Open Access

Eventually Periodic Points of Infra-Nil Endomorphisms

Fixed Point Theory and Applications20102010:721736

https://doi.org/10.1155/2010/721736

Received: 14 August 2009

Accepted: 19 February 2010

Published: 28 February 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
(2.1)

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 [610] 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
(3.1)
It is well known ([6], [9, page 473] or [10]) that the sequence
(3.2)
forms a central series with . It follows that it is possible to choose a generating set
(3.3)

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:

(3.4)

(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:
(3.5)

Lemma 3.1.

Let be an automorphism satisfying that . Then has a block matrix, with respect to any preferred basis of , of the form
(3.6)

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,
(3.7)
It is easy to show (see [13, Proposition ]) that
(3.8)

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, 1418].)

Via the exponential map
(3.9)
we see that every automorphism on is given as follows:
(3.10)
where . Consider the subgroups , , of :
(3.11)
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 :
(3.12)

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
(3.13)
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
(3.14)
of in such a way that is the group generated by and . Then any element is uniquely expressible as a product:
(3.15)
and we can regard as the Mal'cev completion of :
(3.16)

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
(3.17)
is a polynomial function with rational coefficients, which sends into itself. The polynomial function is associated to the homomorphism
(3.18)

on given in Example 3.3.

We recall the famous Campbell-Baker-Hausdorff formula:
(3.19)
where
(3.20)
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
(3.21)

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
(3.22)
where denotes a linear combination of -fold brackets in and with coefficients in the ring . To see this, let us make the following computation:
(3.23)
From this it follows that
(3.24)

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
(3.25)
By a repeated use of formulas (3.22) and (3.25) it is now easy to see that
(3.26)

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
(3.27)
We now compute
(3.28)
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
(3.29)
Now using (3.27) we derive that
(3.30)

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:
(3.31)
Now we note that for some in the refined central series of . Thus and have central series
(3.32)

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
(3.33)
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
(3.34)
Then in for some ; for some . By Lemma 3.6, has coordinates in . Furthermore, for some . Let . Then
(3.35)
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 ,
(3.36)
for some . Since with coordinates in , there is such that . Since , for all . Hence , or for some . Therefore
(3.37)

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 :
(3.38)
Then
(3.39)
Thus the point
(3.40)

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
(4.1)

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:
(4.2)
Moreover, the automorphism on induces a surjection so that the following diagram is commuting:
(4.3)

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:
(4.4)
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
(4.5)

This proves that is dense in .

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, Sogang University

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Copyright

© Ku Yong Ha et al. 2010

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