HTML.gif)
(ii)Conversely, for any homomorphism of Lie algebras, there exists a unique homomorphism
of Lie groups, making the above diagram commuting.
If is a preferred basis of
, then
can be regarded as a basis for the vector space
. We call the basis
of
preferred. In particular, if
is a lattice of
, then every preferred basis
of
becomes a preferred basis
for the vector space
.
We first generalize the concept of toral automorphisms to that of nil endomorphisms and show that every nil endomorphism has eventually dense periodic points.
Let be a nilmanifold and let
be an automorphism satisfying that
. Then the automorphism
induces a surjection
on the nilmanifold
and the following diagram is commuting:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ42_HTML.gif)
Lemma 3.1.
Let be an automorphism satisfying that
. Then
has a block matrix, with respect to any preferred basis of
, of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ6_HTML.gif)
where the diagonal blocks 's are integral matrices, and
. In particular, the automorphism
on
restricts to an automorphism on a lattice of
if and only if its differential
has determinant ±
.
The proof of this lemma is rather straight forward and so we omit the proof. See, for example, [11, Lemma ] and [12, Proposition
].
Definition 3.2.
Let be a nilmanifold and let
be an automorphism with
. Then
induces a surjective map
on the nilmanifold
, which is one of the following two types.
(I) has determinant of modulus 1. In this case
is called a nil automorphism.
(II) has determinant of modulus greater than 1. In this case
is called a nil endomorphism.
If, in addition, is hyperbolic (i.e.,
has no eigenvalues of modulus 1), then we say that the nil automorphism or endomorphism
is hyperbolic.
Example 3.3.
Let be the
-dimensional Heisenberg group with its Lie algebra
. That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ7_HTML.gif)
It is easy to show (see [13, Proposition ]) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ8_HTML.gif)
Thus we see that the differential of any automorphism on
has determinant
and eigenvalues
and
. Thus if
, then
has an eigenvalue of modulus 1. Therefore, there are no hyperbolic nil automorphisms on any nilmanifold
. (There are examples of hyperbolic nil, nontoral, automorphisms. In fact, we can find such examples from many literatures. For example, we refer to [2, 14–18].)
Via the exponential map
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ9_HTML.gif)
we see that every automorphism on
is given as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ10_HTML.gif)
where . Consider the subgroups
,
, of
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ11_HTML.gif)
These are lattices of , and every lattice of
is isomorphic to some
. The following matrices
give simple examples which induce hyperbolic nil endomorphisms on the nilmanifold
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ12_HTML.gif)
Note that the first one has eigenvalues of modulus all greater than , and the second one has determinant of modulus greater than 1, and there is at least one eigenvalue with modulus less than 1.
Corollary 3.4.
If is a nil automorphism, then the automorphism
induces a nil automorphism which is
. In particular,
is a diffeomorphism of
.
By refining the central series of explained in the paragraph above Lemma 3.1, we can find a central series
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ13_HTML.gif)
with , for each
. (We are assuming that
is
-dimensional, and using the same symbol for terms of a refinement of the previous central series.) We can choose a generating set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ14_HTML.gif)
of in such a way that
is the group generated by
and
. Then any element
is uniquely expressible as a product:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ15_HTML.gif)
and we can regard as the Mal'cev completion of
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ16_HTML.gif)
We refer to this preferred basis as a canonical basis of
. Given
, we use
to denote the element of
whose canonical coordinate is
. Thus, we have an identification
sending
to
.
Among interesting properties of this identification, we recall the following ([7, Theorem 2.1.(3)]): for any homomorphism , there exists a polynomial function with rational coefficients
such that
for all
. Moreover, any homomorphism of
extends to a homomorphism of
by using the same polynomial.
Example 3.5.
The map given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ17_HTML.gif)
is a polynomial function with rational coefficients, which sends into
itself. The polynomial function
is associated to the homomorphism
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ18_HTML.gif)
on given in Example 3.3.
We recall the famous Campbell-Baker-Hausdorff formula:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ19_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ20_HTML.gif)
Here stands for a rational combination of
-fold Lie brackets in
and
. Since our Lie algebra is nilpotent, the sum involved in
is always finite. Throughout this paper, we shall use
whenever
is the set of all prime factors of the denominators of the reduced rational coefficients appearing in the Campbell-Baker-Hausdorff formula. For example, if
is
-step nilpotent, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ21_HTML.gif)
and hence .
Lemma 3.6.
For any homomorphism , the associated polynomial function
has coefficients in
.
Proof.
Suppose that is a
-dimensional connected, simply connected nilpotent Lie group. The first thing to notice is that for any
in the Lie algebra
of
, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ22_HTML.gif)
where denotes a linear combination of
-fold brackets in
and
with coefficients in the ring
. To see this, let us make the following computation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ23_HTML.gif)
From this it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ24_HTML.gif)
which is of the form (3.22) claimed above.
Now, let be a canonical basis of
(We mean
where the
form a canonical basis of
). Since
, we have from (3.22) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ25_HTML.gif)
By a repeated use of formulas (3.22) and (3.25) it is now easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ26_HTML.gif)
where is a polynomial with coefficients in
. We will use this fact below.
Finally, let be the Lie group homomorphism of
which extends uniquely the given
. Let
be a term of the canonical basis of
, then
for some
. Using the Campbell-Baker-Hausdorff formula, it is then easy to see (look also at the computation below) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ27_HTML.gif)
We now compute
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ28_HTML.gif)
Here stands for a term which is a linear combination of
-fold brackets of the
and where the coefficients are polynomials in the variables
over the ring
. By continuing this computation, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ29_HTML.gif)
Now using (3.27) we derive that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ30_HTML.gif)
where the are polynomials with coefficients in
. Therefore, using (3.26), this implies that the polynomial
is as required.
Remark 3.7.
Our original proof was longer treating the case where is a
-step nilpotent Lie group. This one was provided by one of the referees.
Now we fix a canonical basis of
. A point
of the nilmanifold
is said to have rational coordinates or simply
has rational coordinates if
for some
. First we show that if
and
with
, then
for some
. We recall the following ([7, Theorem
.(
)]): there exists a polynomial function with rational coefficients
satisfying
for all
. The group product on
is defined using this polynomial
. Now, suppose that
and
with
. Then
for some
. Since
,
for some
. Hence we have
. Since
, and
is a polynomial function with rational coordinates, we must have
. This proves our assertion. Therefore the points of
with rational coordinates are well defined. Consequently for a subring
of
with
, the points of
with coordinates in
are well defined.
It is known that every (infra-)nil automorphism has dense periodic points (see the proof of [4, Lemma ]). Now we will generalize this to the case of (infra-)nil endomorphisms. The proof below is exactly the same as that of Lemma 2.2, except that the coefficients involved are different and hence Lemma 3.6 is essential.
Theorem 3.8.
Let be a nil endomorphism of the nilmanifold
. Let
be a ring obtained from
by adding finitely many primes
, that is,
. Then every point with coordinates in
is an eventually periodic point of
. Moreover, if
for all
, then every point with coordinates in
is a periodic point of
.
Proof.
We will show this by induction on the nilpotency class of
. If
, then
is a torus and this case is proved in Lemma 2.2.
Now let and assume that the assertion is true for any connected, simply connected nilpotent Lie group
of nilpotency class
and for any ring obtained from
by adding finitely many primes.
Consider and
, and the principal fiber bundle
where
,
is a torus and
is a nilmanifold of dimension less than that of
. Since the automorphism
maps
into itself, its induced map
is fiber-preserving. That is, the following diagram is commuting:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ43_HTML.gif)
Now we note that for some
in the refined central series of
. Thus
and
have central series
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ31_HTML.gif)
The canonical basis of
induces the canonical bases
and
of
and
, respectively, where
stands for the image of
in
under the natural surjection
. Hence the points in
with rational coordinates are well defined. Furthermore the points in
with rational coordinates are also well-defined.
For , write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ32_HTML.gif)
Then and
. Since
,
is a point of
with coordinates in
. (Note that the ring
when working over the group
is a subring of
and so
.) By induction hypothesis
for some
and
. On the other hand, since
is a point of the torus
with coordinates in
, by Lemma 2.2,
for some
and
. We may assume that
and
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ33_HTML.gif)
Then in
for some
;
for some
. By Lemma 3.6,
has coordinates in
. Furthermore,
for some
. Let
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ34_HTML.gif)
Simply taking , we may assume that
where
and
with coordinates in
. Hence Lemma 2.2 can be used to conclude that
for some
,
, and
. Thus
for some
. We note further that for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ35_HTML.gif)
for some . Since
with coordinates in
, there is
such that
. Since
,
for all
. Hence
, or
for some
. Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ36_HTML.gif)
which implies that is an eventually periodic point of
.
Moreover, if for all
, then by Lemma 2.2 and induction hypothesis, we can choose
and so
. Thus
is a periodic point of
.
Corollary 3.9.
Every nil endomorphism of the nilmanifold
has dense eventually periodic points.
Proof.
Using the fact that the points of with coordinates in
are dense in
, we obtain the result.
Example 3.10.
Let be the (hyperbolic) nil endomorphism on the nilmanifold
induced by the automorphism on
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ37_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ38_HTML.gif)
Thus the point
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ39_HTML.gif)
is not a periodic point, but an eventually periodic point of with least period
(i.e., an eventually fixed point). Note here that
and
is the coefficient coming from the nilpotent Lie group
.
At this moment, we donot know whether Corollary 3.9 is true for periodic points in the general case, that is, the case where for some
. We now propose naturally the following problem.
Question 1.
Every nil endomorphism has dense periodic points.
Corollary 3.11.
Every nil automorphism of the nilmanifold
has dense periodic points.
Proof.
The proof follows from that .
4. Infra-Nil Endomorphisms
Let be a connected, simply connected nilpotent Lie group and let
be a maximal compact subgroup of
. A discrete and cocompact subgroup
of
is called an almost crystallographic group. Moreover, if
is torsion-free, then
is called an almost Bieberbach group and the quotient space
an infra-nilmanifold. In particular, if
, then
is a nilmanifold. Recall from [19] that
is the maximal normal nilpotent subgroup of
with finite quotient group
, called the holonomy group of
.
Definition 4.1.
Let be an infra-nilmanifold and let
be an automorphism which is weakly
-equivariant; that is, there is a homomorphism
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ40_HTML.gif)
Then induces a surjection
, which is one of the following types.
(I) has determinant of modulus 1. In this case
is called an infra-nil automorphism.
(II) has determinant of modulus greater than 1. In this case
is called an infra-nil endomorphism.
If, in addition, is hyperbolic, then we say that the infra-nil automorphism or endomorphism
is hyperbolic.
Let be an infra-nilmanifold with surjection
. Let
be the pure translations of
. Then it is not difficult to see that there exists a fully invariant subgroup
of
with finite index. For example, one can take
(see also [20, Lemma
]). Thus
is a nilmanifold which is a finite regular covering of
and has
as the group of covering transformations. The homomorphism
associated with
induces a homomorphism
and in turn induces a homomorphism
so that the following diagram is commuting:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ44_HTML.gif)
Moreover, the automorphism on
induces a surjection
so that the following diagram is commuting:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ45_HTML.gif)
Since for all
, we have
for all
. Hence
is the unique extension of the homomorphism
of the lattice
of
. If
is an isomorphism, then
is also an isomorphism. Conversely, assume that
is an isomorphism. Using the fact that
is torsion-free, we can show that
is injective. This fact implies that
is also injective on the finite group
and hence
must be an isomorphism. Therefore,
is an isomorphism. (The converse was suggested by a referee.) If
is an infra-nil automorphism, then being
implies by Lemma 3.1 that
is an isomorphism and thus
is a nil automorphism, and vice versa. Note also that
is an infra-nil endomorphism if and only if
is a nil endomorphism.
Let denote the set of eventually periodic points of a self-map
.
Theorem 4.2.
Every infra-nil endomorphism has dense eventually periodic points.
Proof.
Consider the following commuting diagram:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ46_HTML.gif)
where is an infra-nil endomorphism, and hence
is a nil endomorphism. First we observe that
. The inclusion
is obvious. For the converse, let
and
. Then
for some
and
. Clearly
and
is a permutation on the finite set
. Hence
for some
. The reverse inclusion
is proved. Now by the continuity of
and by Corollary 3.9, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F721736/MediaObjects/13663_2009_Article_1332_Equ41_HTML.gif)
This proves that is dense in
.
References
Devaney RL: An Introduction to Chaotic Dynamical Systems, Addison-Wesley Studies in Nonlinearity. 2nd edition. Addison-Wesley, Redwood City, Calif, USA; 1989:xviii+336.
Smale S: Differentiable dynamical systems. Bulletin of the American Mathematical Society 1967, 73: 747–817. 10.1090/S0002-9904-1967-11798-1
Franks J: Anosov diffeomorphisms. In Global Analysis, Proceedings of Symposia in Pure Mathematics. Volume 14. American Mathematical Society, Providence, RI, USA; 1970:61–93.
Manning A: There are no new Anosov diffeomorphisms on tori. American Journal of Mathematics 1974, 96: 422–429. 10.2307/2373551
Gromov M: Groups of polynomial growth and expanding maps. Institut des Hautes Études Scientifiques 1981, (53):53–73.
Dekimpe K: Almost-Bieberbach Groups: Affine and Polynomial Structures, Lecture Notes in Mathematics. Volume 1639. Springer, Berlin, Germany; 1996:x+259.
Dekimpe K, Lee KB: Expanding maps on infra-nilmanifolds of homogeneous type. Transactions of the American Mathematical Society 2003,355(3):1067–1077. 10.1090/S0002-9947-02-03084-2
Dekimpe K, Malfait W: Affine structures on a class of virtually nilpotent groups. Topology and Its Applications 1996,73(2):97–119. 10.1016/0166-8641(96)00069-7
Passman DS: The Algebraic Structure of Group Rings, Pure and Applied Mathematics. John Wiley & Sons, New York, NY, USA; 1977:xiv+720.
Segal D: Polycyclic Groups, Cambridge Tracts in Mathematics. Volume 82. Cambridge University Press, Cambridge, UK; 1983:xiv+289.
Kim SW, Lee JB: Anosov theorem for coincidences on nilmanifolds. Fundamenta Mathematicae 2005,185(3):247–259. 10.4064/fm185-3-3
Malfait W: An obstruction to the existence of Anosov diffeomorphisms on infra-nilmanifolds. In Crystallographic Groups and Their Generalizations, Contemporary Mathematics. Volume 262. American Mathematical Society, Providence, RI, USA; 2000:233–251.
Ha KY, Lee JB: Left invariant metrics and curvatures on simply connected three-dimensional Lie groups. Mathematische Nachrichten 2009,282(6):868–898. 10.1002/mana.200610777
Auslander L, Scheuneman J: On certain automorphisms of nilpotent Lie groups. In Global Analysis, Proceedings of Symposia in Pure Mathematics. Volume 14. American Mathematical Society, Providence, RI, USA; 1970:9–15.
Dani SG, Mainkar MG: Anosov automorphisms on compact nilmanifolds associated with graphs. Transactions of the American Mathematical Society 2005,357(6):2235–2251. 10.1090/S0002-9947-04-03518-4
Lauret J, Will CE: On Anosov automorphisms of nilmanifolds. Journal of Pure and Applied Algebra 2008,212(7):1747–1755. 10.1016/j.jpaa.2007.11.011
Malfait W: Anosov diffeomorphisms on nilmanifolds of dimension at most six. Geometriae Dedicata 2000,79(3):291–298. 10.1023/A:1005264730096
Shub M: Endomorphisms of compact differentiable manifolds. American Journal of Mathematics 1969, 91: 175–199. 10.2307/2373276
Lee KB, Raymond F: Rigidity of almost crystallographic groups. In Combinatorial Methods in Topology and Algebraic Geometry, Contemporary Mathematics. Volume 44. American Mathematical Society, Providence, RI, USA; 1985:73–78.
Lee JB, Lee KB: Lefschetz numbers for continuous maps, and periods for expanding maps on infra-nilmanifolds. Journal of Geometry and Physics 2006,56(10):2011–2023. 10.1016/j.geomphys.2005.11.003
Acknowledgments
The authors would like to thank the referees for pointing out some errors and making careful corrections to a few expressions in the original version of the paper. The authors also would like to thank both referees for suggesting the apt title. The first author was partially supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2005-206-C00004), and the third author was supported in part by KOSEF Grant funded by the Korean Government (MOST) (no. R01-2007-000-10097-0).
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Ha, K., Kim, H. & Lee, J. Eventually Periodic Points of Infra-Nil Endomorphisms. Fixed Point Theory Appl 2010, 721736 (2010). https://doi.org/10.1155/2010/721736
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DOI: https://doi.org/10.1155/2010/721736