The rotation class of a flow

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Topology and its Applications 152 (2005) 201–208

www.elsevier.com/locate/topo

The rotation class of a flow✩

Alex Clark

Department of Mathematics, University of North Texas, Denton, TX 76203-1430, USA

Abstract

Generalizing a construction of A. Weil, we introduce a topological invariant for flows on comconnected, finite dimensional, Abelian, topological groups. We calculate this invariant for somamples. 2004 Elsevier B.V. All rights reserved.

MSC: 37B05; 37B45

Keywords: Flow; Invariant; Solenoid

1. Introduction

For our purposes aflow is a continuous group action of(R,+). We consider the flowφ on X and the flowψ on Y to be topologically equivalent when there is a homeomophismh :X → Y which takes orbits ofφ onto orbits ofψ in such a way that the orientatio

of orbits is preserved, in which case we writeh: φtop≈ ψ . Following the lead of A. Weil

as outlined in [8,9], we introduce therotation class of a flow: an invariant of topological equivalence for flows on finite dimensional, compact, connected, Abelian, topolgroups—hereafter referred to asn-solenoids, see Definition 1 or [3]. A modern accounWeil’s torus invariant and its application to toral flows may be found in [2], [1, ChapteFor eachn-solenoid we shall describe a covering space and a compactification of thering space, thecompactification of perspective. The remainder of this compactification

✩ This work was funded in part by a faculty research grant from the University of North Texas.E-mail address: [email protected] (A. Clark).

0166-8641/$ – see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.topol.2004.10.019

202 A. Clark / Topology and its Applications 152 (2005) 201–208

onding

er

ss is a

nce

homeomorphic toSn−1. Informally stated, we shall show that ifφ andψ are topologicallyequivalent flows on then-solenoid

∑M and if the lifted flow ofφ has an orbit withω-limit

set a point in the remainder of the compactification of perspective, then any corresporbit in the lifted flow ofψ will have a correspondingω-limit point in the remainder of thecompactification. The group structure of

∑M determines which points in the remaind

may correspond.We then calculate the rotation class in some examples for which the rotation cla

useful invariant. Since time changes do not change the structure ofω-limit sets of pointsof flows, it is clear that this invariant is well-suited for the study of topological equivalebut is not as useful in studying classification up to conjugacy.

2. Background and the compactification of perspective

In what follows,πn : Rn → Tn = (R/Z)n is the quotient covering map. IfM is ann×n

matrix with integer entries, the homeomorphismTn → Tn represented by M is the uniquemapf satisfyingf ◦ πn = πn ◦ M .

Definition 1. For a fixed and a sequenceM = (M1,M2, . . .) of n × n matricesMi with in-teger entries and non-zero determinants, we define the topological group

∑M with identity

eM to be the inverse limit of the inverse sequence{Xj , fij }, whereXj = Tn for all j ∈ N

andfj+1j is the epimorphism represented by the matrixMj :∑M

def={

〈xj 〉∞j=1 ∈∞∏

j=1

Tn | f j+1j (xj+1) = xj for all j ∈ N

},

and we define such an inverse limit∑

M to be ann-solenoid.

We shall only consider the case thatn ∈ {2,3, . . .}. As in [3] we have the fibration withunique path lifting

πM : Rn →∑

M

πM(s) = (πn(s),πn

(M−1

1 (s)), . . . , πn

(M−1

n ◦ · · · ◦ M−11 (s)

), . . .

).

If pi denotes the projection of∑

M onto theith Tn factor, thenp−11 (e) is a Cantor set or a

finite discrete space. We may then form the following covering of∑

M .

Definition 2.

EM

def= Rn × p−11 (e)

and

ΠM :EM →∑

M

is the covering map given by

ΠM(s, x) = πM(s) + x.

A. Clark / Topology and its Applications 152 (2005) 201–208 203

ng

See [6] for a similarly defined covering map.With hn : Rn → Dn = {x ∈ Rn | ‖x‖ < 1} denoting the homeomorphism

x = (x1, . . . , xn) → 1√1+ x2

1 + · · · + x2n

x

and identifying a pointx of Rn with its hn image inRn, we may considerDn = {x ∈ Rn |‖x‖ � 1} to be a compactification ofRn. Similarly we may considerDn × p−1

1 (e) to be acompactification ofEM via hn × id

p−11 (e)

. With Sn−1 = {x ∈ Rn | ‖x‖ = 1}, we form the

equivalence relation≈M on Dn × p−11 (e) defined by[

(s, x) ≈M (s′, x′)] ⇐⇒ [

s = s′ ∈ Sn−1].We then have the quotient mapping

qM : Dn × p−11 (e) → (

Dn × p−11 (e)

)/≈M,

the image of which is also a compactification ofEM , compactification of perspective. Theremainder of this compactification,

RM

def= [(Dn × p−1

1 (e))/≈M

] − [qM ◦ (

hn × idp−1

1 (e)

)(EM)

]is canonically homeomorphic toSn−1[

(t, x)]≈M

� t,

where[(t, x)]≈Mdenotes the≈M -class of a point inRM . Hereafter we identifyRM with

Sn−1 as above.In [3] the linear flows on

∑M

FM = {Φω

M| ω ∈ Rn − {0}} given by

Φω

M: R ×

∑M

→∑

M, Φω

M(t, x) = πM(t · ω) + x

were introduced. Each linear flowΦω

Mlifts to the flowΦω

MonEM given by

Φω

M

(t, (s, x)

) = (tω + s, x).

Sincehn(tω + s) → ω‖ω‖ as t → ∞, the image of anyΦω

M-orbit in the compactification

of perspective tends asymptotically toward the same pointω‖ω‖ in the remainderSn−1. See

Fig. 1 for the case of the two-dimensional torus.This leads naturally to the following.

Definition 3. The pointx ∈ Sn−1 is a point of perspective of the flowφ on∑

M if and onlyif φ has an orbit which lifts to an orbitO in the compactification of perspective satisfyiO(t) → x ast → ∞.

Thus, the linear flowΦω has precisely one point of perspective:ω .
M ‖ω‖


nd two

ismssby

to the

Fig. 1. The compactification of perspective of the torus, along with the dotted integer coordinate grid aorbits of the same lifted irrational flow.

3. The rotation class of a flow

In [3] the linear flows were classified up to topological equivalence. The automorphof

∑M play a key role in this classification. Ifα :

∑M → ∑

M is an automorphism, it ishown in [3] thatα lifts to an automorphismRn → Rn, which may then be representeda matrixA(α), as indicated in the following commutative diagram

Rn A(α)

πM

Rn

πM∑M

α ∑M

Any automorphismA : Rn → Rn induces the mapA :Sn−1 → Sn−1

A(x) = A(x)

‖A(x)‖ .

Translating the classification of linear flows into the language of perspective leadsfollowing notions.


l

d as

f

Definition 4. ForX ⊂ Sn−1

M(X)def=

⋃{A(α)(X) | α ∈ Aut

(∑M

)},

and therotation class of a flowφ on∑

M is given by

ρ(φ)def= M

({points of perspective ofφ}).Remarks. Sincex → −x is an automorphism of

∑M for anyM , we could work in rea

projective space just as well as inSn−1. When applied to the torus,ρ(φ) is called therotation orbit ofφ in [1, Chapter 6].

The topological classification of linear flows given in [3] may then be rephrasefollows:[

Φω

Mis topologically equivalent toΦω′

M

] ⇐⇒ [ρ(Φω

M

) = ρ(Φω′

M

)].

We now investigate how this can be generalized to other flows.

Lemma 5. If φ and ψ are flows on∑

M and h: φtop≈ ψ with h homotopic to translation by

some τ ∈ ∑M , then

{points of perspective of φ} = {points of perspective of ψ}.Proof. Supposeω ∈ Sn−1 is a point of perspective ofφ and thatO is a liftedφ-orbit ofx0 ∈∑

M with O(t) → ω ast → ∞ in the compactification of perspective. Generally ast → ∞the orbits of the lifted linear flowΦω

Mapproachω asymptotically in the compactification o

perspective. LetRn ×{c} be the path component ofEM containingO(R), which is foliatedby Φω

M. With eachx ∈ EM we associate the ray

Axdef= Φω

M

([0,∞), x).

Notice thatAx , the closure ofAx in the compactification of perspective, has onlyω in itsremainder. ThenAO(t) → ω ast → ∞ in the Hausdorff metric sinceO(t) → ω.

If τ(φ) denotes the flow

R ×∑

M→

∑M

, τ(φ)(t, x) = φ(t, x − τ) + τ

and ifO′ is a liftedτ(φ)-orbit of τ +x0 ∈ ∑M in Rn ×{c′}, then we have(s0, c) = O(0) ∈

Rn × {c} and(s′0, c

′) = O′(0) ∈ Rn × {c′} satisfying

πM(s0) + c = x0 and πM(s′0) + c′ = x0 + τ, implying

πM(s′0 − s0) = c − c′ + τ,

which in turn implies that ifτ(s, c)def= (s + s′

0 − s0, c′), then

Rn × {c} τ

(πM+c)

Rn × {c′}(πM+c′)∑ +τ ∑

M M


ing

-

sisction

tcts.tis

d

is a commutative diagram of maps andτ carries orbits ofΦω

Mto orbits ofΦω

Mand takes

AO(t) in Rn × {c} to a corresponding rayA′t of a Φω

M-orbit in Rn × {c′}. The raysA′

t aretranslates of the raysAO(t) by s′

0 − s0, and this difference is negligible under the mapp

hn ast → ∞. Again taking closures in the compactification of perspective,A′t → ω in the

Hausdorff metric, implying thatO′(t) → ω ast → ∞.

Now the mapδ(x)def= h(x) − (τ + x) is homotopic to the constant map

∑M → {eM},

and so there is liftδ making the following diagram commute

Rn

πM∑M

δ

δ ∑M

Since∑

M is compact,‖δ(x)‖ has a maximum valuem. Also, with

h(s, c)def= (

s + s′0 − s0 + δ

(πM(s) + c

), c′) = “

(δ(πM(s) + c

), c′) + τ (s, c)” ,

the following diagram commutes

Rn × {c} h

(πM+c)

Rn × {c′}(πM+c′)∑

Mh ∑

M

Thus, identifyingRn × {c′} with Rn, ‖h(s, c) − τ (s, c)‖ � m. For any preassigned numberK , we may chooseT sufficiently large so that for allt � T {inf ‖τ (a)‖ | a ∈ AO(t)} > K

in Rn ×{c′}. By the construction ofhn and the uniform bound‖h(s, c)− τ (s, c)‖ � m, we

conclude thath(At ) → ω at t → ∞ in the compactification of perspective. The analyis very similar to that of the torus case [1, Chapter 6,1.5], where the difference funis not only uniformly bounded but also doubly periodic. Eachh(AO(t)) contains a poinh(O(t)) = O′′(t ′′) in a lift O ′′ of theψ -orbit of h(x0). While there is no reason to expethatt ′′ = t , we may concludet ′′ → ∞ ast → ∞ sinceh preserves the orientation of orbitHence,ω is a point of perspective ofψ as well. The other set containment follows mutamutandis. �Lemma 6.

If a ∈ Aut(∑

M) and α: φtop≈ ψ , then ρ(φ) = ρ(ψ).

Proof. Let ω ∈ Sn−1, O, x0 ∈ ∑M be as above. Withω′ = A(α)(ω), α maps orbits ofΦω

M

to orbits ofΦω′M

after (possibly) rescaling time (see [3, 3.5]), andα lifts to an affine mapRn × {c} → Rn × {c}, so that the above argument forτ(φ) can be slightly adapted anapplied here to show thatω′ is a point of perspective ofψ . SinceM({ω}) = M({ω′}), weconclude thatρ(φ) ⊂ ρ(ψ). Similarly, withα−1 in place ofα, ρ(φ) ⊃ ρ(ψ). �Theorem 7. If the flows φ and ψ on

∑are topologically equivalent, then ρ(φ) = ρ(ψ).
M


ed andificant

e-

Proof. Suppose thath: φtop≈ ψ and letω ∈ Sn−1, O, x0 ∈ ∑

M be above. Withτ :∑

M →∑M denoting translation byh(eM), the homeomorphismg = τ−1 ◦ h fixes the identity

elementeM . From [7] (or see [3, 3.8]), it follows that there is a homotopy

H : [0,1] ×∑

M→

∑M

from g = H0 to an automorphismα = H1. Then

h′ def= h ◦ α−1 = τ ◦ g ◦ α−1

is homotopic toτ via τ ◦ Ht ◦ α−1, implying that h = h′ ◦ α, whereh′ is homotopicto a translation. Then by the above lemmas it follows thatω′ = A(α)(ω) is a point ofperspective ofψ . SinceM({ω}) = M({ω′}), we conclude thatρ(φ) ⊂ ρ(ψ) and similarlyρ(φ) ⊃ ρ(ψ). �

4. Examples of the rotation class

Recall that a non-empty setM ⊂ X is aminimal set of the flowφ on X if and only ifM is closed and invariant and has no proper, non-empty subset which is also closinvariant. In the cases where the rotation class is most useful, the orbits of all the signminimal sets of the flow all approach the same point of perspective.

Given a countable subgroupE of (R,+), we now proceed to find a flow with a corrsponding point of perspective on the solenoid which is the Pontryagin dual toE . First wefind a maximal rationally independent subset

A = {αi}ni=1 ⊂ E

(n is the torsion-free rank ofE , see [5]) with

{αi}n−1i=1 ⊂ R − Q (it is possible thatA ⊂ R − Q).

If n = 1 we obtain the desired flow by a linear flow on a 1-solenoid. Whenn > 1, foreachi ∈ {1, . . . , n − 1} we construct a Denjoy homeomorphismhi of S1 with rotationnumberαi . Then for eachi there is a monotone mapσi :S1 → S1 isotopic to idS1 whichprovides a semiconjugacy betweenhi and rotation byπ1(αi) in S1. Each suchhi has aminimal Cantor setCi . Then

C =n−1∏i=1

Ci

is yet another Cantor set and

h((xi)

n−1i=1

) def= (hi(xi)

)n−1i=1

is a homeomorphism ofTn−1 that hasC as an invariant set since eachCi is hi -invariant.Also

σ((xi)

n−1) def= (σi(xi)

)n−1
i=1 i=1


.

the

a

stw on

se ofhese

d

stems1996.

torus,

–209.. Math.

is isotopic to idTn−1 and provides a semiconjugacy betweenh andR, rotation inTn−1 by(π1(αi))

n−1i=1 , which is minimal by the rational independence of theαi .

The semiconjugacy ofh|C to R shows thath|C is minimal [10, Chapter IV.6.1(b)]Recall that ifh :X → Y provides a continuous semiconjugacy between a flowφ onX anda flow ψ on Y , φ is said to be analmost 1 : 1-extension of ψ when there is somex ∈ X

with a denseφ-orbit satisfyingh−1(h(x)) = {x}, see, e.g., [10, Chapter IV.6.1]. Thenh|Cis an almost 1 : 1 extension ofR. The suspension ofh will then be a flow onTn having aone-dimensional minimal set havingC as a 0-dimensional cross-section. By adjustingtime scale on this flow to have a return time toC of 1/αn, we obtain a minimal flowφ ona one-dimensional subcontinuum ofTn.

By enumerating the elements ofE − 〈A〉 = {g1, g2, . . .} (possibly empty), we obtaindirect limit representation ofE

〈A〉 ↪→ ⟨A ∪ {g1}

⟩↪→ ⟨

A ∪ {g1, g2}⟩↪→ ·· ·E

and dual to this direct sequence is an inverse sequence

TnE1� Tn

E2� TnE3� · · ·

∑M

where eachEi is an epimorphism and the inverse limit is ann-solenoid. Pulling the flowφ back by the variousEi as described in [4], we obtain a flow on

∑M . This flow has a

one-dimensional invariant setM which is minimal [10, Chapter IV.6.1(b)] and is an almo1 : 1 extension of the corresponding linear flow obtained by pulling back the linear floTn to a linear flow on

∑M . This construction in the casen = 2 is considered in detail in

[4] where such minimal sets (called denjoids) are classified topologically. In the caa general denjoidE is not finitely generated and has torsion-free rank 2. In general, tflows will have only one point of perspective:

α = (α1, . . . , αn)

‖(α1, . . . , αn)‖since the lifted flow lines inRn ×{c} differ from a lifted irrational flow by a map (obtainefrom σ(x) − x) homotopic to a constant, implying the existence of a boundedδ as inLemma 5.

References

[1] S.Kh. Aranson, G.R. Belitsky, E.V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Syon Surfaces, Transl. Math. Monographs, vol. 153, American Mathematical Society, Providence, RI,

[2] S.Kh. Aranson, E.V. Zhuzhoma, The topological classification of singular dynamical systems on theIzv. Vyssh. Uchebn. Zaved. Mat. 5 (168) (1976) 104–107.

[3] A. Clark, Linear flows onκ-solenoids, Topology Appl. 94 (1999) 27–49.[4] A. Clark, Solenoidalization and denjoids, Houston J. Math. 26 (2000) 661–692.[5] L. Fuchs, Infinite Abelian Groups, Academic Press, New York, 1970.[6] M.C. McCord, Inverse limit sequences with covering maps, Trans. Amer. Math. Soc. 114 (1965) 197[7] W. Scheffer, Maps between topological groups that are homotopic to homomorphisms, Proc. Amer

Soc. 33 (2) (1972) 562–567.[8] A. Weil, Les families de courbes sur le tore, Mat. Sb. 1 (1936) 779–781.[9] A. Weil, On systems of curves on a ring-shaped surface, J. Indian Math. Soc. 19 (1931) 109–114.

[10] J. de Vries, Elements of Topological Dynamics, Kluwer Academic, Dordrecht, 1993.

The rotation class of a flow

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