Eventually Periodic Points of Infra-Nil Endomorphisms

Fixed Point Theory and Applications Volume 2010, Article ID 721736,15pages doi:10.1155/2010/721736

Research Article

Eventually Periodic Points of Infra-Nil Endomorphisms

Ku Yong Ha, Hyun Jung Kim, and Jong Bum Lee

Department of Mathematics, Sogang University, Seoul 121-742, South Korea

Correspondence should be addressed to Ku Yong Ha,[email protected] Received 14 August 2009; Revised 26 December 2009; Accepted 19 February 2010 Academic Editor: Hichem Ben-El-Mechaiekh

Copyrightq2010 Ku Yong Ha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

LetAbe ann×nnonsingular integer matrix. ThenAinduces a mapL_A : Tⁿ → Tⁿ on the n-torusTⁿZⁿ\Rⁿ. IfAis hyperbolic, we say thatL_Ais a hyperbolic toral endomorphism.

If, in addition, detA ±1, then A 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.

See1for details.

A continuous surjectionf:X → Xof a topological spaceXis said to be topologically transitive if, for any pair of nonempty open setsUandV inX, there existsk > 0 such that f^kU∩V /∅. Intuitively, a topologically transitive map has points which eventually move under iteration from one arbitrary small neighborhood to any other. The continuous map f :X → Xof the metric spaceX, dis said to have sensitive dependence on initial conditions if there existsδ >0 such that, for anyx∈Xand any neighborhoodN ofx, there existy ∈N andn ≥ 0 such thatdfⁿx, fⁿy > δ. Intuitively, a map possesses sensitive dependence on initial conditions if there exist points arbitrarily close toxwhich eventually separate from xby at leastδunder iteration off.

The following proposition shows that a hyperbolic toral automorphism LA is dynamically quite different from its linear counterpart.

Proposition 1.1see1, Theorem II.4.8. A hyperbolic toral automorphismL_A is chaotic onTⁿ. That is,

1the set of periodic points ofL_Ais dense inTⁿ; 2L_Ais topologically transitive;

3LAhas sensitive dependence on initial conditions.

Anosov diffeomorphisms play an important role in dynamics. In2, Smale raised the problem of classifying the closed manifoldsup to homeomorphismwhich admit an Anosov diffeomorphism. Franks3and Manning4proved that every Anosov diffeomorphism on an infra-nilmanifold is topologically conjugate to a hyperbolic infra-nil automorphism. In5, 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 knowncf.4, Lemma 3.

2. Toral Endomorphisms

Now we show that every toral endomorphism has dense periodic points. This generalizes1, Proposition II.4.2 in which it is shown that every toral automorphism has dense periodic points.

Definition 2.1. For a self-mapf :X → X, a pointxofXis called an eventually periodic point offiff^mtx f^txfor somem > 0, t≥ 0. Ift 0, then it becomes a periodic point off with periodm.

Note that if {p1, . . . , p_t} is a nonempty set of prime numbers, then the set S {pⁿ₁¹· · ·pⁿ_t^t | ni ∈ Z, ni ≥ 0, i 1, . . . , t}is a multiplicative subset ofZ. LetS⁻¹Zbe the ring of quotients ofZbyS. We denoteS⁻¹ZbyZ_p1,...,pt. Clearly,Z_p⊂Z_p,qandZ_pZ_qZ_p,q. Lemma 2.2. Let L_A : T^d → T^d be a toral endomorphism of the torus T^d induced by the automorphism A : R^d → R^d and let {p1, . . . , pr} be a nonempty set of prime numbers. Then every point with coordinates in R Z_p1,...,pr is an eventually periodic point of L_A. Moreover, if pi,|detA| 1 for alli, then every point with coordinates inRis a periodic point ofL_A.

Proof. Letx be a point ofT^dZ^d\R^dwith coordinates inR. Finding a common denominator, we may assume thatx is of the formn1/k, . . . , n_d/k∈R^dwheren_iandkare integers. Write x n1/k, . . . , n_d/k∈T^d. Then there are exactlyk^dpoints inT^dof the formn1/k, . . . , n_d/k with 0≤ni< k.

The image of any such point under L_A may also be written in this form, since the entries of A are integers. Thus x is an eventually periodic point of L_A. Moreover, if pi,|detA| 1 for all 1≤i≤r,LAis injective on these points and henceLAis a permutation ofk^d such points. In fact, ifL_An1/k, . . . , n_d/k L_An₁/k, . . . , n_d/k, then we see that

An1/k, . . . , nd/k−An₁/k, . . . , n_d/k∈Z^d, orn1−n₁/k, . . . ,nd−n_d/k∈A⁻¹Z^d. SinceA⁻¹Z^d:Z^d Z^d:AZ^d |detA|andk,|detA| 1, we must have

n1−n₁

k , . . . ,n_d−n_d k

∈Z^d. 2.1

Hencen1/k, . . . , nd/k n₁/k, . . . , n_d/k. Therefore,x is a periodic point ofLA.

Corollary 2.3. Every toral endomorphismLA:T^d → T^dof the torusT^dhas dense periodic points.

Proof. Letpbe a prime number withp,|detA| 1 and letR Zp. Then byLemma 2.2, the points with coordinates inRare periodic. Moreover,Z^d\R^d, the set of points inT^dwith coordinates inR, is a dense subset of the torusT^d.

3. Nil Endomorphisms

In this section, we first recall from6–10some definitions about nilpotent Lie groups and give some basic properties which are necessary for our discussion.

Let G be a connected, simply connected nilpotent Lie group. A discrete cocompact subgroupΓofGis said to be a lattice ofG, and in this case, the quotient spaceΓ\Gis said to be a nilmanifold.

LetΓbe a lattice ofG. ThenΓis a finitely generated torsion-free nilpotent group. Recall that the lower central series ofΓis defined inductively byγ₁Γ Γandγ_i1Γ γiΓ,Γ.

Suppose thatΓisc-step nilpotent, that is,γ_cΓ/1, butγ_c1Γ 1. The isolator of a subgroup HofΓ, denoted by√^Γ

H, is the set

√Γ

H

x∈Γ|x^k∈H for some integerk >0

. 3.1

It is well known6,9, page 473or10that the sequence Γ Γ1 ^Γ

γ₁Γ⊃Γ2^Γ

γ₂Γ⊃ · · · ⊃Γc^Γ

γ_cΓ⊃Γc1 1 3.2

forms a central series withΓi/Γ_i1 ∼Z^ki. It follows that it is possible to choose a generating set

a1,a2, . . . ,ac 3.3

ofΓin such a way thatΓi is the group generated byai {ai1, a_i2, . . . , a_in}andΓ_i1 for each i1,2, . . . , c. We refer toa{a1,a2, . . . ,ac}as a preferred basis ofΓ.

We useGto indicate the Lie algebra ofG. This Lie algebraGhas the same dimension and nilpotency class asG. Moreover, in the case of connected, simply connected nilpotent Lie groups it is known that the exponential map exp :G → Gis a diffeomorphism. We denote its inverse by log. IfG is another connected, simply connected nilpotent Lie group, with Lie algebraG, then we have the following properties.

iFor any homomorphism φ : G → G of Lie groups, there exists a unique homomorphism dφ : G → G differential of φ of Lie algebras, making the following diagram commuting:

G

log

φ

G

log

G

exp

dφ G

exp 3.4

iiConversely, for any homomorphism dφ : G → G of Lie algebras, there exists a unique homomorphism φ : G → G of Lie groups, making the above diagram commuting.

Ifa is a preferred basis ofΓ, then loga{loga1,loga2, . . . ,logac} ⊂Gcan be regarded as a basis for the vector space G. We call the basis loga ofGpreferred. In particular, if Γis a lattice ofR^d, then every preferred basisa of Γbecomes a preferred basis loga a for the vector spaceR^d.

We first generalize the concept of toral automorphisms to that of nil endomorphisms and show that every nil endomorphism has eventually dense periodic points.

LetΓ\Gbe a nilmanifold and let ϕ : G → G be an automorphism satisfying that ϕΓ⊂Γ. Then the automorphismϕinduces a surjectionϕ_Γon the nilmanifoldΓ\Gand the following diagram is commuting:

G ^ϕ G

Γ\G ^ϕΓ Γ\G

3.5

Lemma 3.1. Letϕ : G → G be an automorphism satisfying thatϕΓ ⊂ Γ. Then dϕhas a block matrix, with respect to any preferred basis ofG, of the form

dϕ

⎡

⎢⎢

⎢⎢

⎢⎢

⎣

N₁ 0 . . . 0

∗ N₂ . . . 0 ... ... . .. ...

∗ ∗ . . . N_c

⎤

⎥⎥

⎥⎥

⎥⎥

⎦

, 3.6

where the diagonal blocksNi’s are integral matrices, and|detdϕ| Γ: ϕΓ. In particular, the automorphismϕonGrestricts to an automorphism on a lattice ofGif and only if its differentialdϕ has determinant±1.

The proof of this lemma is rather straight forward and so we omit the proof. See, for example,11, Lemma 3.1and12, Proposition 3.1.

Definition 3.2. Let Γ\ G be a nilmanifold and let ϕ : G → G be an automorphism with ϕΓ ⊂ Γ. Thenϕinduces a surjective mapϕ_Γ on the nilmanifoldΓ\G, which is one of the following two types.

Idϕhas determinant of modulus 1. In this caseϕ_Γis called a nil automorphism.

IIdϕ has determinant of modulus greater than 1. In this case ϕ_Γ is called a nil endomorphism.

If, in addition,ϕis hyperbolici.e.,dϕhas no eigenvalues of modulus 1, then we say that the nil automorphism or endomorphismϕ_Γis hyperbolic.

Example 3.3. Let Nil be the 3-dimensional Heisenberg group with its Lie algebra nil. That is,

Nil

⎧⎪

⎪⎨

⎪⎪

⎩

⎡

⎢⎢

⎣ 1 x z 0 1 y 0 0 1

⎤

⎥⎥

⎦|x, y, z∈R

⎫⎪

⎪⎬

⎪⎪

⎭, nil

⎧⎪

⎪⎨

⎪⎪

⎩

⎡

⎢⎢

⎣ 0 a c 0 0 b 0 0 0

⎤

⎥⎥

⎦| a, b, c∈R

⎫⎪

⎪⎬

⎪⎪

⎭. 3.7 It is easy to showsee13, Proposition 2.2that

Autnil∼

⎧⎪

⎪⎨

⎪⎪

⎩

⎡

⎢⎢

⎣

α γ 0

β δ 0

η μ αδ−βγ

⎤

⎥⎥

⎦| αδ−βγ /0, η, μ∈R

⎫⎪

⎪⎬

⎪⎪

⎭. 3.8

Thus we see that the differential of any automorphismϕon Nil has determinant detdϕ αδ−βγ²and eigenvaluesαδ−βγand1/2{αδ±

α−δ²4βγ}. Thus if|detdϕ|

1, then dϕ has an eigenvalue of modulus 1. Therefore, there are no hyperbolic nil automorphisms on any nilmanifoldΛ\Nil.There are examples of hyperbolic nil, nontoral, automorphisms. In fact, we can find such examples from many literatures. For example, we refer to2,14–18.

Via the exponential map

exp :

⎡

⎢⎢

⎣ 0 a c 0 0 b 0 0 0

⎤

⎥⎥

⎦∈nil−→

⎡

⎢⎢

⎢⎣

1 a c ab 2

0 1 b

0 0 1

⎤

⎥⎥

⎥⎦∈Nil, 3.9

we see that every automorphismϕon Nil is given as follows:

⎡

⎢⎢

⎣ 1 x z 0 1 y 0 0 1

⎤

⎥⎥

⎦−→

⎡

⎢⎢

⎣

1 αxγy z

0 1 βxδy

0 0 1

⎤

⎥⎥

⎦, 3.10

wherez αδ−βγzβγxy αβ/2x²ηxμy γδ/2y². Consider the subgroupsΛk, k∈Z, of Nil:

Λk

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

⎡

⎢⎢

⎢⎣ 1 m

k 0 1 n 0 0 1

⎤

⎥⎥

⎥⎦|m, n, ∈Z

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

. 3.11

These are lattices of Nil, and every lattice of Nil is isomorphic to someΛk. The following matrices dϕ give simple examples which induce hyperbolic nil endomorphisms on the nilmanifoldΛ2k\Nil:

⎡

⎢⎢

⎣ 2 1 0 1 3 0 n m 5

⎤

⎥⎥

⎦,

⎡

⎢⎢

⎣ 3 1 0 1 1 0 n m 2

⎤

⎥⎥

⎦. 3.12

Note that the first one has eigenvalues of modulus all greater than 1, 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 ϕ_Γ is a nil automorphism, then the automorphismϕ⁻¹ : G → G induces a nil automorphism which isϕ⁻¹_Γ . In particular,ϕ_Γis a diffeomorphism ofΓ\G.

By refining the central series ofΓexplained in the paragraph aboveLemma 3.1, we can find a central series

Γ Γ1 ⊃Γ2⊃Γ3⊃ · · · ⊃Γd⊃Γ_d11 3.13

withΓi/Γi1∼Z, for eachi1,2, . . . , d.We are assuming thatGisd-dimensional, and using the same symbol for terms of a refinement of the previous central series.We can choose a generating set

a{a1, a₂, a₃, . . . , a_d} 3.14 ofΓin such a way thatΓiis the group generated byai andΓ_i1. Then any element γ ∈ Γis uniquely expressible as a product:

γaⁿ₁¹aⁿ₂²· · ·aⁿ_d^d, with→−n n1, n2, . . . , nd∈Z^d, 3.15 and we can regardGas the Mal’cev completion ofΓ:

G

a^r₁¹a^r₂²· · ·a^r_d^d | −→r r₁, r₂, . . . , r_d∈R^d

. 3.16

We refer to this preferred basisa {a1, a2, . . . , ad}as a canonical basis of Γ. Given→−n ∈ Z^d, we useγ→−nto denote the element ofΓwhose canonical coordinate is→−n. Thus, we have an identificationγ:Z^d → Γsending→−ntoγ→−n.

Among interesting properties of this identification, we recall the following 7, Theorem 2.1.3: for any homomorphismκ:Γ → Γ, there exists a polynomial function with rational coefficientsψ_κ :Z^d → Z^d such thatκγ→−n γψκ→−nfor all→−n ∈ Z^d. Moreover, any homomorphism ofΓextends to a homomorphism ofGby using the same polynomial.

Example 3.5. The mapψ : Nil → Nil given by ψ

x, y, z

3xy, xy,2zxy3

2x²nxmy1 2y²

3.17 is a polynomial function with rational coefficients, which sends Λ2 into Λ2 itself. The polynomial functionψis associated to the homomorphism

⎡

⎢⎢

⎣ 3 1 0 1 1 0 n m 2

⎤

⎥⎥

⎦ 3.18

on Nil given inExample 3.3.

We recall the famous Campbell-Baker-Hausdorffformula:

loga·b loga∗logb ∀a, b∈G, 3.19 where

A∗BAB1

2A, B ^∞

m3

CmA, B. 3.20 HereC_mA, Bstands for a rational combination ofm-fold Lie brackets inAandB. Since our Lie algebra is nilpotent, the sum involved inA∗Bis always finite. Throughout this paper, we shall useQZ_q1,...,qswhenever{q1, . . . , q_s}is the set of all prime factors of the denominators of the reduced rational coefficients appearing in the Campbell-Baker-Hausdorffformula. For example, ifGis 3-step nilpotent, then

loga·b logalogb1 2

loga,logb

1 12

loga,logb ,logb

− 1 12

loga,logb ,loga

,

3.21

and henceQZ_2,3{k/2^m3ⁿ|k∈Z, m, n∈Z}.

Lemma 3.6. For any homomorphism κ : Γ → Γ, the associated polynomial function ψκ has coefficients inQZ_q1,...,qs.

Proof. Suppose thatGis ad-dimensional connected, simply connected nilpotent Lie group.

The first thing to notice is that for anyX, Yin the Lie algebraGofG, we have that

expXY expXexpYexp

m2

D_mX, Y

, 3.22

whereDmdenotes a linear combination ofm-fold brackets inXandYwith coefficients in the ringQ. To see this, let us make the following computation:

expXexpY expX∗Y

expXYexp−X−YexpX∗Y because expXYexp−X−Y 1 expXYexp−X−Y∗X∗Y.

3.23

From this it follows that

expXY expXexpYexp−−X−Y∗X∗Y, 3.24

which is of the form3.22claimed above.

Now, letA1, A2, . . . , Ad be a canonical basis ofGWe meanAi logaiwhere theai

form a canonical basis ofΓ. SinceD_mY, X −DmX, Y, we have from3.22that

expYexpX expXexpYexp

2

m2

D_mX, Y

. 3.25

By a repeated use of formulas3.22and3.25it is now easy to see that expα1A₁α₂A₂· · ·α_dA_d exp

p₁α1, . . . , α_dA1

· · ·exp

p_dα1, . . . , α_dAd

, 3.26

wherepiα1, . . . , αdis a polynomial with coefficients inQ. We will use this fact below.

Finally, letκbe the Lie group homomorphism ofGwhich extends uniquely the given κ : Γ → Γ. Leta_i be a term of the canonical basis ofΓ, thenκai a^β₁¹a^β₂²· · ·a^β_d^d for some βjβjai∈Z. Using the Campbell-Baker-Hausdorffformula, it is then easy to seelook also at the computation belowthat

dκ

logai

logκai

d j1

γ_jlog a_j

for someγ_j∈Q. 3.27

We now compute κ

a^x₁¹a^x₂²· · ·a^x_d^d

exp log κ

a^x₁¹a^x₂²· · ·a^x_d^d expdκ

log

a^x₁¹a^x₂²· · ·a^x_nⁿ expdκ

x₁loga1∗log

a^x₂²· · ·a^x_d^d expdκ

x1loga1∗

x2loga2∗log

a^x₃³· · ·a^x_d^d expdκ

_c

m1

Em

loga1,loga2, . . . ,logad .

3.28

HereE_mstands for a term which is a linear combination ofm-fold brackets of the logaiand where the coefficients are polynomials in the variablesx_jover the ringQ. By continuing this computation, we see that

κ

a^x₁¹a^x₂²· · ·a^x_d^d exp

m1

Em

dκ

loga1 , dκ

loga2

, . . . , dκ

logad

. 3.29

Now using3.27we derive that κ

a^x₁¹a^x₂²· · ·a^x_d^d exp

q₁x1, x₂, . . . , x_dloga1 · · ·q_dx1, x₂, . . . , x_dlogad

, 3.30 where theqiare polynomials with coefficients inQ. Therefore, using3.26, this implies that the polynomialψ_κis as required.

Remark 3.7. Our original proof was longer treating the case whereGis a 2-step nilpotent Lie group. This one was provided by one of the referees.

Now we fix a canonical basis{a1, . . . , a_d}ofΓ. A pointΓxof the nilmanifoldΓ\Gis said to have rational coordinates or simplyxhas rational coordinates ifxγ→−q a^q₁¹a^q₂²· · ·a^q_d^d for some→−q ∈ Q^d. First we show that ifΓx Γyandx γ→−qwith→−q ∈Q^d, theny γ→−p for some→−p ∈ Q^d. We recall the following7, Theorem 2.1.1: there exists a polynomial function with rational coefficientsμ:Z^d×Z^d → Z^dsatisfyingγ→−m·γ→−n γμ→−m,→−nfor all→−m,→−n∈Z^d. The group product onGis defined using this polynomialμ. Now, suppose that Γx Γyand x γ→−qwith→−q ∈ Q^d. Theny zxfor somez ∈ Γ. Sincez ∈ Γ,z γ→−n for some→−n ∈ Z^d. Hence we havey zx γ→−n·γ→−q γμ→−n,→−q. Since→−n ∈Z^d,→−q ∈Q^d, andμis a polynomial function with rational coordinates, we must haveμ→−n,→−q ∈Q^d. This proves our assertion. Therefore the points ofΓ\Gwith rational coordinates are well defined.

Consequently for a subringRofQwithZ⊂R, the points ofΓ\Gwith coordinates inRare well defined.

It is known that everyinfra-nil automorphism has dense periodic pointssee the proof of4, Lemma 3. Now we will generalize this to the case ofinfra-nil endomorphisms.

The proof below is exactly the same as that ofLemma 2.2, except that the coefficients involved are different and henceLemma 3.6is essential.

Theorem 3.8. Let ϕ_Γ : Γ \G → Γ \G be a nil endomorphism of the nilmanifold Γ \G. Let R be a ring obtained from Q by adding finitely many primes p_j, that is, R Q Z_p1,...,pr. Then every point with coordinates in R is an eventually periodic point of ϕ_Γ. Moreover, if qi,|detdϕ| pj,|detdϕ| 1 for alli, j, then every point with coordinates inRis a periodic point ofϕ_Γ.

Proof. We will show this by induction on the nilpotency classcofG. Ifc1, thenΓ\Gis a torus and this case is proved inLemma 2.2.

Now let c > 1 and assume that the assertion is true for any connected, simply connected nilpotent Lie groupG of nilpotency class≤c−1 and for any ring obtained from Q by adding finitely many primes.

ConsiderG γ_cGand Γ ^Γ

γ_cΓ, and the principal fiber bundleT → M → B whereM Γ\G,T Γ\G is a torus andB Γ\G is a nilmanifold of dimension less than that ofM. Since the automorphismϕ:G → GmapsΓinto itself, its induced mapϕ_Γ is fiber-preserving. That is, the following diagram is commuting:

T ^ϕˆ T

M ^ϕΓ M

B ^ϕ¯ B

3.31

Now we note thatΓ Γ i for someΓi in the refined central series ofΓ. ThusΓandΓ have central series

Γ Γ i⊃Γi1⊃ · · · ⊃Γd⊃Γd11,

Γ Γ/Γi Γ1/Γi⊃Γ2/Γi⊃ · · · ⊃Γi−1/Γi⊃Γi/Γi1.

3.32

The canonical basisa {a1, a2, . . . , ad}ofΓinduces the canonical basesa {ai, a_i1, . . . , ad} anda {a1, a₂, . . . , a_i−1}ofΓandΓ, respectively, whereastands for the image ofa∈ΓinΓ under the natural surjectionΓ → Γ. Hence the points inT Γ\Gwith rational coordinates are well defined. Furthermore the points inB Γ\Gwith rational coordinates are also well- defined.

For→−q q1, q2, . . . , qd∈R^d, write

xa^q₁¹a^q₂²· · ·a^q_d^d, ya^q₁¹a^q₂²· · ·a^q_i−1ⁱ⁻¹, za^q_iⁱa^q_i1ⁱ¹· · ·a^q_d^d. 3.33

Thenx yzandz ∈G. Since x y∈ G,x y Γyis a point ofΓ\Gwith coordinates in R.Note that the ringQwhen working over the groupGis a subring ofQand soQ ⊂R.

By induction hypothesisϕ^tky ϕ^tyfor somek ≥ 1 andt ≥0. On the other hand, since

z Γzis a point of the torusΓ\Gwith coordinates inR, byLemma 2.2,ϕ^τ z ϕ^τzfor some ≥1 andτ ≥0. We may assume thattτ andk so that

ϕ^tky ϕ^ty,

ϕ^tkz ϕ^tz. 3.34

Thenϕ^tky ξϕ^tyinGfor someξ∈Γ;ϕ^tky ξϕ^tywfor somew ∈G. By Lemma 3.6,

w ∈ G has coordinates inR. Furthermore, ϕ^tkz γϕ^tz ϕ^tzγ for some γ ∈ Γ. Let x₁ϕ^tx. Then

ϕ^kx1 ϕ^ktx ϕ^kt y

ϕ^ktz

ξϕ^t y

w

γϕ^tz

ξwϕ^tx ξwx1

for somew ∈G.

3.35

Simply taking ψ ϕ^k, we may assume thatψx1 ξwx₁ where ξ ∈ Γ andw ∈ G with coordinates inR. HenceLemma 2.2can be used to conclude thatψ^muw γψ^uwfor some m≥ 1,u ≥0, andγ ∈Γ. Thus ψ^imuw γ_iψ^uwfor someγ_i ∈Γ. We note further that for anyn >0,

ψ^nmx1 νψ^nm−1wψ^nm−2w· · ·ψ²wψwwx1

ν

⎧⎨

⎩ m−1

j0

ψ^j

ψ^n−1mw· · ·ψ^mww⎫

⎬

⎭x1, ψ^nmux1 ψ^u

ν⎧

⎨

⎩

m−1

j0

ψ^j

ψ^n−1muw· · ·ψ^muwψ^uw⎫

⎬

⎭ψ^ux1 ν

⎧⎨

⎩

m−1

j0

ψ^j

ψ^uwⁿ⎫

⎬

⎭ψ^ux1 ν

⎧⎨

⎩

m−1

j0

ψ^j

ψ^uwⁿ⎫

⎬

⎭ψ^ux1

3.36

for someν, ν∈Γ. Sincew ∈G R^kwith coordinates inR, there isn >0 such thatwⁿ∈Γ Z^k. SinceψΓ ⊂ Γ,ψ^jψ^uwⁿ∈Γfor allj 0,1, . . . , m−1. Henceψ^nmux1 νψ^ux1, or ϕ^knmkutx νϕ^kutxfor someν∈Γ. Therefore

ϕ^knmkut_Γ x ϕ^kut_Γ x, 3.37

which implies thatx is an eventually periodic point ofϕ_Γ.

Moreover, if qi,|detdϕ| pj,|detdϕ| 1 for alli, j, then byLemma 2.2and induction hypothesis, we can choosetu0 and soϕ^knm_Γ x x. Thus x is a periodic point ofϕ_Γ.

Corollary 3.9. Every nil endomorphismϕ_Γ : Γ\G → Γ\Gof the nilmanifold Γ\Ghas dense eventually periodic points.

Proof. Using the fact that the points ofΓ\Gwith coordinates inRare dense inΓ\G, we obtain the result.

Example 3.10. Let ϕ_Λ

2 be the hyperbolic nil endomorphism on the nilmanifold Λ2 \ Nil induced by the automorphism on Nil:

ϕ

⎡

⎢⎢

⎣ 3 1 0 1 1 0 0 0 2

⎤

⎥⎥

⎦:

⎡

⎢⎢

⎣ 1 x z 0 1 y 0 0 1

⎤

⎥⎥

⎦−→

⎡

⎢⎢

⎢⎣

1 3xy 2z3

2x²xy1 2y²

0 1 xy

0 0 1

⎤

⎥⎥

⎥⎦. 3.38

Then

ϕ

⎛

⎜⎜

⎜⎜

⎝

⎡

⎢⎢

⎢⎢

⎣ 1 1

2 3 8 0 1 1 2 0 0 1

⎤

⎥⎥

⎥⎥

⎦

⎞

⎟⎟

⎟⎟

⎠

⎡

⎢⎢

⎢⎣ 1 2 3

2 0 1 1 0 0 1

⎤

⎥⎥

⎥⎦≡

⎡

⎢⎢

⎣ 1 0 0 0 1 0 0 0 1

⎤

⎥⎥

⎦ modΛ2. 3.39

Thus the point

⎡

⎢⎢

⎢⎢

⎣ 1 1

2 3 8 0 1 1 2 0 0 1

⎤

⎥⎥

⎥⎥

⎦∈Λ2\Nil 3.40

is not a periodic point, but an eventually periodic point ofϕ_Λ2 with least period 1 i.e., an eventually fixed point. Note here that detdϕ 2² and 1/2 is the coefficient coming from the nilpotent Lie group Nil.

At this moment, we donot know whetherCorollary 3.9is true for periodic points in the general case, that is, the case whereqi,detdϕ/1 for somei. We now propose naturally the following problem.

Question 1. Every nil endomorphism has dense periodic points.

Corollary 3.11. Every nil automorphismϕ_Γ : Γ\G → Γ\Gof the nilmanifold Γ\Ghas dense periodic points.

Proof. The proof follows from that|detdϕ|1.

4. Infra-Nil Endomorphisms

LetGbe a connected, simply connected nilpotent Lie group and letCbe a maximal compact subgroup of AutG. A discrete and cocompact subgroupΠofGC⊂AffG GAutGis called an almost crystallographic group. Moreover, ifΠis torsion-free, thenΠis called an almost Bieberbach group and the quotient spaceΠ\Gan infra-nilmanifold. In particular, ifΠ⊂G, then Π\G is a nilmanifold. Recall from 19that Γ Π∩G is the maximal normal nilpotent subgroup ofΠwith finite quotient groupΨ Π/Γ, called the holonomy group ofΠ\G.

Definition 4.1. Let Π\G be an infra-nilmanifold and let ϕ : G → Gbe an automorphism which is weaklyΠ-equivariant; that is, there is a homomorphismθθ_ϕofΠsuch that

ϕαx θαϕx, α∈Π, x∈G. 4.1

Thenϕinduces a surjectionϕ_Π :Π\G → Π\G, which is one of the following types.

Idϕhas determinant of modulus 1. In this caseϕ_Πis called an infra-nil automorphism.

IIdϕhas determinant of modulus greater than 1. In this caseϕ_Π is called an infra-nil endomorphism.

If, in addition,ϕis hyperbolic, then we say that the infra-nil automorphism or endomorphism ϕ_Π is hyperbolic.

LetΠ\Gbe an infra-nilmanifold with surjectionϕ_Π:Π\G → Π\G. LetΓ Π ∩ G be the pure translations ofΠ. Then it is not difficult to see that there exists a fully invariant subgroupΛ⊂ΓofΠwith finite index. For example, one can takeΠ^Π:Γsee also20, Lemma 3.1. ThusΛ\Gis a nilmanifold which is a finite regular covering ofΠ\G and hasΠ/Λ as the group of covering transformations. The homomorphismθ : Π → Πassociated with ϕ_Π induces a homomorphismθ:Λ → Λand in turn induces a homomorphismθ:Π/Λ → Π/Λso that the following diagram is commuting:

1 Λ

θˆ

Π

θ

Π/Λ

θ¯

1

1 Λ Π Π/Λ 1

4.2

Moreover, the automorphismϕonG induces a surjectionϕ_Λ : Λ\G → Λ\G so that the following diagram is commuting:

Λ\G ^ϕΛ Λ\G Π\G ^ϕΠ Π\G

4.3

Sinceϕλx θλϕx for allλ ∈ Λ, x ∈ G, we have ϕλ θλ for all λ ∈ Λ. Hence ϕ : G → G is the unique extension of the homomorphismθ : Λ → Λ of the lattice Λ

ofG. Ifθis an isomorphism, thenθis also an isomorphism. Conversely, assume thatθis an isomorphism. Using the fact thatΠis torsion-free, we can show thatθis injective. This fact implies thatθis also injective on the finite groupΠ/Λand henceθmust be an isomorphism.

Therefore,θis an isomorphism.The converse was suggested by a referee.Ifϕ_Πis an infra-nil automorphism, then being|detdϕ|1 implies byLemma 3.1thatθis an isomorphism and thusϕ_Λis a nil automorphism, and vice versa. Note also thatϕ_Πis an infra-nil endomorphism if and only ifϕ_Λis a nil endomorphism.

Let ePerfdenote the set of eventually periodic points of a self-mapf:X → X.

Theorem 4.2. Every infra-nil endomorphismϕ_Π :Π\G → Π\Ghas dense eventually periodic points.

Proof. Consider the following commuting diagram:

Λ\G

p ϕΛ

Λ\G

p

Π\G ^ϕΠ Π\G

4.4

where ϕ_Π is an infra-nil endomorphism, and hence ϕ_Λ is a nil endomorphism. First we observe that ePerϕ_Λ p⁻¹ePerϕ_Π. The inclusion ⊆ is obvious. For the converse, let

x ∈p⁻¹ePerϕΠandpx x ∈ePerϕΠ. Thenϕ^mt_Π x ϕ^t_Πxfor somem > 0 andt≥0.

Clearlypϕ^t_Λx ϕ^t_Πxandϕ^m_Λ :p⁻¹ϕ^t_Πx → p⁻¹ϕ^t_Πxis a permutation on the finite set p⁻¹ϕ^t_Πx. Henceϕ^m_Λ ϕ^t_Λx ϕ^t_Λxfor some . The reverse inclusion⊇is proved. Now by the continuity ofpand byCorollary 3.9, we have

Π\GpΛ\G p

ePerϕ_Λ

⊂pePerϕ_Λ pp⁻¹ePerϕ_Π ePerϕ_Π. 4.5

This proves that ePerϕ_Πis dense inΠ\G.

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 GovernmentMOST no. R01-2007-000-10097-0.

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