arXiv:1710.00627v1 [] 2 Oct 2017

Equicontinuity, orbit closures and invariant
compact open sets for group actions on
zero-dimensional spaces
Colin D. Reid∗
University of Newcastle, School of Mathematical and Physical
Sciences, Callaghan, NSW 2308, Australia
colin@reidit.net
October 3, 2017

Abstract
Let X be a locally compact zero-dimensional space, let S be an
equicontinuous set of homeomorphisms such that 1 ∈ S = S −1 , and
suppose that Gx is compact for each x ∈ X, where G = hSi. We show
in this setting that a number of conditions are equivalent: (a) G acts
minimally on the closure of each orbit; (b) the orbit closure relation is
closed; (c) T
for every compact open subset U of X, there is F ⊆ G finite
such that g∈F g(U ) is G-invariant. All of these are equivalent to a
notion of recurrence, which is a variation on a concept of Auslander–
Glasner–Weiss. It follows in particular that the action is distal if and
only if it is equicontinuous.

1

Introduction

Let X be a compact metrizable zero-dimensional space and let G be a finitely
generated group of homeomorphisms of G. Define the orbit closure relation
R on X by setting (x, y) ∈ R if and only if y ∈ Gx. Then by a result of
J. Auslander, E. Glasner and B. Weiss ([1, Theorem 1.8]), G has minimal
orbit closures (that is, R is a symmetric relation) if and only if R is a closed
subset of X×X; indeed both conditions are equivalent to a kind of recurrence
defined in terms of a word metric on G. It follows ([1, Corollary 1.9]) that
the action of G is distal if and only if it is equicontinuous.
∗
The author is an ARC DECRA fellow. Research supported in part by ARC Discovery
Project DP120100996.

1

We prove similar consequences of minimal orbit closures in a more general setting. In particular, we generalize from finitely generated group actions to group actions that are equicontinuously generated, meaning
that the group G has a symmetric generating set that is equicontinuous
with respect to some compatible uniformity on the space. For this purpose
we use a variation of the notion of recurrence given in [1], with modifications arising from the change from finitely generated to equicontinuously
generated groups.
Definition 1.1. Let G be a group with a specified generating set S, and
define the associated word metric | · | with respect to the generating set
S ′ = {1} ∪ S ∪ S −1 . Given g ∈ G, define K(g) = (S ′ )|g|−1 g.
Suppose G acts on by homeomorphisms on a locally compact Hausdorff
space X and let x ∈ X. Then x is S-recurrent for the action if for every
net (gi )i∈I in G such that |gi | → ∞ and every neighbourhood U of x, there
is an integer n, a subnet (gi(j) )j∈J and cj ∈ K(gi(j) ), such that cj x ∈ U and
|cj | ≤ n.
The point x is almost periodic for G if for any neighbourhood U of x,
and letting N = {h ∈ G | hx ∈ U }, then there is a finite subset F of G such
that F N = G.
A locally compact space is zero-dimensional if it is Hausdorff and has
a base consisting of clopen sets.
Theorem 1.2. Let X be a locally compact zero-dimensional space, let S be
a family of homeomorphisms of X with 1 ∈ S and S = S −1 , such that S is
equicontinuous with respect to some compatible uniformity, and let G = hSi.
Let U be an open subset of X, and write U0 for the union of all compact
G-invariant subsets of U . Then the following are equivalent:
(i) U0 is open and consists of S-recurrent points.
(ii) U0 is open and consists of almost periodic points for G.
(iii) For every T
compact openTsubset V of U , there is a finite subset F of G
such that g∈G g(V ) = g∈F g(V ).
(iv) Given x ∈ U and y ∈ U0 such that y ∈ Gx, then x ∈ Gy.
(v) There is an open neighbourhood W of U0 such that the G-orbit closure
relation restricted to W × W is closed.
(vi) U0 is open and there is a G-equivariant quotient map φ : U0 → Y ,
where Y is a locally compact zero-dimensional space such that G acts
trivially on Y and minimally on each fibre of φ.
As a consequence, we obtain a situation where an equicontinuously generated distal action is actually equicontinuous, generalizing for instance [2,
Theorem 2].
2

Corollary 1.3. Let X be a locally compact zero-dimensional space equipped
with a compatible uniformity, let S ⊆ Homeo(X) with 1 ∈ S and S = S −1 ,
let G = hSi, and let x ∈ X. Suppose that G is distal and Gx is compact for
every x ∈ X. Then G is equicontinuous if and only if S is equicontinuous.
Remark 1.4. The original motivation for this article was to develop tools
for understanding the dynamics of groups of automorphisms of totally disconnected locally compact (t.d.l.c.) groups on coset spaces. In particular,
suppose G is a t.d.l.c. group; S is a set of automorphisms of G such that
1 ∈ S = S −1 , S is equipped with a compact topology and (s, g) 7→ s(g) is a
continuous map from S × G to G; and K is a closed subgroup of G such that
s(K) = K for all s ∈ S. Then we have an action by homeomorphisms of
H = hSi on the locally compact zero-dimensional space G/K such that S is
equicontinuous with respect to a natural uniformity, and thus Theorem 1.2
applies to this situation. The consequences will be explored in a second
article.

Acknowledgement
I thank Eli Glasner for bringing the article [1] to my attention, which inspired
the present article.

2

Preliminaries

Notation. Given a group G, a subset X of G and a set of automorphisms
Y , we say that X is Y -invariant if y(X) = X for all y ∈ Y . Given subsets
X and Y of a group G, we define X −1 := {x−1 | x ∈ X}, XY := {xy | x ∈
X, y ∈ Y }, and X n = {x1 x2 . . . xn | x1 , x2 , . . . , xn ∈ X}. All group actions
are left actions unless otherwise stated: given a group G acting on a set X,
we write gx for the image of x ∈ X under g ∈ G, Gx for the set {gx | g ∈ G}
and GY for the set {gy | g ∈ G, y ∈ Y }.

2.1

Equicontinuous sets of homeomorphisms

Definition 2.1. Let X be a locally compact Hausdorff space with a compatible uniformity U , let x ∈ X and let S ⊆ Homeo(X). The set S is bounded
at x if Sx is compact; it is equicontinuous at x if for every entourage
E ∈ U , there is a neighbourhood U of x such that
∀s ∈ S ∀y ∈ U : (sx, sy) ∈ E.
We say S is equicontinuous if S is equicontinuous at every x ∈ X, and
pointwise bounded if it is bounded at every x ∈ X.
We remark that every locally compact Hausdorff space admits a compatible uniformity; for our applications, the choice of uniformity will not be
3

significant in practice. In particular, it is easily seen that if Sx is compact,
then all compatible uniformities are equivalent for determining whether or
not S is equicontinuous at x.
Equicontinuity is preserved by finite unions, and finite products of pointwise bounded equicontinuous families are pointwise bounded equicontinuous.
(To see why equicontinuity on its own is not preserved by products, let f be
a homeomorphism of Q2 such that for all x ∈ Q2 rZ2 and y ∈ Z2 , f (x+y) =
f (x) + y/|x|2 , and consider the family S = {x 7→ f (x) + 2−n | n ∈ N} of
homeomorphisms of Q2 . In this case, S is equicontinuous but S 2 is nowhere
equicontinuous.)
Lemma 2.2. Let X be a locally compact Hausdorff space equipped with some
uniformity U , let S, T ⊆ Homeo(G) and let x ∈ X.
(i) If S and T are equicontinuous at x, then so is S ∪ T .
(ii) Suppose that T x is compact, T is equicontinuous at x, and T x consists
of equicontinuous points for S on X. Then ST is equicontinuous at
x ∈ X. If in addition, Sy is compact for every y ∈ T x, then ST x is
compact.
Proof. (i) Let E be an entourage. Since S and T are equicontinuous at x,
there are neighbourhoods V1 and V2 of x such that (s.x, s.y) ∈ E for all
s ∈ S and y ∈ V1 , and also for all s ∈ T and y ∈ V2 . Thus (sx, sy) ∈ E for
all s ∈ S ∪ T and y ∈ V1 ∩ V2 .
(ii) Let E be an entourage and let z ∈ T x. Since S is equicontinuous at
z, there is an open neighbourhood Vz of z with compact closure such that
(s.y, s.w) ∈ E for all s ∈ S and y, w ∈ Vz . In turn there is a smaller open
neighbourhood Wz of z and an entourage Ez such that
∀y ∈ X ∀w ∈ Wz : (y, w) ∈ Ez ⇒ y ∈ Vz .
Since T x is compact, we can choose finitely many points z1 , . . . , T
zn in T x
′
such that {Wz1 , . . . , Wzn } covers T x, and obtain an entourage E = ni=1 Ezi .
Since T is equicontinuous at x, there is a neighbourhood W of x such that
(tx, ty) ∈ E ′ for all y ∈ W . Given t ∈ T and y ∈ W , then ty ∈ Wzi for
some i, so t.x ∈ Vzi , and hence (stx, sty) ∈ E for all s ∈ S. Thus ST is
equicontinuous at x.
Now suppose in addition that Sy is compact for every y ∈ T x. For each
y ∈ T x choose an entourage Ey such that the set Ky := {(z, w) ∈ Ey |
z ∈ Sy, w ∈ X} has compact closure, and let Uy be a neighbourhood of
y such that for S
all s ∈ S, z ∈ Uy ⇒ (sy, sz) ∈ Ey . Since T x is compact,
we haveST x ⊆ y∈F Uy for a finite subset F of Sy. It then follows that
ST x ⊆ y∈F Ky , so ST x has compact closure.
We also note the following application of the Arzelà–Ascoli theorem.
4

Lemma 2.3. Let X be a locally compact Hausdorff topological space, let
S ⊆ Homeo(X) and let U ⊆ X be a compact open set, such that both SU
and S −1 SU are compact and SU consists of equicontinuous points for S −1
with respect to some compatible uniformity. Then {sU | s ∈ S} is finite.
Proof. Let Y = SU and let C(Y, X) be the space of continuous functions
from Y to X with the topology of uniform convergence. By restricting the
domain, let us regard S −1 as a collection of functions from Y to X. Then
this family of functions is equicontinuous and uniformly bounded. Thus
by the Arzelà–Ascoli theorem (see [4, Theorem 43.15]), S −1 has compact
closure in C(Y, X). Since U is clopen in X, we then have a continuous
map from C(Y, X) to C(Y, {0, 1}), given by f 7→ 1U ◦ f . Note moreover
that C(Y, {0, 1}) carries the discrete topology, as any uniformly convergent
sequence of functions to {0, 1} is eventually constant. Thus the set {1U ◦s−1 |
s ∈ S} is finite. The function 1U ◦ s−1 determines the subset sU of Y , so
the set {sU | s ∈ S} is also finite.

2.2

Distal and almost periodic actions

Definition 2.4. Let G be a group, let X be a Hausdorff topological space
and let G act on X via the homomorphism θ : G → Homeo(X). A proximal
pair for the action is a pair (x, y) ∈ X × X such that, for some z ∈ X, we
have (z, z) ∈ {(θ(g).x, θ(g).y) | g ∈ G}. An element x ∈ X is a distal point
if, whenever (x, y) is a proximal pair, then y = x. The action is distal if
every point is a distal point.
We equip the space X X of functions from X to X with the topology of
pointwise convergence. The enveloping semigroup T of the action θ of G
on X is the closure of θ(G) in X.
It is straightforward to verify that T 2 ⊆ GT ⊆ T , so the enveloping
semigroup T is indeed a semigroup under composition; if in addition, Gx
is compact for every x ∈ X, then T is compact by Tychonoff’s theorem.
However in general, T is not a group.
Theorem 2.5 (Ellis [2]). Let G be a group of homeomorphisms of a Hausdorff space X such that Gx is compact for every x ∈ X. Then the following
are equivalent:
(i) The closure of G in X X is a group.
(ii) For every cardinal κ > 0, G is pointwise almost periodic on X κ .
(iii) There exists a cardinal κ ≥ 2 such that G is pointwise almost periodic
on X κ .
(iv) G is distal.

5

Distality also implies a certain orbit closure property, and almost periodic points in a locally compact Hausdorff space can be characterized in
terms of orbit closures.
Definition 2.6. Let G be a group acting on a topological space X. The
orbit closure relation R for the action is given by (x, y) ∈ R if y ∈ Gx.
An action of a group G on a topological space X is minimal if every
orbit is dense. More generally, the action has minimal orbit closures if
G acts minimally on Gx for every x ∈ X; equivalently, the orbit closure
relation R is symmetric.
Lemma 2.7. Let H be a group acting by homeomorphisms on a Hausdorff
topological space X. Let x, y ∈ X be such that Hx is compact and consists
of distal points for the action. Then H acts minimally on Hx. Indeed, we
have Hy = Hx whenever y ∈ X is such that Hx ∩ Hy 6= ∅.
Proof. Let C = Hx and let T be the enveloping semigroup of the action
of H on C. By Theorem 2.5, T is a group; it is also compact, since C C is
compact. Let (hi ) be a net in H such that (hi (y)) converges to c ∈ C. By
passing to a subnet we may assume that the image of (hi ) in T converges to
some limit t ∈ T . Letting c′ = t−1 (c), then hi (c′ ) converges to t(t−1 (c)) = c.
In particular, (y, c′ ) is a proximal pair for the action of H. Since C consists
of distal points, we must have y ∈ C, and thus Hy ⊆ Hx. In particular, Hy
is compact and consists of distal points for the action, so the same argument
applied to x implies that in fact Hy = Hx.
Lemma 2.8 (See also e.g. [3, p31]). Let X be a locally compact Hausdorff
space, let G ≤ Homeo(X) and let x ∈ X. Then the following are equivalent:
(i) Gx is compact and G acts minimally on Gx;
(ii) x is an almost periodic point for G.
Proof. Let Y = Gx.
Suppose that Y is compact and that G acts minimally on Y , let V be
a neighbourhood of x and let N = {h ∈ G | hx ∈ V }. By minimality,
Sn Y is
covered by the G-translates of V ; since Y is compact, we have Y ⊆ i=1 gi V
for some g1 , . . . , gn ∈ G. Given g ∈ G, then gx ∈ gi V for some 1 ≤ i ≤ n,
so gi−1 gx ∈ V , that is, gi−1 g ∈ N . Thus G = F N where F = {g1 , . . . , gn },
showing that x is an almost periodic point.
Conversely, suppose that x is an almost periodic point. Let Z be an
nonempty open G-invariant subset of Y ; since Gx is dense in Y , we have
Gx ∩ Z 6= ∅ and hence x ∈ Z. Let V be a compact neighbourhood of x such
that V ∩ Y ⊆ Z. Then G = F N where F is a finite set and N = {h ∈ G |
hx ∈ V }. Hence
∀g ∈ G ∃f ∈ F : gx ∈ f V ∩ Y = f (V ∩ Y );
6

or in other words, Gx ⊆ F (V ∩ Y ). Since F is finite, the set F (V ∩ Y )
is compact and in particular, closed in Y . Since Gx is dense in Y , we
conclude that Y = F (V ∩ Y ) = Z. Thus Y is compact and there is no
proper nonempty open G-invariant subset of Y , that is, G acts minimally
on Y .

3

Recurrence for equicontinuously generated actions

Here is our substitute for [1, Proposition 1.5].
Lemma 3.1. Let X be a locally compact Hausdorff space, let S ⊆ Homeo(X)
with 1 ∈ S and S = S −1 and let G = hSi. Let x ∈ X and let F be a finite
subset of G. Then there is a positive integer n such that for all g ∈ G with
|g| ≥ n, there is c ∈ G such that |c| = n and F c ⊆ K(g).
Proof. Since 1 ∈ S = S −1 and S generates G, there is some n such that
F ⊆ S n−1 . Suppose |g| ≥ n, write g as g = hc where |c| = n and |g| = |h|+|c|
and let f ∈ F . Then
|f cg−1 | = |f h−1 | ≤ (n − 1) + |h| < |g|,
so f cg −1 ∈ S |g|−1 and hence f c ∈ K(g). Thus F c ⊆ K(g).
We are now ready to prove the main theorem.
Proof of Theorem 1.2. Let us first note that for every x ∈ U0 , the set Gx is
a compact subset of U0 . Indeed, given x ∈ U0 , then x ∈ V ⊆ U0 for some
compact G-invariant set V , and then Gx is a closed subset of V . It follows
by Lemma 2.2 and induction on n that S n is equicontinuous at every x ∈ U0 .
Suppose (vi) holds. Since U0 is open, we can take W = U0 . For each
y ∈ Y , we see that φ−1 (y) is a minimal G-set. As a consequence, given
x, y ∈ W , we have (x, y) ∈ R if and only if φ(x) = φ(y). By continuity of φ,
this defines a closed relation on W . Thus (vi) implies (v).
Suppose (v) holds for the neighbourhood W of U0 , and let x ∈ U and
y ∈ U0 such that y ∈ Gx. In particular, since W is a neighbourhood of y,
gx ∈ W for some g ∈ G. We also have gi x → y for some net (gi ) in G. We
see that (gi x, gx) ∈ R ∩ (W × W ) eventually as i → ∞, so (y, gx) ∈ R, since
the restriction of R to W × W is closed. Thus gx ∈ Gy and hence x ∈ Gy.
Thus (iv) holds.
Suppose (iv) holds, and let V be
. Let V0 = V
T a compact open subset of UT
and for each n ≥ 0, let Vn+1 = s∈S s−1 (Vn ) and let V∞ = n≥0 Vn . By
Lemma 2.2 and induction
on n, S n+1 is equicontinuous at every z ∈ Vn . We
T
also see that V∞ = g∈G g(V ); additionally V∞ is compact.

7

Let Pn = S n Vn . By Lemma 2.3, the set {gVn | g ∈ S n } is finite,
so Pn is compact. By construction, we have Pn+1 ⊆ Pn ⊆ V0 for all n.
Suppose
T that for all n, we have Pn 6⊆ V1 . Then by compactness, there exists
x ∈ n≥0 (Pn rV1 ). Then x ∈ S n Vn for all n, in other words, there is gn ∈ S n
such that gn−1 x ∈ Vn . By compactness (gn−1 x) accumulates at some y ∈ V∞ ,
so y ∈ Gx. Since V∞ is compact and G-invariant, Gy is a compact subset
of U and in particular y ∈ U0 . Thus by (iv), we have x ∈ Gy; thus x ∈ V∞ .
This is impossible as we chose x not to be an element of V1 . From this
contradiction, it follows that in fact Pn ⊆ V1 for some n, and hence Pm ⊆ V1
for all m ≥ n.T In other words, for all m ≥ n and g ∈ S m we have g(Vm ) ⊆ V1 .
Then Vm ⊆ g∈S m g −1 (V1 ) = Vm+1 , so in fact Vn = Vm for all m ≥ n, and
hence Vn = V∞ . Thus V∞ is a compact open G-invariantTneighbourhood
of x; by the compactness of V r V∞ , it follows that V∞ = g∈F g(V ) for a
finite subset F of G. We have now proved (iii).
Suppose (iii) is satisfied. Taking x ∈ U0 , then there is a compact open
neighbourhood
V of x contained in U . Condition (iii) ensures that V∞ =
T
g(V
)
is
a
G-invariant set for some finite F ⊆ G; clearly V∞ is then
g∈F
a compact open G-invariant neighbourhood of x contained in U0 . Thus U0
is open. Suppose that some y ∈ U0 is not almost periodic for G. Then by
Lemma 2.8 there is z ∈ Gy such that y 6∈ Gz. There is then a compact open
neighbourhood W of Gz not containing y, and hence by (iii), there is an
open G-invariant neighbourhood W∞ of Gz not containing y. But then Gy
is disjoint from W∞ , so z 6∈ Gy, a contradiction. Thus (iii) implies (ii).
Suppose that (ii) is satisfied. Let y ∈ U0 , let V be a neighbourhood of
y and let N be the set of h ∈ G such that hy ∈ V . Then there is a finite
subset F of G such that F N = G, in other words N intersects F −1 g for
all g ∈ G. Given a net (gi ) in G such that |gi | → ∞, then by Lemma 3.1
there is a subnet (gi(j) )j∈J such that for all j ∈ J, there is dj such that
F −1 dj ⊆ K(gi(j) ), and moreover |dj | is bounded. We then choose fj ∈ F −1
such that fj dj ∈ N and set cj = fj dj ; then |cj | is bounded and cj y ∈ V for
all j ∈ J. Hence y is S-recurrent, so (ii) implies (i).
Now suppose that (i) is satisfied. We will first use (i) to prove (v), and
then, since (v) has been shown to imply (i)–(iv), we can use conditions (iii)
and (iv) to prove (vi) and thereby complete the cycle of implications between
the conditions (i)–(vi).
Since U0 is open, we simply set W = U0 . Suppose (yi , zi )i∈I is a net of
pairs of points in R such that (yi , zi ) → (y, z) ∈ U0 × U0 but (y, z) 6∈ R,
that is, z 6∈ Gy. There is then a compact open neighbourhood V of z that
is contained in U0 and disjoint from Gy; by passing to a subnet we may
assume yi ∈ U0 and zi ∈ V for all i ∈ I.
Given i ∈ I, let gi be an element of G of least possible word length such
that gi yi ∈ V . Such elements gi exist since zi ∈ Gyi ∩ V for all i. Since
V is compact, by passing to a subnet we may assume gi yi → w for some

8

w ∈ V . Suppose that |gi | 6→ ∞. Then we can pass to a subnet on which gi is
bounded, say |gi | ≤ n. Since S n is equicontinuous at y and yi → y, for every
entourage E there is iE ∈ I such that for all i > iE , we have (gi y, gi yi ) ∈ E.
It then follows that gi y → w, a contradiction, since w ∈ V whereas Gy is
disjoint from V . From this contradiction we conclude that in fact |gi | → ∞,
and hence |gi−1 | → ∞.
Since w is an S-recurrent point, on passing to a subnet we may ensure
the existence an integer n and elements ci of K(gi−1 ), such that ci w ∈ V
and |ci | ≤ n for all i ∈ I. Since S n is equicontinuous at U0 , the elements
{ci | i ∈ I} form an equicontinuous family at w. Since V is compact open,
there is an entourage E such that
w1 ∈ V, (w1 , w2 ) ∈ E ⇒ w2 ∈ V ;
let WE be a neighbourhood of w such that (ci w, ci w′ ) ∈ E for all w′ ∈
WE and i ∈ I. Then for all sufficiently large i, we have gi yi ∈ WE , so
(ci w, ci gi yi ) ∈ E, and hence ci gi yi ∈ V . On the other hand, for each i, we
can write ci as ci = hi gi−1 where |hi | < |gi |; thus ci gi yi = hi yi 6∈ V by the
minimality of |gi |. From this contradiction, we conclude that in fact the
restriction of R to U0 × U0 is closed, so (i) implies (v).
Condition (iv) ensures that given x, y ∈ U0 , we have (x, y) ∈ R if and only
if Gx = Gy, so R restricts to a closed equivalence relation R′ on U0 . Write
[x] for the R′ -class of x, let Y be the set of R′ -classes and let φ : U0 → Y
be given by x → [x]. We see that φ(gx) = φ(x) for all x ∈ U0 , so φ is
G-equivariant with respect to the trivial action on Y , and moreover G acts
minimally on each fibre. We equip Y with the quotient topology induced by
φ. The fibres of φ are of the form Gx for x ∈ U0 , so they are compact. Let
τ be the topology of Y and let L be the set of compact open G-invariant
subsets of U0 . Given V ∈ L we see that V is a union of R′ -classes, so
φ−1 (φ(V )) = V and in particular φ(V ), Y r φ(V ) ∈ τ . On the other hand,
given x ∈ U0 and a neighbourhood W of x containing Gx, then W contains
a compact open neighbourhood of Gx, and by (iii) we obtain V ∈ L such
that φ−1 (y) ⊆ V ⊆ W , so {φ(V ) | V ∈ L} is a base for τ and τ is Hausdorff.
We conclude that Y is locally compact and zero-dimensional, completing
the proof of (vi).
Remark 3.2. The implications (v) ⇒ (iv) and (iii) ⇒ (ii) ⇒ (i) do not use
the assumption that S is equicontinuous.
Let X be a locally compact zero-dimensional space, let G ≤ Homeo(X)
be equicontinuously generated, let Y be the interior of the set of almost
periodic points of X and let U be a G-invariant open subset of X. As a
consequence of Theorem 1.2, we see that if U0 ⊆ Y , then U0 = U ∩ Y and
U satisfies all of the conditions (i)–(vi) of the theorem, whereas if U0 6⊆ Y ,
then U fails to satisfy any of the conditions (i)–(vi) of the theorem. Thus

9

the set Y can be characterized in several equivalent ways, using each of the
conditions (i)–(vi) individually.
As a consequence of Theorem 1.2, we obtain a more general version of
[1, Corollary 1.9]. We state a version here where a single orbit is assumed
to have compact closure; Corollary 1.3 will then follow immediately as a
special case.
Corollary 3.3. Let X be a locally compact zero-dimensional space equipped
with a compatible uniformity, let S ⊆ Homeo(X) with 1 ∈ S and S = S −1 ,
let G = hSi, and let x ∈ X. Suppose that Gx is compact and that there
is a neighbourhood of Gx in X consisting of distal points for G. Then the
following are equivalent:
(i) There is a closed G-invariant neighbourhood U of x consisting of equicontinuous points for S.
(ii) There is a compact G-invariant neighbourhood V of x on which G acts
equicontinuously.
Proof. Clearly (ii) implies (i). Suppose that (i) holds. Then we are in the
situation of Theorem 1.2, and moreover by Lemma 2.7, Theorem 1.2(iv) is
satisfied. We also note that x ∈ U0 . Thus using Theorem 1.2(iii), we obtain
a compact G-invariant neighbourhood V of x on which G acts distally, such
that V ⊆ U . By Theorem 2.5, the action of G on V × V has minimal
orbit closures, and it is clear that S is equicontinuous on V × V . Applying
Theorem 1.2(iii) again, for any compact
T open subset
T E of V × V , there is a
finite subset F of G such that E∞ := g∈G g(E) = g∈F g(E). In particular,
every neighbourhood E of the diagonal in V × V contains a compact open
neighbourhood E ′ of the diagonal, which in turn contains a G-invariant
neighbourhood (E ′ )∞ of the diagonal. Thus G acts equicontinuously on V ,
proving (ii).

References
[1] J. Auslander, E. Glasner and B. Weiss, On recurrence in zero dimensional flows, Forum Math. 19 (2007), 107–114.
[2] R. Ellis, Distal transformation groups, Pacific J. Math. 8 (1958), 401–
405.
[3] W. Gottschalk and G. Hedlund, Topological dynamics, Amer. Math.
Soc. Colloquium Publications, vol. 36, Providence, 1955.
[4] S. Willard, General topology, Addison–Wesley, 1970.

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