On Virtual Conjugacy of Generalized Odometers
Marı́a Isabel Cortez∗
Departamento de Matemática y Ciencia de la Computación,
Universidad de Santiago de Chile
maria.cortez@usach.cl
Konstantin Medynets†
Department of Mathematics, United States Naval Academy
medynets@usna.edu
Dated: August 7, 2015
Abstract
The paper is focused on the study of continuous orbit equivalence
for generalized odometers (profinite actions). We show that two generalized odometers are continuously orbit equivalent if and only if the
acting groups have finite index subgroups (having the same index)
whose actions are piecewise conjugate. This result extends M. Boyle’s
flip-conjugacy theorem originally established for Z-actions. As a corollary we obtain a dynamical classification of the restricted isomorphism
between generalized Bunce-Deddens C ∗ -algebras. We also show that
the full group associated with a generalized odometer is amenable if
and only if the acting group is amenable.
1
Introduction
Let (X, G) and (Y, H) be dynamical systems with the phase spaces X and Y
being the Cantor set. We say that (X, G) and (Y, H) are continuously orbit
equivalent if there is a homeomorphism ψ : X → Y establishing a bijection
between G-orbits and H-orbits such that for any x ∈ X and g ∈ G there is
h ∈ H with g = ψ −1 ◦ h ◦ ψ on a clopen neighborhood of x.
The study of continuous orbit equivalence of dynamical systems is primarily motivated by its applications to the classification theory of crossed
∗
The research of the first author was supported by Anillo Research Project 1103
DySyRF and Fondecyt Research Project 1140213.
†
The second author was supported by NSA grant H98230CCC5334.
1
product C ∗ -algebras and geometric group theory. Denote by Cr (X, G)
the reduced crossed product C ∗ -algebra associated to a dynamical system
(X, G). It turns out that the C ∗ -algebras Cr (X, G) and Cr (Y, H) are isomorphic via an isomorphism mapping C(X) onto C(Y ) if and only if the
systems (X, G) and (Y, H) are continuously orbit equivalent. This result
was originally established by T. Giordano, I. Putnam, C. Skau in [GPS1,
Theorem 2.4] for minimal Z-actions and later generalized to topologically
free Z-system by J. Tomiyama [T, Theorem 2] and to arbitrary topologically
free dynamical systems by J. Renault [R, Proposition 4.13].
To any Cantor dynamical system (X, G), with G a countable group,
we can associate a countable group [[G]] defined as the set of all homeomorphisms X that locally coincide with elements of G. The group [[G]]
is termed the topological full group of (X, G). No topology is assumed on
[[G]]. The “topological” is a historical term used to differentiate from full
groups arising in ergodic theory. In [Me], the author proved that, under
some minor assumptions, two systems are continuously orbit equivalent if
and only if their topological full groups are isomorphic as abstract groups.
This result, in particular, says that topological full groups encode a lot of
dynamical information about the underlying system. A detailed discussion
on the interplay between algebraic properties of topological full groups and
properties of the systems when the acting group is Z can be found in [GM].
The topological full groups were recently used to construct first examples of
simple finitely generated amenable groups [JM].
The goal of this paper is to study the continuous orbit equivalence for
equicontinuous systems. We show that every minimal equicontinuous system (X, G) is topologically conjugate to some generalization of a profinite
action, which we call a subdometer ([CzP], [Cz]). The terminology is motivated by the fact if G = Z, then the minimal equicontinuous system is
automatically conjugate to an odometer. We also show that to admit an
effective equicontinuous action the group G must necessarily be residually
finite (Theorem 3.5).
Given a residually finite group G and a nested sequence of finite-index
(not necessarily normal) subgroups {Gn }n≥0 . The G-subodometer is defined as the action of G on the inverse limit X = lim(G/Gi , πi ), where
πi : G/Gi+1 → G/Gi is the natural quotient map (see Section 2 for more
details). The following are the main results of the paper (see Section 4 for
the proofs). Recall that a dynamical system (X, G) is called free if g · x = x,
x ∈ X, g ∈ G, implies that g = e, the group identity. The following are the
main results of the paper (see Section 4 for the proofs).
Theorem 1.1. Let (Y, H) be a free dynamical system. If (Y, H) is continuously orbit equivalent to a free subodometer (X, G) with G a finitely
generated group, then (Y, H) is a subodometer and the groups G and H are
commensurable.
2
Theorem 1.2. Let (X, G) and (Y, H) be free subodometers. Then the following are equivalent:
(1) (X, G) and (Y, H) are continuously orbit equivalent.
(2) The topological full groups [[G]] and [[H]] are isomorphic.
(3) Cr (X, G) and Cr (Y, H) are isomorphic via an isomorphism mapping
C(X) onto C(Y ).
(4) There exist nested sequences of finite index subgroups {Gn }n≥0 and
{Hn }n≥0 determining the structure of (X, G) and (Y, H) as subodometers,
respectively, and a group isomorphism θ : G0 → H0 such that [G : G0 ] =
[H : H0 ] and θ(Gn ) = Hn for every n ≥ 1.
As a corollary, we obtain that if two Zd -actions are continuously orbit
equivalent, then they are virtually piecewise conjugate (Theorem 4.3).
One of the big open problems in the topological orbit equivalence theory
is to describe the systems whose topological full groups are amenable. All
topological full groups of minimal Z-systems are amenable [JM]. However,
there are minimal Z2 -systems with non-amenable full groups. We also refer
the reader to the paper [JNS] establishing the amenability of full groups for
a big class of dynamical systems. Theorem 1.3 shows that equicontinuous
minimal systems have amenable topological full groups whenever the acting group is amenable. This, in particular, implies that the full groups of
products of Z-odometers are amenable.
Theorem 1.3. The topological full group of a free minimal equicontinuous
system is amenable if and only if the acting group is amenable.
The crossed products C ∗ -algebras appearing in Theorem 1.2 were earlier studied by Orfanos in [O] and were termed generalized Bunce-Deddens
algebras. These algebras coincide with classical Bunce-Deddens algebras
whenever G = Z [D, Section V.3].
In [B], [BT], M. Boyle and J. Tomiyama showed that the continuous
orbit equivalence in the case G = H = Z is equivalent to the conjugacy
(up to a group isomorphism) of these systems (Definition 2.1). Such results
are often referred to as Rigidity Theorems. Theorem 1.2 can be seen as
an extension of Boyle-Tomiyama’s rigidity theorem to the case of profinite
actions of residually finite groups.
Historically, the orbit equivalence rigidity phenomena were first discovered in the measurable dynamics, see, for example, [I] and references therein.
In [I], Ioana studies the measurable cocycle superrigidity for profinite actions
of property (T ) groups. We note that, in spirit, his results have some similarities with ours, though their scopes and the techniques employed are
completely different. We would also like to mention a recent preprint [Li],
where the author establishes a number of rigidity results for various topological dynamical systems.
3
The structure of the paper is the following. In section 2 we introduce
main notations and detail the construction of subodometers. In section 3
we show that every minimal equicontinuous system is conjugate to a subodometer. The main results of the paper are proven in Section 4. Section 5
is devoted to the study of topological full groups associated with subodometers.
Acknowledgement: This project was started when the second-named
author visited the mathematics department of Universidad de Santiago de
Chile. He would like to thank the department for the hospitality during his
visit.
2
Preliminaries
By a dynamical system we mean a triple (X, T, G), where X is a compact metric space, G is an infinite countable discrete group, and T : G →
Homeo(X) is a homomorphism. We will write T g for T (g). Note that
T gh = T g T h for g, h ∈ G. A group action T is called effective if T g ̸= T h for
distinct g and h. When the action T is clear from the context, we will omit
the symbol T in (X, T, G). In this case, the action of an element g ∈ G on
x ∈ X, will be denoted by g · x.
A dynamical system (X, T, G) is called minimal if every G-orbit is dense
in X. A subset Y ⊂ X is called a minimal component if Y is G-invariant
and (Y, T, G) is minimal.
Definition 2.1. (1) Dynamical systems (X1 , T1 , G) and (X2 , T2 , G) are
called conjugate if there exists a homeomorphism ψ : X1 → X2 such that
ψ(T1g x) = T2g ψ(x) for every g ∈ G and x ∈ X1 .
(2) Dynamical systems (X1 , T1 , G1 ) and (X2 , T2 , G2 ) are called conjugate
(up to an isomorphism), if there exist a group isomorphism θ : G → H and
θ(g)
a homeomorphism ψ : X1 → X2 such that ψ(T1g x) = T2 ψ(x) for every
g ∈ G and x ∈ X1 . In this case, we will also say that the systems are
θ-conjugate.
Remark 2.2. Consider dynamical systems (Xi , Si ), i = 1, 2, where Si :
Xi → Xi is a homeomorphism. Note that the homeomorphisms {Si } generate Z-actions. The dynamical systems (X1 , S1 ) and (X2 , S2 ) are called flip
conjugate if (X1 , S1 ) is conjugate to (X2 , S2 ) or to (X2 , S2−1 ). Since id and
−id are only automorphisms of the the group Z, the notions of conjugacy
up to an isomorphism and of the flip conjugacy are equivalent.
Let G be a group and {Gi }i≥0 be a decreasing sequence of finite-index
subgroups (not necessarily normal!). Let πi : G/Gi+1 → G/Gi be the nat←
−
ural quotient map. Consider the inverse limit G of the systems (G/Gi , πi ).
∏
←
−
In other words, G consists of tuples (g0 , g1 , g2 , . . .) ∈ ∞
i=1 G/Gi such that
4
←
−
πi (gi+1 ) = gi for all i ≥ 0. We will sometimes denote G by lim(G/Gi , πi ).
←
−
←
−
The topology on G is generated by the clopen sets [a]i = {{gn } ∈ G : gi =
ai }, where ai ∈ G/Gi .
←
−
The group G acts continuously on G by left multiplication, i.e., the
←
−
action is defined as g · {hi } := {ghi }, where g ∈ G and {hi } ∈ G . The proof
of the following proposition is straightforward.
←
−
Proposition 2.3. The dynamical system ( G , G) is minimal.
←
−
We note that the action of G on G is not always free. If the subgroups
←
−
{Gi } are normal in G, then G is a profinite group and there is a natural
←
−
←
−
homomorphism τ : G → G that defines ∩
the action of G on G . In this
case, the action of G is free if and only if ∞
i=1 Γi = {1}. Indeed, τ (g) = 1
iff
∩ for every g ∈ Γi for all i ≥ 1. Thus, τ is an embedding if and only if
i≥1 Γi = {1}. Following [CzP], we give the definitions.
←
−
Definition 2.4. We will call the dynamical system ( G , G) a G-subodometer
or, simply, a subodometer when the group G is clear from the context.
←
−
(2) We will call a subodometer ( G , G) a G-odometer if the finite-index
subgroups {Gi } determining its structure are normal.
We note that subodometers are sometimes referred to as profinite actions [AE], [I]. Every G-subodometer is a factor of a G-odometer [CzP,
Proposition 1].
←
−
Consider a subodometer ( G , G). Denote by e the element {en }n≥1 ∈
←
−
G , where en is the coset in G/Gn corresponding to the group Gn . Set
Cn = [e]n . Note that Cn = Gn · e. Furthermore, the group Gn is precisely
the set of return times to Cn . In other words, Gn = {g ∈ G : g(x) ∈
Cn } for any x ∈ Cn . Note also that the induced system (Cn , Gn ) is a
subodometer determined by the sequence of subgroups {Gi }i≥n+1 . The
following lemma shows how one can recover the original subodometer from
an induced subodometer. We will use the result later in the paper.
Lemma 2.5. Let (X, G) be a Cantor system. Suppose that there exist a
finite-index subgroup G0 ⊂ G, a system {f0 , . . . , fn−1 } of representatives for
G/G0 , and a clopen set C ⊂ X such that
(i) G0 = {g ∈ G : g(x) ∈ C} for every x ∈ C;
(ii) the system (C, G0 ) is a G0 -subodometer;
(iii) the family {f0 · C, . . . , fn−1 · C} is a clopen partition of X.
Then (X, G) is conjugate to a G-subodometer.
Proof. Fix a decreasing sequence of finite-index subgroups {Gn }n≥1 of G0
that determine the structure of the G0 -subodometer (C, G0 ). Consider a
←
−
G-subodometer ( G , G) corresponding to the sequence of groups {Gn }n≥0 .
←
−
We claim that the systems (X, G) and ( G , G) are conjugate.
5
Note that (C, G0 ) and ([e]1 , G0 ) are conjugate. Denote the homeomorphism implementing the conjugacy between the systems by φ. Extend it to
←
−
a homeomorphism φ : X → G as follows: for y ∈ X find unique x ∈ C and
fi , i = 0, . . . , n − 1, with fi · x = y and set φ(fi · x) = fi · φ(x).
Fix g ∈ G. Find h ∈ G0 and fj , j = 0, . . . , n − 1, such that gfi = fj h.
Then
φ(g·y) = φ(gfi ·x) = φ(fj h·x) = fj ·φ(h·x) = fj h·φ(x) = gfi ·φ(x) = g·φ(y).
This shows that φ is G-equivariant, which completes the proof.
Definition 2.6. Suppose that dynamical systems (X, G) and (Y, H), with
groups acting freely, are continuously orbit equivalent (Section 1). We can
define a function f : G × X → H by
f (g, x) · θ(x) = θ(g · x) for every g ∈ G and x ∈ X,
termed an orbit cocycle. Note that f satisfies the cocycle identity
f (g1 g2 , x) = f (g1 , g2 · x)f (g2 , x) for every g ∈ G and x ∈ X.
By construction, the function f : G × X → H is continuous and for
every x ∈ X f (·, x) : G → H is a bijection. Note that the “dual” cocycle
q : H × Y → G is also continuous.
Let (X, G) be a Cantor minimal system. Denote by [[G]] the group of
homeomorphisms s : X → X such that for every x ∈ X there exists a clopen
neighbourhood U of x and an element g ∈ G such that s(y) = g(y) for every
y ∈ U.
Definition 2.7. The group [[G]] is called the topological full group of (X, G).
We refer the reader to the paper [GM] surveying algebraic properties of
full groups. The following result shows that the topological full group [[G]]
is a complete invariant of continuous orbit equivalence [Me, Remark 2.11].
We note that the original result [Me] was established under much weaker
assumptions than those presented here.
Theorem 2.8. Let (X, G) and (Y, H) be Cantor minimal systems. Then
(X, G) and (Y, H) are continuously orbit equivalent if and only if the topological full groups [[G]] and [[H]] are isomorphic as abstract groups.
Furthermore, for every group isomorphism α : [[G]] → [[H]] there exists
a homeomorphism Λ : X → Y such that α(g) = Λ ◦ g ◦ Λ−1 for all g ∈ [[G]].
Definition 2.9. Let (X, G) be a dynamical system, with X being the Cantor
set.
(1) For a subset U ⊂ X and point x ∈ X, the set of return times of the
point x to U is defined as TU (x) = {g ∈ G : g · x ∈ U }.
6
(2) A point x ∈ X is said to be regularly recurrent if for every clopen
neighborhood U of x there exists a finite-index subgroup K ⊂ G such that
K ⊂ TU (x).
The following result will be needed in Section 5 to show that the topological full groups “remember” the set of regularly recurrent points.
Proposition 2.10 (Corollary 1 in [CzP]). Let (X, G) be a dynamical system
and let x ∈ X. The point x is regularly recurrent if and∩only if there exists
a decreasing sequence (Cn )n≥0 of clopen sets such that n≥0 Cn = {x} and
for every y ∈ Cn , the set of return times of y to Cn is a finite index subgroup
of G, independent of y.
3
Equicontinuous systems
This section is devoted to the general theory of equicontinuous systems. Our
goal is to understand what discrete groups admit effective equicontinuous
actions and understand the basic structure of equicontinuous systems. In
particular, we will prove that if (X, G) is a free minimal equicontinuous
system, where G is a finitely generated group, then G must be residually
finite and (X, G) is conjugate to a subodometer.
Definition 3.1. A dynamical system (X, G, T ) is called equicontinuous if
the collection of maps defined by the action of G is uniformly continuous,
i.e., if for every ε > 0 there exists δ > 0 such that d(x, y) ≤ δ implies
d(T g (x), T g (y)) < ε, for every g ∈ G.
Every equicontinuous system is a disjoint union of its minimal (equicontinuous) components, see Corollary 10 in Ch.1 and Theorem 2 in Ch.2 in
[Au].
In [AGW] the authors showed that every distal action of a finitely generated group on a compact zero-dimensional metric space is equicontinuous.
The following example gives a universal model for equicontinuous systems.
Example 3.2. Consider a discrete group G. Suppose that there exists a
homomorphism φ : G → K into a compact group K such that φ(G) is dense
in K. Fix a closed subgroup H of K. Define an action of G on the left
cosets K/H as follows
T g (kH) = φ(g)kH, for every k ∈ K and g ∈ G.
The system (K/H, T, G) is minimal and equicontinuous [Au, Page 39]. According to Theorem 6 in [Au, Ch.3], every minimal equicontinuous system
arises in such a way.
The following result shows that every minimal equicontinuous system on
a Cantor set is conjugate to a subodometer.
7
Proposition 3.3. Let (X, T, G) be a minimal Cantor system. Then (X, T, G)
is equicontinuous if and only if (X, T, G) is conjugate to a subodometer.
Proof. We first note that every subodometer is minimal and equicontinuous
[Au, Ch.2].
Consider an arbitrary minimal equicontinuous Cantor system (X, T, G).
It is conjugate to a system (K/H, T, G) as described in Example 3.2. Note
that the group G acts minimally on the group K by translations [Au].
It follows from the arguments in [Au, Ch. 3, Thm 6] that the group K
arises as a closed subset of X X . Thus, K is a zero-dimensional topological
group. Let U be a clopen neighborhood of 1K in K. Theorem 7.7 in [HR]
implies that there exists a clopen subgroup L of K contained in U . By
minimality, for every
∪ k ∈ K there exists g ∈ G such that φ(g)k ∈ L. It
follows that K = g∈G φ(g)L.
Since K is compact and∪the sets φ(g)L are open, there exists a finite
set F ⊂ G such that K = g∈F φ(g)L = φ(F )L. Set Λ = L ∩ φ(G). If
φ(g) = φ(f )l for some g ∈ G, f ∈ F ⊂ G and l ∈ L, then l ∈ Λ. Hence,
φ(G) = φ(F )Λ.
It follows that
G = Ker(φ)F φ−1 (Λ) = F Ker(φ)φ−1 (Λ).
This shows that G′ = Ker(φ)φ−1 (Λ) is a finite-index subgroup of G, which
is contained in the set of return times of 1K to U . Thus, 1K is a regularly
recurrent point. Since K is a group, every element k ∈ K is regularly
recurrent.
Since (K/H, T, G) is a factor of (K, T, G), every point in G/H is regularly
recurrent. Applying Corollary 2 from [CzP], we conclude that (K/H, T, G)
is topologically conjugate to a subodometer.
Corollary 3.4. (1) Every equicontinuous dynamical system on a Cantor
set is topologically conjugate to a disjoint union of subodometers.
(2) Every equicontinuous dynamical system is measure-theoretically conjugate to a subodometer.
Proof. (1) Using Corollary 10 in Ch.1 and Theorem 2 in Ch.2 from [Au], we
see that every equicontinuous system is a disjoint union of its minimal components. Proposition 3.3 implies that every minimal component is conjugate
to a subodometer.
(2) It follows from (1) that that every ergodic measure must be supported
by a minimal component, which is conjugate to a subodometer.
Let G be a discrete group. A bounded complex-valued function f on
G is called almost periodic if the collection of the G-translates of f is relatively compact under the uniform norm. Most statements of the following
8
result are well-known, though, scattered in the literature. We sketch the
proofs for the reader’s convenience. Groups satisfying one of the equivalent
statements of the theorem are called maximally almost periodic. We note
that the assumption that the group is finitely-generated is only needed for
establishing (2) ⇒ (3).
Theorem 3.5. Let G be a finitely generated discrete group. The following
statements are equivalent:
(1) The family of almost periodic functions on G separate points in G.
(2) G embeds into a compact group.
(3) The group G is residually-finite.
(4) The group G admits an effective equicontinuous action on a compact
(Cantor) space.
Proof. According to [Lo, p. 168], for any discrete group G there exists a
compact group K, termed the Bohr compactification of G, and a homomorphism α : G → K such that a function f on G is almost periodic if and only
if there is a continuous function h on K such that f (g) = h(α(g)) for all
g ∈ G. The homomorphism α is injective if and only if the almost periodic
functions separate points in G. This shows that (1) implies (2).
Assume (2). We note that the Bohr compactification K is a universal
object with respect to group homomorphisms. Hence, the group G can be
embedded in K. It now follows that (2) implies (1).
By the Peter-Weyl theorem, finite dimensional representations separate
elements in compact groups. Thus, (2) implies that the group G is a finitelygenerated linear group. It now follows from Mal’cev’s theorem that G is
residually finite. Thus, (2) implies (3).
Every residually-finite group G embeds in its profinite competition Ḡ.
Note that G acts Ḡ by left translations. This shows that (3) implies (4), see
[Au] for more details.
The equivalence of (1) and (4) was originally established in [Hu]. For
the reader’s convenience, we will present an alternative proof establishing
that (4) implies (2). We will make use of the structure of equicontinuous
systems established in Proposition 3.3.
Suppose that (X, T, G) is an effective equicontinuous system. Let (Xi )i∈I
be the collection of minimal components of X. Since the restriction of T to
every Xi is equicontinuous, for every i ∈ I there exists a Hausdorff compact
topological group Ki , a closed subgroup Hi of Ki , and a homomorphism
φi : G → Ki such that φi (G) is dense in Ki and such that (Xi , T |Xi , G) is
conjugate to (Ki /Hi , T, G), where T g (kHi ) = φi (g)kH
∏ i , for every g ∈ G and
k ∈ Ki (see Example 3.2). Consider the group K = i∈I Ki and φ : G → K
defined by φ(g) = (φi (g))i∈I . If g ∈ Ker(φ), then φi (g) = 1Ki for every
i ∈ I. It follows that T g = id. The effectiveness of the action implies that
g = 1G . Hence, φ is an embedding of G into a compact group.
9
We will also present an alternative proof of (4) ⇒ (3) not requiring the
assumption that G is finitely generated. Suppose that G admits an effective
equicontinuous action on a Cantor set. By Corollary 3.4, we can write this
action as
∏
(X, T, G) =
(Xi , T |Xi , G),
i∈I
where (Xi , T |Xi , G) is a subodometer, for every i ∈ I. For each i, let (Γi,j )j∈N
be the decreasing sequence of finite index subgroups of Γ that defines the
subodometer (Xi , T |Xi , G).
Let g ∈ G \ {1G }. Since the action is effective, there exist i ∈ I and
x ∈ Xi such that T g (x) ̸= x. It follows that we can choose j ∈ N such that
gxj and xj lie in distinct cosets G/Γi,j , where xj is the j-th coordinate of
x. Hence, g ∈
/ xj Γi,j x−1
j . Using the fact that every finite index subgroup
contains a normal finite index subgroup, fix Γ ⊆ Γi,j , a finite index normal
subgroup of G. Note that g ∈
/ Γ. Thus, we have shown that for every nontrivial g ∈ G, there exists a finite index subgroup Γ ⊂ G such that g ∈
/ Γ.
This implies that the group G is residually finite.
4
Rigidity Theorems
In this section we extend Boyle’s flip conjugacy theorem [B], [BT] to the case
of free minimal equicontinuous systems. In view of Proposition 3.3 and Theorem 3.5, we can assume that the systems of interest are free subodometers
and all acting groups are residually finite.
Definition 4.1. Let (X, G) and (Y, H) be subodometers. We say that
(X, G) and (Y, H) are structurally conjugate if there exist decreasing sequences of finite index subgroups {Gn }n≥0 and {Hn }n≥0 that determine
(X, G) and (Y, H), respectively, and an isomorphism θ : H0 → G0 such that
θ(Hn ) = Gn , n ≥ 1, and [G : G0 ] = [H : H0 ].
The following theorem shows that the structure of equicontinuous systems is so rigid that any continuous orbit equivalence automatically turns
into a structural conjugacy.
Theorem 4.2 (Rigidity Theorem). Let (Y, H) be a subodometer with H
being a finitely generated residually finite group and let (X, G) be a free
dynamical system. Then (X, G) and (Y, H) are continuously orbit equivalent
if and only if (1) (X, G) is a free subodometer and (2) (X, G) and (Y, H)
are structurally conjugate.
Proof. (1) Assume that (X, G) and (Y, H) are continuously orbit equivalent.
By conjugating the system (Y, H), we can assume that both groups G and
H act on the same space X. Denote by f : H × X → G the orbit cocycle
10
defined by f (h, x) · x = h · x. Notice that for a given h ∈ H, f (h, ·) : X → G
is a continuous function.
Fix a symmetric set of generators {s1 , · · · , sr } for H. Find a clopen
partition O1 ⊔ O2 ⊔ . . . ⊔ Op = X such that the cocycle f (si , ·)|Oj = const for
every i and j. Let δ > 0 be Lebesgue’s number of the partition {O1 , . . . , Op }.
Since the system (X, H) is equicontinuous, we can find a clopen refinement
U1 ⊔ . . . ⊔ Uk = X of the partition {O1 , . . . , Op } such that if x, y ∈ Ui for
some i, then d(h · x, h · y) < δ for every h ∈ H. Here d is a metric compatible
with the topology. Therefore, if x, y ∈ Ui and h · x ∈ Oj , h ∈ H, then
h · y ∈ Oj . It follows that if x, y ∈ Ui , then for any h ∈ H and sj , we have
that f (sj , h · x) = f (sj , h · y).
Consider an arbitrary element h = si1 · · · sim ∈ H. If x, y ∈ Ui , then
f (h, x) =
m
∏
f (sil , sil+1 sil+2 · · · sim ·x) =
l=1
m
∏
f (sil , sil+1 sil+2 · · · sim ·y) = f (h, y).
l=1
Let {Hn }n≥1 be a sequence of subgroups determining the structure of the
subodometer (Y, H). Choose n > 0 such that the partition {C0 , . . . , Cq−1 }
into cosets H/Hn refines {U1 , · · · , Uk }. We will assume that the set C0
corresponds to [e]n . Set H ′ = Hn . Note that the set C0 is H ′ -invariant.
Note also that f (h, x) = f (h, y) for every h ∈ H and x, y ∈ C0 .
Fix x ∈ C0 . Set θ(h) = f (h, x). The definition of θ is independent of
x ∈ C0 . If h1 , h2 ∈ H ′ , then
θ(h1 h2 ) = f (h1 h2 , x) = f (h1 , h2 ·x)f (h2 , x) = f (h1 , x)f (h2 , x) = θ(h1 )θ(h2 ).
Set G′ = θ(H ′ ). Note that if θ(h) = e, then, in view of freeness of the
action, h = e. It follows that θ : H ′ → G′ is an isomorphism and that
(C0 , G′ ) is a G′ -subodometer. Fix representatives {h0 , . . . , hq−1 } for cosets
in H/H ′ . Then
G = f (H,
(⊔ x)
)
q−1
′, x
= f
h
·
H
j
j=0
⊔q−1
′
′
=
j=0 f (hj , H · x) f (H , x)
⊔q−1
′
=
j=0 f (hj , x) G .
This implies that [H : H ′ ] = [G : G′ ] < ∞. Note that hi · C0 = f (hi , x) · C0 .
Hence, {f (h0 , x)·C0 , . . . , f (hq−1 , x)·C0 } is a clopen partition of X. Applying
Lemma 2.5 we obtain that (X, G) is a G-odometer that is determined by the
sequence of subgroups {Gn , Gn+1 , . . .}, where Gk = θ(Hk ) for every k ≥ n.
This completes the proof of the “only if” part.
(2) Conversely, fix finite-index subgroups {Gn }n≥0 and {Hn }n≥0 of G
and H, respectively, with [G : G0 ] = [H : H0 ] < ∞. Let θ : H0 → G0
be an isomorphism as in Definition 4.1. By conjugating the system (Y, H),
11
we can assume that both groups G and H act on the same space X. Let
{C0 , . . . , Cn−1 } be the collection of clopen sets corresponding to the cosets
G/G0 . Let C0 correspond to the coset G0 . Note that C0 is G0 - (and,
in fact, H0 -) invariant. Since the systems (C0 , H0 ) and (C0 , G0 ) are θconjugate, we get that the topological full group [[(C0 , G0 )]] coincides with
[[(C0 , H0 )]]. Now we will show how to extend this identity isomorphism to
an isomorphism of groups [[G]] and [[H]].
We first observe that elements of G and H permute the clopen sets
{Cj }. Fix two sets of coset representatives FG and FH for G/G0 and H/H0 .
Assume that 1 ∈ FG and 1 ∈ FH . Fix a bijection FG ∋ f 7→ af ∈ FH with
1 7→ 1. Define a homeomorphism φ : X → X by setting
φ|f ·C0 (x) = af f −1 · x
for every f ∈ FG and x ∈ f · C0 . We claim that φ implements a continuous
orbit equivalence and, as a result, gives rise to an isomorphism between the
topological full groups [[G]] and [[H]].
Let x ∈ C0 and g ∈ G. Write g = f g0 , where f ∈ FG and g0 ∈ G0 . Then
φ(g · x) = φ(f g0 · x) = af · (g0 · x) = af θ−1 (g0 ) · x.
(1)
Thus, φ(x) and φ(g·x) lie in the same H-orbit. Therefore φ(G·x) ⊂ H ·φ(x).
Note that, by minimality, the inclusion extends to any x ∈ X. Using the
same argument one can also show that φ(G · x) ⊃ H · φ(x) for every x ∈ X.
Thus, φ implements an orbit equivalence between the systems (X, G) and
(Y, H). Since Equation (1) holds on a clopen neighborhood of x, we conclude
that φ implements a continuous orbit equivalence.
For Zd -odometers, the proof of the previous theorem can be adapted to
show that the systems are virtually piecewise conjugate whenever they are
continuously orbit equivalent.
Theorem 4.3. Let (X, G) and (Y, H) be free odometers, where G = H =
Zd . Let φ : X → Y be a map implementing the continuous orbit equivalence
between (X, G) and (Y, H) and let f : G × X → H be the corresponding
orbit cocycle. Then there exist finite index normal subgroups G0 ⊂ G and
H0 ⊂ H with [G : G0 ] = [H : H0 ], a clopen partition {C0 , . . . , Cq−1 } of Y
into H0 -invariant sets, and isomorphisms θi : H0 → G0 , i = 0, . . . , q − 1,
such that f (h, y) = θi (h) for every h ∈ H0 and y ∈ Ci , i = 0, . . . , q − 1.
Proof. We will use the same notation as in the proof of the “only if ” part
of Theorem 4.2. We will also assume that both groups act on X. Let
{Hn }n≥1 be a sequence of subgroups determining the structure of the odometer (X, H). Choose n > 0 such that the partition {C0 , . . . , Cq −1} into cosets
generated by the group H ′ = Hn refines {U1 , · · · , Uk }. Observe that every
12
set Ci is H ′ -invariant. Note also that f (h, x) = f (h, y) for every h ∈ H and
x, y ∈ Ci .
Fix i = 0, . . . , q − 1 and x ∈ Ci . Set θi (h) = f (h, x). Set G′i = θi (H ′ ). As
in the proof of Theorem 4.2, we obtain that the definition of θi is independent
of x ∈ Ci and θi : H ′ → G′i is an isomorphism of the groups. Furthermore,
[G : G′i ] = [H : H ′ ] < ∞ for every i = 0, . . . , q − 1.
We will show that Gi = Gj for all i and j. Observe that, by construction
of the odometer, the subgroup H ′ possesses the property that H ′ = {h ∈
H : h(x) ∈ Ci } for every i and x ∈ Ci . Let h ∈ H ′ and q ∈ H. Then
f (qhq −1 , x) = f (q, hq −1 · x)f (h, q −1 · x)f (q −1 , x)
= f (q, q −1 · x)f (h, q −1 · x)f (q −1 , x)
= f (q −1 , x)−1 f (h, q −1 · x)f (q −1 , x).
Therefore, if x ∈ Ci and q −1 · x ∈ Cj , q ∈ H, then
G′i = f (H ′ , x) = f (qH ′ q −1 , x) = f (q −1 , x)−1 f (H ′ , q −1 · x)f (q −1 , x) = G′j .
Set G′ = G′i . Thus, for every i = 0, . . . , q − 1, we have that θi : H ′ → G′
is a group isomorphism and f (h, x) = θi (h), for every x ∈ Ci and h ∈
H ′ . In particular, this implies that the systems (Ci , H ′ ) and (Ci , G′ ) are
topologically θi -conjugate. This completes the proof.
As a corollary of Theorems 2.8 and 4.2, we immediately obtain the following result.
Corollary 4.4. Let (X, G) and (Y, H) be free subodometers with G and H
finitely generated residually finite groups. Then the topological full groups
[[G]] and [[H]] are isomorphic as abstract groups if and only if the systems
(X, G) and (Y, H) are structurally conjugate.
The following proposition shows that for Z-odometers the structural and
flip conjugacies are equivalent. The result is an immediate consequence of
Corollary 4.4 and [BT, Theorem 2.3]. Recall that two Z-actions (X, T )
and (Y, T ) are called flip conjugate if (X, T ) is conjugate to (Y, S) or to
(Y, S −1 ). We would like to mention that using the Gottschalk-Hedlung
theorem and ideas from the proof of Theorems 4.2 and 4.3, one can show
that the continuous orbit cocycle is cohomologous to an automorphism of
Z. This can be used as an alternative proof of Proposition 4.5.
Proposition 4.5. Let (X, T ) and (Y, S) are Z-odometers. Then (X, T ) and
(Y, S) are flip conjugate if and only if they are structurally conjugate.
The following result gives a dynamical interpretation of the restricted
isomorphism for generalized Bunce-Deddens algebras introduced in [O]. The
result follows from Corollary 4.4 and [R, Proposition 4.13], see also [Ma,
Theorem 5.1]. We note that the papers cited use the language of groupoids.
13
We refer the reader to [Ma, Section 2] for more information on the groupoids
arising from dynamical systems. Denote by Cr∗ (X, G) the reduced crossed
product C ∗ -algebra arising from a dynamical system (X, G). These crossed
products are called generalized Bunce-Deddens algebras [O]. These algebras
coincide with classical Bunce-Deddens algebras whenever G = Z.
Corollary 4.6. Let (X, G) and (Y, H) be free subodometers with G and H
finitely generated residually finite groups. The following are equivalent:
(1) (X, G) and (Y, H) are structurally conjugate.
(2) There exists an isomorphism π : Cr∗ (X, G) → Cr∗ (Y, H) such that
π(C(X)) = C(Y ).
We finish the section by presenting two structurally conjugate, but not
conjugate Z2 -odometers. Existence of such examples tells us that Proposition 4.5 does not generalize to more general group actions. We note that
similar examples were independently constructed in [Li].
Example 4.7. Let A0 and B0 be the 2 × 2 matrices given by
(
)
(
)
4 1
2 0
A0 =
and B0 =
.
0 1
0 2
Set
An = An0 and Bn = B0 An−1
.
0
Let G = H = Z2 and Gn = An Z2 and Hn = Bn Z2 . Consider the odome←
−
←
−
ters ( G , G) and ( H , H) determined by the sequences of normal subgroups
{Gn }n≥1 and {Hn }n≥1 , respectively. Note that [G : G0 ] = [H : H0 ]. Set
θ = A0 B0−1 . Then θ : G0 → H0 is a group isomorphism. Furthermore,
←
−
←
−
θ(Gn ) = Hn for every n ≥ 1. Thus, the odometers ( G , G) and ( H , H) are
structurally conjugate (Definition 4.1).
←
−
←
−
Assume that the dynamical systems ( G , G) and ( H , H) are topologically
conjugate. This means that there exists a group isomorphism (treated as
←
−
←
−
a matrix) Λ : Z2 → Z2 and a homeomorphism α : G → H such that
←
−
α(g · x) = Λ(g) · α(x) for every g ∈ G and x ∈ G . In other words, the
sequences of subgroups {Λ(Gn )}n≥0 and {Hn }n≥0 define the same (up to a
homeomorphism of the spaces) odometers. These odometers are uniquely
determined by the sequences of matrices {ΛAn }n≥0 and {Bn }n≥1 . By [Cz,
Lemma 2] there exist a matrix P ∈ GL2 (Z) and n ≥ 0 such that B0 P = ΛAn .
Hence 2Λ−1 P = An0 . It follows that the entries of An0 are divisible by 2, which
←
−
←
−
is a contradiction. Therefore, ( G , G) and ( H , H) cannot be topologically
conjugate.
5
Topological Full Groups
In this section we describe the algebraic structure of topological full groups
of G-subodometers. The main result of the section is the proof of the fact
14
that the topological full group of a G-subodometer is amenable if and only if
G is amenable. In particular, this implies that the topological full group of a
product of Z-odometers is amenable since such systems can be obtained as
Zd -odometers using the diagonal scale matrices (see [Cz] for more details.)
5.1
Toeplitz Systems
Our first result is that the Toeplitz property is encoded in the algebraic
properties of topological full groups. We refer the reader to the paper [Do]
surveying dynamical properties of Z-Toeplitz systems, see also [CzP] for
more details on generalized Toeplitz systems.
Definition 5.1. A dynamical system (X, G) is called Toeplitz if it is an almost one-to-one extension of a G-odometer, i.e., if there exists a G-odometer
(Y, G), a G-equivariant map π : X → Y such that the set of y ∈ Y with
π −1 (y) being a singleton is a residual set of Y .
If (X, G) is an almost one-to-one extension of an odometer (Y, G) with
π : X → Y being a factor map, then the set of regular recurrent points
in X is precisely the π-pre-image of the set of points in Y which have one
pre-image under π [CzP, Theorem 2].
Proposition 5.2. Let (X, G) and (Y, Γ) be two free minimal Cantor systems. Suppose that (X, G) and (Y, Γ) are continuously orbit equivalent via
the map h : X → Y . Then x ∈ X is regularly recurrent if and only if h(x)
is regularly recurrent.
Proof. Let y0 ∈ Y be a regularly recurrent point. Then there exist a decreasing sequence of clopen set (Cn )n≥0 and∩a decreasing sequence of finite
index subgroups (Γn )n≥0 such that {y0 } = n≥0 Cn and such that for every n ≥ 0 and y ∈ Cn , the set of return times of y to Cn is equal to Γn
(Proposition 2.10).
For every n ≥ 0, let Bn = h−1 (Cn ) and
Gn = {g ∈ G : f (g, x) ∈ Γn , for every x ∈ Bn },
where f (g, y) ∈ Γ is the cocycle determined by h(g · x) = f (g, x) · h(x) for
every x ∈ X and g ∈ G.
Let x ∈ Bn and g ∈ Gn . Then g · x = h−1 (f (g, x) · h(x)) ∈ Bn . If
g1 , g2 ∈ Gn , then, by the cocycle identity,
f (g1 g2 , x) = f (g1 , g2 · x)f (g2 , x) ∈ Γn .
Therefore, Gn is a group. If x ∈ Bn and g ∈ G is such that g · x ∈ Bn , then
f (g, x) ∈ Γn , which implies that g ∈ Gn . In other words Gn is the set of
recurrent times to Bn . Lemma 4 in [CzP] implies that Gn has finite index
in G.
15
−1
is the unique element in
∩ Thus, we have shown that x0 = h (y0 ), which
−1 instead of h, we get that
B
,
is
a
regularly
recurrent
point.
Using
h
n≥0 n
x ∈ X is regularly recurrent if and only if h(x) ∈ Y is regularly recurrent.
The following results are easy consequences of Proposition 5.2 and [CzP,
Theorem 2].
Corollary 5.3. Let (X, G) and (Y, Γ) be two free minimal Cantor systems.
Suppose that [[G]] and [[Γ]] are isomorphic. Then (X, G) is equicontinuous
if and only if (Y, Γ) is equicontinuous.
Corollary 5.4. Let (X, G) and (Y, Γ) be two free minimal Cantor systems.
Suppose [[T ]] and [[S]] are isomorphic. Then (X, G) is Toeplitz if and only
if (Y, Γ) is Toeplitz.
5.2
Amenability
Let (X, G) be a free minimal equicontinuous system on the Cantor set
X. By Proposition 3.3, we can assume that (X, G) is a G-subodometer
lim(G/Gi , πi ), where (Gi )i≥0 is a decreasing sequence of finite index subgroups of G.
For every n ≥ 0, let Cn be the subset of all x = (xk )k≥0 ∈ X such that
xn = en (see Section 3 for more details). The collection Pn = {T g (Cn ) :
g ∈ Fn } is a clopen partition of X, where Fn is any set of representatives of
G/Γn . The partition Pn is independent of the choice of Fn . The family of
partitions (Pn )n≥0 spans the topology of X. Note also that the elements of
G permutes the atoms of Pn .
Let [[G]] be the topological full group of (X, G). For each γ ∈ [[G]] and
x ∈ X, let f (γ, x) ∈ G be such that f (γ, x) · x = γ · x. Since the group
G acts freely, the function f : [[G]] × X → G is well-defined. Note that f
satisfies the cocycle identity:
f (γ1 γ2 , x) = f (γ1 , γ2 · x)f (γ2 , x) for every γ1 , γ2 ∈ G
(2)
For every n ≥ 0, denote by [[G]]n the set of all γ ∈ [[G]] such that the
cocycle f (γ, ·) is Pn -compatible, i.e., constant on atoms of Pn . Using the
cocycle identity (2) and the fact that the group G permutes the atoms of Pn ,
we see that [[G]]n is a group. Setting P0 = {X}, we have that [[G]]0 = G.
The following result follows from the fact that (Pn )n≥0 is a nested sequence
of partitions spanning the topology of X.
∪
Proposition 5.5. [[G]] = n≥0 [[G]]n and [[G]]n ⊂ [[G]]n+1 for every n ≥ 0.
Denote by Sp the symmetric group on p elements.
Proposition 5.6. Suppose that (X, G) is a free odometer defined by a sequence of normal subgroups {Gn }.
16
[G:G ]
(1) The group [[G]]n is isomorphic to the semidirect product Gn n o
S[G:Gn ] .
(2) The topological full group [[G]] is isomorphic to an inductive limit
lim(Gn[G:Gn ] o S[G:Gn ] , τn ).
−→
Proof. Let {Cn } and {Pn } be as above. Recall that the group G permutes
atoms of Pn . Consider s ∈ [[G]]n . Then the orbit cocycle f (s, ·) is constant
on every atom of Pn . Fix a family of representatives Fn defining the partition
On .
For g ∈ Fn and x ∈ Cn , denote f (s, g · x) by ug ∈ G. Then
s · (g · x) = f (s, gx)g · x = ug g · x = g ′ q · x,
where g ′ ∈ Fn and q ∈ Gn . This shows that s permutes atoms of Pn . Denote
by in : [[G]]n → SPn the induced homomorphism. Note that SPn ∼
= SG/Gn .
The kernel of in is the subgroup of [[G]]n that stabilizes every atom of
Pn . Since the groups {Gn } are normal, Gn is the set of return times to
g · Cn , g ∈ Gn . Thus, for every s ∈ Ker(ψ) and every q · Cn , q ∈ Fn , we have
that s|q(Cn ) = qf |q(Cn ) for some qf ∈ Gn . It follows that
n]
Ker(ψ) ∼
.
= G[G:G
n
[G:G ]
Therefore, [[G]]n is isomorphic to the semidirect product Gn n o S[G:Gn ] .
The second conclusion of the theorem immediately follows from Proposition
5.5.
The following result is a corollary of Proposition 5.6. For more details
on properties of amenable groups, see [CC, Section 4.5].
Corollary 5.7. The topological full group [[G]] of a free odometer (X, G) is
amenable if and only if G is amenable.
We will need the following folklore result.
Lemma 5.8. Let (X, G) be a Cantor system. Suppose that (Y, G) is a
factor of (X, G). Then the topological full group of (Y, G) embeds into the
topological full group of (X, G).
Proof. Let π : (X, G) →
⊔ (Y, G) be a factor map. For s ∈ [[(Y, G)]], find a
clopen partition Y = Yi and elements {gi } ⊂ G such that s|Ai = gi |Ai .
Define π̄(s) as an element of [[(X, G)]] such that
π̄(s)|π−1 (Yi ) = gi |π−1 (Yi ) .
Since π is G-equivariant, π(s) is well-defined. Now it is routine to check
that π̄ is a group embedding.
17
Corollary 5.9. Let (X, G) an equicontinuous free minimal Cantor system.
Then [[G]] is amenable if and only if G is amenable.
Proof. By Proposition 3.3, (X, G) is conjugate to a free subodometer. Then
(X, G) is a factor of an odometer (see [CzP, Proposition 1]).
Suppose that G is amenable. By Lemma 5.8, [[G]] is isomorphic to a
subgroup of the topological full group of the odometer. Hence, Corollary
5.7 implies that [[G]] is amenable.
Conversely, if [[G]] is amenable, then G is amenable as a subgroup of
[[G]].
References
[AE]
M. Abért, G. Elek, Dynamical properties of profinite actions, Ergodic Theory Dynam. Systems 32 (2012), no. 6, 1805–1835.
[Au]
J. Auslander, Minimal flows and their extensions. North-Holland
Mathematics Studies, 153. Notas de Matemtica [Mathematical
Notes], 122. North-Holland Publishing Co., Amsterdam, 1988.
[AGW] J. Auslander, E. Glasner, B. Weiss, B. On recurrence in zero dimensional flows. Forum Math. 19 (2007), no. 1, 107–114.
[B]
M. Boyle, Topological orbit equivalence and factor maps in symbolic
dynamics, Ph.D. Thesis, University of Washington, Seattle (1983).
[BT]
M. Boyle and J. Tomiyama, Bounded topological orbit equivalence
and C ∗ -algebras. J. Math. Soc. Japan 50 (1998), no. 2, 317–329.
[CC]
T. Ceccherini-Silberstein, M. Coornaert, Cellular automata and
groups. Springer Monographs in Mathematics. Springer-Verlag,
Berlin, 2010.
[Cz]
M.I. Cortez, Zd -Toeplitz arrays. Discrete Contin. Dyn. Syst. 15
(2006), no. 3, 859–881.
[CzP]
M.I. Cortez, S. Petite, G-odometers and their almost one-to-one
extensions. J. Lond. Math. Soc. (2) 78 (2008), no. 1, 1–20.
[D]
K.R. Davidson, C ∗ -algebras by example, Fields Institute Monographs, (1996).
[Do]
T. Downarowicz, Survey of odometers and Toeplitz flows. Algebraic and topological dynamics, 7–37, Contemp. Math., 385, Amer.
Math. Soc., Providence, RI, 2005.
18
[EM]
G. Elek, N. Monod, On the topological full group of a minimal
Cantor Z2 -system. Proc. Amer. Math. Soc. 141 (2013), no. 10,
3549–3552.
[GPS1] T. Giordano, I. Putnam, C. Skau. Topological orbit equivalence and
C ∗ -crossed products. J. Reine Angew. Math. 469 (1995), 51–111.
[GM]
R.I. Grigorchuk, K. Medynets, On the algebraic properties of topological full groups. Sb. Math. 205 (2014), no. 5–6, 843–861.
[HR]
E. Hewitt, K.A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations. Die Grundlehren der mathematischen Wissenschaften, Bd.
115 Academic Press, Inc., Publishers, New York; Springer-Verlag,
Berlin-G´’ottingen-Heidelberg, 1963.
[Hu]
T.-J. Huang, A characterization of maximally almost periodic
groups, Proc. of the AMS, 75, 1979(1), 59–62.
[I]
I. Ioana, Cocycle superrigidity for profinite actions of property (T)
groups. Duke Math. J. 157 (2011), no. 2, 337–367.
[JM]
K. Juschenko and N. Monod, Cantor systems, piecewise translations and simple amenable groups. Ann. of Math. (2) 178 (2013),
no. 2, 775–787.
[JNS]
K. Juschenko, V. Nekrashevych, M. de la Salle, Extensions of
amenable groups by recurrent groupoids, preprint (2014).
[Li]
X. Li, Continuous Orbit Equivalence Rigidity, ArXiv:1503.01704
(2015).
[Lo]
L.H. Loomis, An introduction to abstract harmonic analysis. D. Van
Nostrand Company, Inc., Toronto-New York-London, 1953.
[Ma]
H. Matui, Homology and topological full groups of tale groupoids on
totally disconnected spaces. Proc. Lond. Math. Soc. (3) 104 (2012),
no. 1, 27–56.
[Me]
Medynets, K. Reconstruction of orbits of Cantor systems from full
groups. Bull. Lond. Math. Soc. 43 (2011), no. 6, 1104–1110.
[O]
S. Orfanos, Generalized Bunce–Deddens algebras. Proc. Amer.
Math. Soc. 138 (2010), no. 1, 299–308.
[R]
J. Renault, Cartan subalgebras in C ∗ -algebras. Irish Math. Soc.
Bull. No. 61 (2008), 29–63.
19
[T]
J. Tomiyama, Topological full groups and structure of normalizers
in transformation group C ∗ -algebras. Pacific J. Math. 173 (1996),
no. 2, 571–583.
20