Pseudo-Orbit Tracing and Algebraic actions of countable amenable groups

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by  Tom Meyerovitch

PSEUDO-ORBIT TRACING AND ALGEBRAIC ACTIONS OF
COUNTABLE AMENABLE GROUPS
7
1 TOMMEYEROVITCH
0
2

b
e
F Abstract. Consideracountableamenablegroupactingbyhomeomorphisms
on a compact metrizable space. Chung and Li asked if expansiveness and
6 positive entropy of the action imply existence of an off-diagonal asymptotic
pair. Foralgebraicactionsofpolycyclic-by-finitegroups,ChungandLiproved
] it does. We provide examples showing that Chung and Li’s result is near-
S
optimal in the sense that the conclusion fails for some non-algebraic action
D
generatedbyasinglehomeomorphism,andforsomealgebraicactionsofnon-
. finitely generated abelian groups. On the other hand, we prove that every
h
expansive action of an amenable group with positive entropy that has the
t
a pseudo-orbit tracing property must admit off-diagonal asymptotic pairs. Us-
m ing Chung and Li’s algebraic characterization of expansiveness, we prove the
pseudo-orbit tracing property for aclass of expansive algebraic actions. This
[
classincludeseveryexpansiveprincipalalgebraicactionofanarbitrarycount-

2 ablegroup.
v
8
1. Introduction
1
3
This paper is partly motivated by relatively recent work of Chung and Li [3]
1
aboutthe dynamics ofcountable subgroupsofautomorphismsfor compactgroups,
0
. and algebraic actions in particular. Part of the paper [3] is an investigation of the
1
relation between topological entropy and asymptotic behavior of orbits for actions
0
ΓyX whereX isacompactmetrizablegroupandΓactsbygroupautomorphisms
7
1 [3]. The following question was posed and left open by Chung and Li [3]:
:
v Question 1.1. Let a countable amenable group Γ act by homeomorphisms on a
Xi compact metric space X. Suppose the action ΓyX is expansive and has positive
topological entropy. Must there be an off-diagonal asymptotic pair in X?
r
a
Under the additional assumptions that Γ is a polycyclic-by-finite group, that
X is a compact abelian group and that Γ y X is an expansive action of Γ by
automorphisms, Chung and Li obtained an affirmative answer to Question 1.1 [3,
Theorem1.2]. Ontheotherhand,asremarkedin[3],thereisamucholderexample
duetoLindandSchmidtofnon-expansiveZ-actions(infact,toralautomorphisms)
with positive entropy where all asymptotic pairs are on the diagonal [7, Expample
3.4].
2000 Mathematics Subject Classification. 22D40,37B05,37B40.
Keywords and phrases. Algebraicactions,expansiveness,groupactions,pseudo-orbittracing
property,subshiftoffinitetype.
TheresearchleadingtotheseresultshasreceivedfundingfromthePeopleProgramme(Marie
CurieActions)oftheEuropeanUnion’sSeventhFrameworkProgramme(FP7/2007-2013) under
REAgrantagreementno. 333598andfromtheIsraelScienceFoundation(grantno. 626/14).
1
In addition to expansiveness, the two main assumptions in [3, Theorem 1.2]
concernthe algebraicnature ofthe action(actionby automorphismsona compact
group)andthe grouptheoreticconditiononΓ(polycyclic-by-finite). We showthat
theseassumptionsarenotjustartifactsoftheproofmethod. Specifically,weprove:
Theorem 1.2. There exists a totally disconnected compact abelian group X and a
countable amenable subgroup Γ⊂Aut(X) so that ΓyX is expansive, has positive
topological entropy and no off-diagonal asymptotic pairs. In particular, the group
Γ can be an abelian group (for instance (Z/2Z)).
n∈Z
Theorem 1.3. There exists an expansivLe homeomorphism T :X →X of a totally
disconnected compact metrizable space X with positive topological entropy and no
off-diagonal asymptotic pair.
Theorem 1.3 might be considered a distant cousin of a striking result due to
Ornstein and Weiss stating that every transformation is bilaterally deterministic
[9].
In attempt to understand when Chung and Li’s question has an affirmative
answer,we are led to explore the pseudo-orbit tracing property. This fundamental
dynamicalpropertyturnsouttoensureanaffirmativeanswertoQuestion1.1. The
pseudo-orbittracing property was first introduced andstudied by R. Bowen[1] for
Z-actions,motivatedbythestudyofAxiomAmaps. Waltersandotherscontinued
this study and obtained further consequences of pseudo-orbit tracing [13]. Chung
andLeerecentlyconsideredthepseudo-orbittracingpropertyforactionsof(finitely
generated) countable amenable groups and showed that topological stability and
otherimportantconsequencesofthepseudo-orbittracingpropertyholdinthismore
general setting [2]. In relation with Question 1.1 above we have the following:
Theorem 1.4. Let Γ be a countable amenable group. Every expansive Γ-action on
a compact metrizable space that satisfies the pseudo-orbit tracing property and has
positive topological entropy admits an off-diagonal asymptotic pair.
The pseudo-orbit tracing property is of interest in the context of algebraic ac-
tions. Aparticularinstanceofoneofourresultappliestoprincipalalgebraicactions,
a well-known class of algebraic actions:
Theorem 1.5. Let Γ be a countable group. Every expansive principal algebraic
Γ-action satisfies the pseudo-orbit tracing property.
In the paper [3] Chung and Li also provided an affirmative answer to Question
1.1forprincipalalgebraicactionsforanycountableamenablegroupΓ(infactthey
proved much more, see Remark 3.6 below). This is also a conclusion of Theorem
1.5 combined with Theorem 1.4.
The organization of the paper is as follows: In section 2 we recall the pseudo-
orbit tracing property, subshifts and subshifts of finite type in particular. We also
prove Theorem 1.4. In section 3 we discuss the pseudo-orbit tracing for algebraic
actions and derive some consequences, in particular the proof of Theorem 1.5. In
section4weproveTheorem1.2,providinganegativeanswertoQuestion1.1within
the class of algebraic actions. In section 5 we prove Theorem 1.3.
Acknowledgements: IthankNishantChandgotia,Nhan-PhuChungandHan-
feng Li for valuable comments on an early version of this paper.
2
2. The pseudo-orbit tracing property for actions of countable
groups
Throughout this paper, Γ will be a countable group and X will be a compact
metrizablespace,equippedwithametricd:X×X →R . WedenotebyAct(Γ,X)
+
the space of all continuous actions Γ y X. The space Act(Γ,X) inherits a Polish
topology from Homeo(X)Γ. Specifically, given an enumeration of Γ = {g ,g ...},
1 2
we have the following metric on Act(Γ,X):
∞
1
ρ(α,β):= supd(α (x),β (x)).
2n gn gn
x∈X
n=1
X
In this paper when the action α ∈ Act(Γ,X) is clear from the context we will
sometimes write g·x instead of α (x).
g
Definition 2.1. (Compare with [2, Definition 2.5]) Fix S ⊂ Γ and δ > 0. A
(S,δ) pseudo-orbit for α ∈ Act(Γ,X) is a Γ-sequence (x ) in X such that
g g∈Γ
d(α (x ),x ) < δ for all s ∈ S and g ∈ Γ. We say that a pseudo-orbit (x ) is
s g sg g g∈Γ
ǫ-traced by x∈X if d(α (x),x )<ǫ for all g ∈Γ.
g g
Definition 2.2. An action α ∈ Act(Γ,X) has the pseudo-orbit tracing property
(abbreviated by p.o.t.) if for every ǫ > 0 there exists δ > 0 and a finite set S ⊂ Γ
such that every (S,δ) pseudo-orbit is ǫ-traced by some point x∈X.
Definition 2.3. (Compare with [2, Definition 2.1]) We say that α ∈ Act(Γ,X)
is topologically stable if for every ǫ > 0 there exists an open neighborhood U ⊂
Act(Γ,X) of α such that for every β ∈ U there exists a continuous f : X → X so
that α ◦f =f ◦β for every g ∈Γ and
g g
supd(f(x),x)≤ǫ.
x∈X
Chung and Lee [2, Theorem 2.8] proved the following:
Theorem 2.4. If an action α∈Act(Γ,X) is expansive and satisfies p.o.t, then it
is topologically stable. Moreover if η >0 is an expansive constant for α then:
(1) For every 0 < ǫ < η there exists an open neighborhood α ∈U ⊂ Act(Γ,X)
so that for every β ∈ U there exists a unique map f : X → X so that
α ◦f =f ◦β for every g ∈Γ and sup d(f(x),x)≤ǫ.
g g x∈X
(2) If furthermore β as above is expansive with expansive constant 2ǫ, then the
conjugating map f above is injective.
Here is a quick sketch of proof for Theorem 2.4: If β ∈ Act(Γ,X) is suffi-
ciently close to α then every β-orbit is an (S,δ) pseudo-orbit for α. By p.o.t this
pseudo-orbit is ǫ-traced by the α-orbit of some point y. If ǫ is sufficiently small,
expansiveness of α implies that y as above is unique. Furthermore, expansiveness
implies that the function f : X → X sending a point x ∈ X to the unique point
y whose α-orbit traces the β-orbit of x is continuous. Uniqueness of the map f
impliesitisΓ-equivariant. If2ǫisanexpansiveconstantforβ,thenitisimpossible
for a single y to ǫ-trace the β-orbits of two distinct points, so f is injective.
Remark 2.5. Strictly speaking, Chung and Lee restricted attention to finitely
generatedgroupsin[2]. Theextensiontogeneralcountablegroupsdoesnotrequire
anynewideas. Notethatthefollowingsimplificationoccursinthefinitelygenerated
3
case: If Γ is generated by a finite set S, α ∈ Act(Γ,X) satisfies p.o.t if for every
ǫ > 0 there exists δ > 0 such that every (S,δ) pseudo-orbit is ǫ-traced by some
point x∈X.
Definition 2.6. Given α ∈ Act(Γ,X), (x,y) ∈ X ×X is called a α-asymptotic
pair if lim d(α (x),α (y)) = 0. In other words, if for every ǫ > 0 there are
Γ∋g→∞ g g
at most finitely many g ∈Γ so that d(α (x),α (y))>ǫ.
g g
Proof of Theorem 1.4. Suppose α ∈Act(Γ,X) is expansive, satisfies p.o.t and has
positive topological entropy.
Fixǫ>0sothat2ǫisanexpansiveconstantforα. Becauseαsatisfiesp.o.tthere
exists δ > 0 and a finite set S ⊂ Γ so that every (δ,S) pseudo-orbit is ǫ/2-traced
by some x ∈ X. By further increasing S we can safely assume that S = S−1 and
that S contains the identity.
Let (F )∞ be a left-Følner sequence in Γ. Denote
n n=1
∂ F :={g ∈Γ : Sg∩F 6=∅ and Sg∩Fc 6=∅}.
S n n n
Because (F )∞ is a left-Følner sequence in Γ, it follows that
n n=1
|∂ F |
S n
(1) lim =0.
n→∞ |Fn|
Given a finite F ⊂Γ and δ >0, a set Y ⊂X is (F,δ)-separated if
maxd(α (x),α (y))≥δ for every distinct x,y ∈Y.
g g
g∈F
Let sep (X,d) denote the maximal cardinality of an (F,δ)-separated set in X.
δ,F
StandardargumentgivethatforeveryfiniteF ⊂Γandeveryδ >0thefollowing
holds:
logsep (X,d)≤|F|logsep (X,d).
δ,F δ/2,{1}
Thus by (1):
(2) logsep (X,d)=o(|F |),
δ,∂SFn n
Foreveryn>0,letX ⊂X bean(2ǫ,F )-separatedsetofmaximalcardinality.
n n
Because 2ǫ is an expansive constant for α, the topological entropy of α is equal to
1
(3) h(α)= lim log|X |>0.
n
n→∞|Fn|
By (2) and (3),
sep (X,d)=o(|X |).
δ,∂SFn n
In particular, for large enough n there exists distinct x,x′ ∈X so that
n
(4) max d(α (x),α (x′))<δ.
g g
g∈∂SFn
Define (y ) ∈XΓ as follows:
g g∈Γ
α (x′) g ∈F
g n
(5) y =
g
(αg(x) g ∈Γ\Fn
4
Then (y ) is a (S,δ) pseudo-orbit for α, and by p.o.t it is ǫ-traced by some
g g∈Γ
y ∈ X. Thus d(α (x),α (y)) < ǫ for every g ∈ Γ\F . This implies that (x,y) is
g g n
an α-asymptotic pair. Also
max d(α (x′),α (y))<ǫ,
g g
F∈Fn
Because {x,x′} are (F ,2ǫ)-separated, it follows that there exists g ∈ F so that
n n
d(α (x′),α (x)) > 2ǫ. By the triangle inequality, there exists g ∈ Γ so that
g g
d(α (x),α (y))>ǫ, and in particular x6=y. (cid:3)
g g
We now recall a class of Γ-actions called Γ-subshifts. These are also referred to
as symbolic dynamical systems.
Definition 2.7. Let A be a discrete finite set. Consider AΓ as a (metrizable)
topologicalspacewiththe producttopology. The(left) shiftactionσ ∈Act(Γ,AΓ)
is given by:
(6) σg ·(x)h =xg−1h.
The pair (AΓ,σ) is called the full shift with alphabet A over the group Γ. A
Γ-subshift is a subsystem of a full shift. In other words, the dynamical systems
(X,σ ) where action σ :=σ | ∈Act(Γ,X), where X ⊂AΓ is closed σ-invariant
X X X
subset of AΓ.
Evidently, every Γ-subshift is expansive. It is also well known that every ex-
pansive action Γ y X on a totally disconnected compact metrizable space X is
isomorphic to a Γ-subshift. An action ΓyX that is isomorphic to a Γ-subshift is
sometimes called a symbolic dynamical system. From this abstract point of view,
symbolicdynamicsisthestudyofexpansiveactionsonatotallydisconnectedcom-
pact metrizable space.
Let us recall a class of systems called subshifts of finite type, arguably the most
important class of systems in symbolic dynamics.
Definition 2.8. A Γ-subshift of finite type (Γ-SFT) is a subshift (X,σ ) of the
x
form:
(7) X = x∈AΓ : (g·x)| ∈L for every g ∈Γ ,
F
where F ⊂Γ is a finite(cid:8)set and L⊂AF. (cid:9)
Subshifts of finite type overa countable groupΓ can be characterizedas the set
of expansive α ∈ Act(Γ,X) that satisfy p.o.t, where X is a totally disconnected
compact metrizable space [2, 8, 13]. Another characterizationof subshifts of finite
typeis giveninterms ofacertaindescending chainconditionthatisreminiscentof
thedefinitionofNoetherianrings[12]. Avariantofthisdescendingchaincondition
appeared back in the early work of Kitchens and Schmidt on automorphisms of
compact groups [6].
Remark 2.9. Theorem 1.4 is a direct extension of the classical observation that
every subshift of finite type with positive entropy has an off-diagonal asymptotic
pair [10, Proposition 2.1].
Aswewillnowsee,inmanyrespects,expansiveactionsonacompactmetrizable
space X that satisfy p.o.t canbe thought ofas “systems of finite type”, evenwhen
X is not totally disconnected.
5
Definition 2.10. Suppose X is a compact metric space and α∈Act(Γ,X). Let
(8) Sub (X):={Y ⊂X : Y is closed and α (Y)=Y for every g ∈Γ}.
α g
The Hausdorff metric on the closed subsets of X induces a topology on Sub (X)
α
thatmakesitacompactmetricspace. WesaythatY ∈Sub (X)isalocalmaximum
α
ifthere exists anopenneighborhoodU ⊂Sub (X)of Y suchthat Z ⊆Y for every
α
Z ∈U.
Proposition 2.11. Let α ∈ Act(Γ,X) be expansive and satisfy p.o.t. If β ∈
Act(Γ,Y) is expansive and Φ : X → Y is continuous, Γ-equivariant and injective
then Φ(X)∈Sub (Y) is a local maximum in Sub (Y).
β β
Proof. Let α ∈ Act(Γ,X), β ∈ Act(Γ,Y) and Φ : X → Y be as above. Then
β | ∈Act(Γ,Φ(X)) is isomorphic to α and in particular satisfies p.o.t. Let ǫ be
Φ(X)
an expansive constant for β. Choose a finite S ⊂ Γ and δ ∈ (0,ǫ) so that every
(S,δ) pseudo-orbit for β | is ǫ/2-traced by some y ∈ Φ(X). Assume with out
Φ(X)
loss of generality that 1∈S. Now let
(9) U := Z ∈Sub (Y): sup inf maxd(β (z),β (Φ(x)))<δ/2 .
β g g
(cid:26) z∈Zx∈X g∈S (cid:27)
Then U is an open neighborhood of Φ(X) in Sub (Y). Now if Z ∈ U, then by
β
definition of U, for every z ∈Z there is (y ) ∈Φ(X)Γ so that
g g∈Γ
d(β (z),β (y ))<δ/2 for every g ∈Γ, h∈S.
hg h g
It follows that d(β (y ),y ) < δ for all s ∈ S and g ∈ Γ, so (y ) is an (S,δ)
s g sg g g∈Γ
pseudo-orbit for β | . Thus it is ǫ/2 traced by some y ∈ Φ(X). But for every
Φ(X)
g ∈Γ we have
d(β (y),β (z))<d(β (y),y )+d(y ,β (z))<ǫ.
g g g g g g
It follows that z =y, so z ∈Φ(X). It follows that Z ⊆Φ(X).
(cid:3)
Remark 2.12. A subsystem Z ∈ Sub (Y) is a local maximum if and only if it
α
satisfies the following stable intersection property : For every decreasing sequence
(Y )∞ ∈Sub (Y)N
n n=1 α
(10) Y ⊇Y ...⊇Y ⊇Y ⊇...
1 n n+1
such that Z = ∞ Y , there exists N ∈ N so that Y = Y for all n ≥ N.
n=1 n n N
The equivalence of these two conditions follows because the relation ⊆ is closed in
T
Sub (Y)×Sub (Y). The observationthat the stable intersectionproperty charac-
α α
terizes subshifts of finite type is due to K. Schmidt [12].
3. Pseudo-orbit tracing for algebraic actions
Inthissectionwediscussthepseudo-orbittracingforalgebraicactionsandsome
consequences. We derive Theorem 1.5 as a particular case of Theorem 3.4 below,
thus establishing pseudo-orbit tracing for a class of algebraic actions.
Let X be a compact metrizable abelian group. We denote by Act (Γ,X) ⊂
alg
Act(Γ,X) the collection of Γ-actions on X by continuous group automorphisms.
Every α ∈ Act (Γ,X) is called an algebraic action. We recall some notation,
alg
essentially following [3]:
6
Denote by ZΓ the group ring of Γ. Denote by ℓ∞(Γ) the Banach space of
all bounded R-valued functions on Γ, equipped with the k·k -norm. Also, de-
∞
note by ℓ1(Γ) the Banach algebra of all absolutely summable R-valued functions
on Γ, equipped with the ℓ1-norm k · k and the involution f 7→ f∗ defined by
1
f s ∗ := f s−1.
s∈Γ s s∈Γ s
For k ∈ N and p ∈ [1,∞], we write ℓp(Γ,Rk) := (ℓp(Γ))k, equipped with the
(cid:0)P (cid:1) P
suitablek·k -norm. Wedenotebyℓp(Γ,Zk)theintegervaluedelementsofℓp(Γ,Rk).
p
For k ∈N, let M (ℓ1(Γ)) denote the Banach algebra of k×k matrices with ℓ1(Γ)-
k
entries, with the norm
k(f ) k := kf k .
i,j 1≤i,j≤n 1 i,j 1
1≤i,j≤n
X
The involution on ℓ1(Γ) also extends naturally to an isometric linear involution on
M (ℓ1(Γ)) given by
k
(f )∗ :=(f∗ ) .
i,j 1≤i≤k,1≤i≤k j,i 1≤i≤k,1≤i≤k
The following is a classical and crucial fact in the theory of algebraic actions:
Pontryagin duality yields a natural one-to-one correspondence between algebraic
actions of Γ and (discrete, countable) ZΓ-modules. Thus, to each ZΓ-module M
corresponds an algebraic action α(M) ∈ Act (Γ,M). The dynamics of an al-
alg
gebraic action α(M) are completely determined by the algebraic properties of the
dualasaZΓ-module. Overthe years,asignificantnucmberofimportantdynamical
propertieshavefound beautiful algebraicinterpretationsin terms ofthe dual mod-
ule. For f ∈ZΓ, we let X denote the dualgroupof the ZΓ-module ZΓ/ZΓf. The
f
correspondingalgebraicactionα(f) ∈Act (Γ,X )iscalledtheprincipal algebraic
alg f
action associated with f.
Definition 3.1. If X is a compact group with identity element 1 ∈ X, and α ∈
Act (Γ,X), a point x ∈ X is called homoclinic with respect to the action α if
alg
(x,1)isanα-asymptoticpair. Inthiscasethehomoclinic group,denotedby∆(X),
is the set of all homoclinic points in X.
It is straightforward to check that ∆(X) is a Γ-invariant subgroup and (x,y) is
a an α-asymptotic pair if and only if xy−1 ∈∆(X).
We set up some more notation:
Let
(11) P :(ℓ∞(Γ))k →((R/Z)k)Γ
denote the canonicalprojectionmap, andlet ρ be the metric on(R/Z)k givenby
∞
(12) ρ v+Zk,w+Zk := min kv−w−mk , v,w∈Rk
∞ ∞
m∈Zk
(cid:0) (cid:1)
Lemma 3.2. For any δ < 1/2 and a˜,˜b ∈ (R/Z)k satisfying ρ (a˜,˜b) < δ, the
∞
following holds: Let a∈Rk be the unique element of [−1,1)k such that a˜=a+Zk.
2 2
Then there exists a unique b∈[−1,1]k so that ˜b=b+Zk and ka−bk <δ.
∞
Proof. Existence and uniqueness of a as above follows from the fact that Rk =
[−1,1)k+n . Similarly, there exists a unique b ∈ [−1,1)k +a so that
n∈Zk 2 2 2 2
b=˜b+Zk. Note that [−1,1)k+a⊆[−1,1]k because a∈[−1,1)k, so b∈[−1,1]k.
U (cid:0) (cid:1) 2 2 2 2
7
From the definition of ρ in (12), there exists n∈Zk so that ka−(b+n)k <δ.
∞ ∞
It follows that
1
knk =kb−(b−n)k <kb−ak +ka−(b+n)k ≤ +δ <1.
∞ ∞ ∞ ∞
2
It follows that n=0, so kb−ak <δ. (cid:3)
∞
Lemma3.3. Supposeδ ∈(0,1/2)andthatK,W ⊂Γarefinitesubsetsthatcontain
the identity element of Γ. Assume (x(g)) ∈((R/Z)k)Γ satisfy
g∈Γ
(13) ρ (x(g) ,x(wg))<δ for every g ∈Γ, f ∈K and w∈K−1W.
∞ w−1f f
Then there exists y(g) ∈ ([−1,1]k)Γ ⊂ (ℓ∞(Γ))k with P(y(g)) = x(g) for every
g ∈Γ so that:
(g) (hg)
(14) ky −y k <2δ for every g ∈Γ, f ∈K and h∈W.
h−1f f ∞
Proof. Let(x(g)) ∈((R/Z)k)Γ satisfy(13). Definey(g) ∈([−1,1]k)Γ ⊂(ℓ∞(Γ))k
g∈Γ
as follows:
First, for every g ∈ Γ let y(g) be the unique a ∈ [−1,1)k so that P(a) = x(g).
1 2 2 1
(g)
Next, for every g ∈Γ, h∈W and f ∈K\{h}, let y be the unique element of
h−1f
[−1,1]k satisfying
ky(g) −y(f−1hg)k <δ.
h−1f 1 ∞
Existence and uniqueness of such elements follow by applying Lemma 3.2 above
with ˜b = x(g) and a˜ = x(f−1hg), noting that ρ (x(g) ,x(f−1hg)) < δ by (13)
h−1f 1 ∞ h−1f 1
because 1 ∈ K. Finally, for every g ∈ Γ, h ∈ Γ\W−1K, let y(g) be the unique
h
a ∈ [−1,1)k so that P(a) = x(g). It now follows that for every g ∈ Γ, f ∈ K and
2 2 h
h∈W,
(g) (hg) (g) (f−1hg) (f−1hg) (hg)
ky −y k≤ky −y k+ky −y k<2δ.
h−1f f h−1f 1 1 f
(cid:3)
\
For A∈M (ZΓ), denote X :=(ZΓ)k/(ZΓ)kA. Explicitly:
k A
(15) X := x∈((R/Z)k)Γ : (xA∗) =Zk for every g ∈Γ
A g
Let α(A) ∈Actalg(Γ,(cid:8)XA) denote the canonical algebraic action on X(cid:9)A.
Theorem1.5isaparticularinstanceofthefollowingmoregeneralresult,inspired
by [3]:
Theorem 3.4. Let k ∈ N and A ∈ M (ZΓ) be invertible in M (ℓ1(Γ)). Then the
k k
canonical action α ∈Act (Γ,X ) is expansive and satisfies p.o.t.
A alg A
Proof. To avoid some subscripts, in this proof we let
(16) X :=X and α:=α ∈Act (Γ,X).
A A alg
By [3, Lemma 3.7], α∈Act (Γ,X) is expansive. Fix a metric d on X.
alg
Let ǫ>0 be arbitrary. By definition of p.o.t, we need to show that there exists
δ′ >0andafinite setW ⊂Γsothatevery(W,δ′)pseudo-orbitisǫ-tracedbysome
point x∈X.
Choose δ >0 small enough so that
1 1
(17) δ ≤min kAk−1, ,ǫ .
4 1 4
(cid:26) (cid:27)
8
From the fact that α is expansive, it follows that by possibly making δ even
smaller we have
(18) supρ (x ,x˜ )<2δ implies x=x˜.
∞ g g
g∈Γ
In fact, the proof of [3, Lemma 3.7] shows that (17) already implies (18), but we
will not need this.
Since A is invertible in M (ℓ1(Γ)) so is A∗. Let B ∈ M (ℓ1(Γ)) ∼= ℓ1(Γ,M )
k k k
denote the inverse of A∗. Using the natural identification of M (ℓ1(Γ)) as a subset
k
of the M (R)-valued functions on Γ, write B = B with B ∈M (R). There
k g∈Γ g g k
exists a finite symmetric set F ⊂Γ containing the identity with the property that
P
δ
(19) kB k< kA∗k−1.
g 2 1
g∈XΓ\F
Choosea finite symmetric setS ⊂Γcontainingthe identity thatsupports A∗ in
the sense that there exists (A∗) ∈(M (Z))S so that
s s∈S k
(yA∗)g = ygs−1A∗s for every y ∈(ℓ∞(Γ))k and g ∈Γ,
s∈S
X
and let
(20) K :=FS−1 ⊂Γ.
Choose δ′ >0 small enough so that
(21) d(x,y)<δ′ implies maxρ (x ,y )<δ.
∞ f f
f∈K
By compactness of X and (18) it follows that there exists a finite set W ⊂ Γ
with the property that:
(22) maxρ∞(xg−1,x˜g−1)<2δ implies d(x,x˜)<min{δ′,ǫ} for all x,x˜∈X.
g∈W
Let W ⊂Γ be such a finite set, so that in addition F−1 ⊂W.
Suppose (x(g)) ∈ XΓ is a (K−1W,δ′) pseudo-orbit. Then by (21), it follows
g∈Γ
that (13) holds.
Let y(g) ∈([−1,1]k)Γ ⊂(ℓ∞(Γ))k be as in the conclusion of Lemma 3.3. Let
(23) z(g) :=y(g)A∗ ∈ℓ∞(Γ,Zk).
Then (14) together with (17) imply
(24) kz(g) −z(hg)k <1 for every g ∈Γ , f ∈K and h∈W.
h−1f f ∞
But z(g) ,z(hg) ∈Zk so
h−1f f
(25) z(g) =z(hg) for every g ∈Γ , f ∈K and h∈W.
h−1f f
Also note that
(26) kz(g)k ≤ky(g)k kA∗k ≤kA∗k .
∞ ∞ 1 1
Define z ∈ℓ∞(Γ,Zk) by
(g)
(27) zg−1 :=z1 for every g ∈Γ.
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Using (25) and the fact that F−1 ⊂W it follows that
(28) α (z)| =z(g) | .
g F F
By (19), (26) and (28):
(29) k(α (z)B) −(z(g)B) k<δ for every g ∈Γ.
g 1 1
Let
(30) y :=zB ∈(ℓ∞(Γ))k and x:=P(y).
It follows that
yA∗ =z ∈ℓ∞(G,Zk).
Thus x∈X , recalling the definition of X in (15).
A A
It follows that from (29) and (13) that
ρ∞((αg(x))f−1,x(fg−)1)<2δ for every g ∈Γ and f ∈W.
By (22) this implies
d(α (x),x(g))<ǫ for every g ∈Γ.
g
We found x∈X that ǫ-traces (x(g)) , so the proof is complete. (cid:3)
A g∈Γ
Theorem1.5followsfromTheorem3.4abovebylettingk =1,soA∈M (ZΓ)∼=
k
ZΓ.
Combining Theorem 1.5 with Theorem 1.4 we recover the following result:
Corollary 3.5. LetΓbeacountableamenablegroup. Everyexpansiveprincipal al-
gebraic Γ-action with positive topological entropy admits a non-diagonal asymptotic
pair. Equivalently, it has a non-trivial homoclinic group.
Remark 3.6. By [3, Corollary 7.9] every non-trivial expansive principal algebraic
action of an amenable group has positive entropy. By [3, Lemma 5.4] algebraic
actions of the form α ∈ Act(Γ,X ) as in the statement of Theorem 3.4 have a
A A
dense set of homolicnic points. In particular, this proves Corollary 3.5 holds. I
thank Nhan-Phu Chung for pointing out this out to me.
Combining Theorem 1.5 with Theorem 2.4 we get:
Corollary 3.7. Every expansive principal algebraic Γ-action is topologically stable.
UsinganalgebraiccharacterizationsofexpansivealgebraicactionsduetoChung
and Li we have:
Corollary3.8. LetΓbeacountablegroup. Supposeα∈Act (Γ,X)isexpansive.
alg
Thenαisalgebraicallyconjugatetoasubsystemofanalgebraicactionthatisexpan-
sive and satisfies p.o.t. In other words, there exists an expansive β ∈Act (Γ,Y)
alg
that satisfies p.o.t and a Γ-equivariant continuous monomorphism Φ:X →Y.
Proof. Suppose α ∈ Act (Γ,X) is expansive. As in [3], we can identify the left
alg
ZΓ-moduleX with (ZΓ)k/J forsomek ∈N andsomeleft ZΓ-submoduleJ. By [3,
Theorem3.1]thereexistsA∈M (ZΓ)invertibleinM (ℓ1(Γ))sothattherowsofA
k k
arecontainedbinJ. Thenαisalgebraicallyconjugatetoasubsystemofthenatural
algebraic Γ-action on the dual of (ZΓ)k/(ZΓ)kA. By Theorem 1.5 this Γ-action
satisfies p.o.t. (cid:3)
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