1 Introduction
Let
$(X,T)$
be a topological dynamical system, by which we mean a compact metrizable space X with an action of a group T by homeomorphisms
$\alpha ^t$
,
$t\in T$
. Here, X has at most cardinality
${\mathfrak c}$
that of the continuum. The Ellis or enveloping semigroup
$E(X,T)$
of the dynamical system is the closure of
$\{\alpha ^t:t\in T\}$
in
$F(X)$
, the semigroup of functions from X to X equipped with composition as multiplication and the topology of point-wise convergence. If no confusion arises, we simply write E or
$E(X)$
for
$E(X,T)$
. Here,
$E(X)$
is closed under composition and so a compact right topological sub-semigroup of
$F(X)$
.
The Ellis semigroup has rich algebraic and topological properties, and these can be used to characterize the dynamical system. A recent survey on this can be found in [Reference Glasner7]. One property that has attracted much attention is tameness [Reference Köhler16]. Here,
$E(X,T)$
(and
$(X,T)$
) is tame if all its elements are Baire class 1 functions, that is, can be obtained as a limit of a sequence of continuous functions [Reference Glasner6]. An equivalent characterization is that
$E(X,T)$
is tame if its cardinality is at most
${\mathfrak c}$
[Reference Glasner and Megrelishvili10]. A third characterization relates the absence of tameness to the existence of an independence sequence [Reference Kerr and Li15].
Like any semigroup,
$E(X,T)$
has an ideal structure and can be decomposed into the equivalence classes of Green’s relations. In the context of non-tame dynamical systems, it is interesting to know which of these parts are especially big. More specifically, we can look at the kernel
$\ker E$
of
$E(X,T)$
(its smallest bilateral ideal). Given any minimal idempotent
$e\in E$
, the kernel is the bilateral ideal generated by it,
$\ker E = EeE$
. It is a completely simple semigroup without zero and therefore has the following algebraic structure [Reference Hindman and Strauss11, Reference Howie12]. Let
$\Lambda $
be the set of minimal left ideals and I the set of minimal right ideals of E, then
$$ \begin{align*}\ker\, E \cong I\times {\mathcal H}\times \Lambda\end{align*} $$
with multiplication
$$ \begin{align*}(i,g,\unicode{x3bb}) (j,h,\mu)\, := (i,g a_{\unicode{x3bb} \, j} h,\mu),\end{align*} $$
where
${\mathcal H}$
is the so-called structure group and
$(a_{\unicode{x3bb} \, i})_{i\in I,\unicode{x3bb} \in \Lambda }$
is a matrix with values in
${\mathcal H}$
. Given a minimal idempotent
$e\in E$
,
${\mathcal H}$
can be taken to be
$$ \begin{align*} {\mathcal H}_e:= eEe\end{align*} $$
different choices of minimal idempotents leading to isomorphic groups.
What can we say about the size of I,
${\mathcal H}$
and
$\Lambda $
? Let us mention some known results for minimal
$(X,T)$
.
-
(1)
$(X,T)$
is distal if and only if E is a group [Reference Auslander2, Reference Hindman and Strauss11]. This implies that
$E = \ker E = {\mathcal H}$
and
$\mathrm {card}(I)=\mathrm {card}(\Lambda ) = 1$
. -
(2) If
$(X,T)$
is even equicontinuous, then the cardinality of
${\mathcal H}=E$
is at most
$\mathfrak {c}$
. -
(3) If
$(X,T)$
is almost automorphic with maximal equicontinuous factor
$X_{\max }$
, then
$\mathrm {card}(\Lambda ) = 1$
and
${\mathcal H} = E(X_{\max })$
[Reference Aujogue, Barge, Kellendonk and Lenz1], we recall the proof below. This shows that
$\mathrm {card}({\mathcal H}) \leq \mathfrak {c}$
. -
(4) Dynamical systems arising from primitive aperiodic bijective substitutions provide examples of non-tame Ellis semigroups for which
$E = \mathbb {Z}\cup \ker E$
with finite I and
$\Lambda $
, but
$\mathrm {card}({\mathcal H}) = 2^{\mathfrak {c}}$
[Reference Kellendonk and Yassawi14]. -
(5) If
$\mathrm {card}{\Lambda }\leq \mathfrak {c}$
, then
$(X,T)$
is a PI-flow [Reference Glasner and Glasner9].
To illuminate the last result, we recall that
$(X,T)$
is a PI-flow if it admits a proximal extension
$(\tilde X,T)$
that itself is a tower of PI-extensions of the one-point system
$(pt,T)$
, and a PI-extension is an extension that is a composition of a proximal with an isometric extension [Reference Glasner and Glasner9]. As is also shown in [Reference Glasner and Glasner9], the converse is not always the case; there exists a PI minimal metrizable system
$(X,T)$
even with
$T=\mathbb {Z}$
for which
$E(X, T)$
contains nevertheless
$2^{\mathfrak {c}}$
minimal left ideals.
In this work, we focus on the cardinality of the structure group
${\mathcal H}$
. The starting point is the observation that this group need not to have the same cardinality as the Ellis semigroup. Indeed, there are almost automorphic non-tame dynamical systems with
$T=\mathbb {Z}$
, and hence their structure group coincides with their maximal equicontinuous factor and so has a cardinality
$\leq {\mathfrak c}$
. To state our result, we need to provide more background.
Recall that
$(X,T)$
is equicontinuous if the family of homeomorphisms
$\{\alpha ^t:t\in T\}$
is equicontinuous. Any dynamical system
$(X,T)$
, equicontinuous or not, admits a maximal equicontinuous factor
$\pi :(X,T)\to (X_{\max },T)$
. The equivalence relation
$x\sim y$
if and only if
$\pi (x)=\pi (y)$
is called the equicontinuous structure relation. A point
$\xi \in X_{\max }$
is called singular if the fibre
$\pi ^{-1}(\xi )$
contains at least two proximal points. Otherwise, it is called regular. By a finite distal fibre of
$(X,T)$
, we mean a fibre
$\pi ^{-1}(\xi )$
that is finite and contains only pairwise distal points. A system is called almost automorphic if it is minimal and there is
$\xi \in X_{\max }$
such that
$\pi ^{-1}(\xi )$
contains a single point.
A recent result states that if a minimal system
$(X,T)$
for which X admits a T-invariant probability measure is tame, then it is almost automorphic and the set of regular points has full Haar measure in
$X_{\max }$
[Reference Glasner8]. The converse need not to be true. Regular Toeplitz systems (minimal almost automorphic extensions of odometers) need not be tame, although their set of regular points has full measure. For instance, of the two symbolic dynamical systems associated to the two substitutions
$$ \begin{align*}\begin{matrix} a & \mapsto & a a b a a\\ b & \mapsto & a b b a a \end{matrix} \qquad \begin{matrix} a & \mapsto & a a b a a\\ b & \mapsto & a b a b a \end{matrix} \end{align*} $$
the one on the left is tame, whereas the other is not tame [Reference Fuhrmann, Kellendonk and Yassawi5]. Note that these substitutions differ only in the order of two letters, in particular, their associated dynamical systems are strong orbit equivalent.
The following is Theorem 3.7 in the main text.
Theorem 1.1. Let
$(X,T)$
be a minimal system with infinite X. If the proximal relation agrees with the equicontinuous structure relation, then the cardinality of the structure group is
${\mathfrak c}$
.
We note that for almost automorphic systems, the proximal relation always agrees with the equicontinuous structure relation, but, as we already said above, this does not imply that
$E(X)$
is tame.
As a partial converse to the above, we obtain the following statement. Recall that the Ellis semigroup
$E(X_{\max })$
of the maximal equicontinuous factor acts on
$X_{\max }$
and by minimality
$ev_{\xi _0}:E(X_{\max })\to X_{\max }$
,
$ev_{\xi _0}(f) = f(\xi _0)$
is a continuous surjection. If the action is free, then
$ev_{\xi _0}$
is a homeomorphism that can be used to push the group structure from
$E(X_{\max })$
to
$X_{\max }$
. This is, for instance, always the case if T is abelian. We then define
${\mathcal L}^{\text {sing}}$
to be the group generated by elements
$\xi \zeta ^{-1}$
, where
$\xi $
,
$\zeta $
are singular points in
$X_{\max }$
(if there are no singular points, we set
${\mathcal L}^{\text {sing}}$
to be the trivial group). The result below is Corollary 3.12 in the main text.
Theorem 1.2. Let
$(X,T)$
be a minimal system with finite coincidence rank for which the action of
$E(X_{\max })$
on
$X_{\max }$
is free. If the proximal relation is not transitive and
$X_{\max }/{\mathcal L}^{\mathrm {sing}}$
uncountable, then the structure group has cardinality
$2^{\mathfrak {c}}$
.
The condition that
$X_{\max }/{\mathcal L}^{\text {sing}}$
is uncountable can be replaced by
${\mathcal L}^{\text {sing}}$
not being open, see Theorem 3.13. Note that this condition is always satisfied if there are only countably many singular points in
$X_{\max }$
, while
$X_{\max }$
is uncountable. We prove this theorem using a method already employed in [Reference Kellendonk and Yassawi14].
2 Preliminaries
We recall here some concepts and results for minimal topological dynamical systems
$(X,T)$
with compact metrizable space X.
Two points
$x,y\in X$
are proximal, written
$x\sim _p y$
, if
$\inf _{t\in T} d(\alpha ^t(x),\alpha ^t(y)) = 0$
, where d is a metric that induces the topology and
$\alpha $
the T-action. This notion does not depend on the choice of metric. A point is distal if it is not proximal with any other point. Additionally,
$(X,T)$
is point distal if it is minimal and contains a distal point.
We denote by
$(X_{\max },T)$
the maximal equicontinuous factor of
$(X,T)$
(which is uniquely determined up to isomorphism) and its factor map by
$\pi $
. The equicontinuous structure relation on X is the relation induced by
$\pi $
:
$x\sim y$
if and only if
$\pi (x)=\pi (y)$
. It is an equivalence relation that always contains the proximal relation.
We say that
$(X,T)$
has a finite distal fibre if there is
$\xi \in X_{\max }$
such that
$\pi ^{-1}(\xi )$
is finite and its points are pairwise non-proximal.
We define the coincidence rank of the fibre
$\pi ^{-1}(\xi )$
to be
$$ \begin{align*} \mathrm{cr}(\xi):=\sup\{l\in\mathbb{N}: \text{ there exists } x_1,\ldots,x_l\in\pi^{-1}(\xi) \text{ for all } {i\neq j}: x_i\not\sim_p x_j\}. \end{align*} $$
By [Reference Barge and Kellendonk3, Lemma 2.10],
$\mathrm {cr}(\xi )$
is the same for all
$\xi $
if
$(X,T)$
is minimal. (While [Reference Barge and Kellendonk3] makes the standing assumption that T is abelian, this is not required for the proof of Lemma 2.10.) We therefore may call
$\mathrm {cr}=\mathrm {cr}(\xi )$
the coincidence rank of the minimal system
$(X,T)$
. We summarize some properties of the coincidence rank. Items (3) and (5) provide criteria for the hypothesis of Theorem 1.1.
Lemma 2.1. Let
$(X,T)$
be minimal.
-
(1)
$(X,T)$
has a finite distal fibre if and only if it is point distal and has finite coincidence rank
$\mathrm {cr}$
. -
(2)
$\mathrm {cr}=1$
if and only if the proximal relation agrees with the equicontinuous structure relation. -
(3) If
$(X,T)$
is almost automorphic, then
$\mathrm {cr}=1$
. -
(4) If
$(X,T)$
is point distal and
$\mathrm {cr}=1$
, then it is almost automorphic. -
(5) Suppose that
$\mathrm {cr}$
is finite and that T contains a compact set K such that any open set containing K generates T. If the proximal relation is closed, then it is transitive and
$\mathrm {cr}=1$
.
Proof. (1) The implication ‘
$\Rightarrow $
’ is direct. For the other direction ‘
$\Leftarrow $
’, suppose that
$(X,T)$
has a distal point. Then, the set of distal points is residual [Reference Ellis4]. We can therefore apply a result of Veech [Reference Veech18], which states that there is
$\xi \in X_{\max }$
such that the distal points of
$\pi ^{-1}(\xi )$
are dense in that fibre. As
$\mathrm {cr}(\xi )$
is finite,
$\pi ^{-1}(\xi )$
contains finitely many distal points. As a finite set is closed,
$\pi ^{-1}(\xi )$
is a finite distal fibre.
(2) We have
$\mathrm {cr}=1$
if and only if all points of a fibre belong to the same proximal class. As the proximal relation is contained in the equicontinuous structure relation, this means that the two relations must coincide.
(3) This is direct, as we can measure
$\mathrm {cr}$
at the point
$\xi $
whose fibre is a singleton.
(4) This follows from 1, because point distal and
$\mathrm {cr}=1$
imply that there is a finite distal fibre, say at
$\xi $
, and since
$\mathrm {cr}(\xi )=1$
, this fibre has a single point.
(5) Suppose that the proximal relation P is topologically closed. Then it is transitive [Reference Auslander2] and hence an equivalence relation, and the quotient space
$X/P$
is compact in the induced metric. As
$\mathrm {cr}$
is finite by hypothesis, there exists
$\delta>0$
such that distal points of X which belong to the same fibre
$\pi ^{-1}(\xi )$
have distance at least
$\delta $
. Hence,
$(X/P,T)$
is a finite covering of
$(X_{\max },T)$
. As shown in [Reference Sacker and Sell17], the condition on T implies that
$(X/P,T)$
is an equicontinuous system. Therefore,
$X/P=X_{\max }$
and hence,
$\mathrm {cr}=1$
.
3 The structure of the kernel of
$E(X)$
If a semigroup admits a smallest bilateral ideal, this ideal is called the kernel of the semigroup. Compact right-topological semigroups admit always a kernel. This kernel, which we denote
$\ker E$
, is a completely simple semigroup (without zero element). This means that
$\ker E$
does not admit a proper bilateral ideal (it is simple) and that it contains at least one idempotent. We now partly describe
$\ker E$
referring the reader to [Reference Howie12] or [Reference Hindman and Strauss11] for more details.
Let
$J_{\min }$
be the set of idempotents in
$\ker E$
. These are called minimal idempotents. Then,
$\ker E$
is the bilateral ideal generated by p for any choice of
$p\in J_{\min }$
. Additionally,
$\ker E$
is partitioned by its left ideals and two idempotents
$p,q$
of the same left ideal satisfy
$pq=p$
. Furthermore,
$\ker E$
is also partitioned by its right ideals and two idempotents
$p,q$
of the same right ideal satisfy
$pq=q$
. The intersection of a left ideal with a right ideal contains a unique idempotent p and is a group, namely
${\mathcal H}_p:=p E p$
. Here, p is the neutral element of this group. All these groups are isomorphic. If
$p,q$
belong to the same left ideal, then the isomorphism is given by left multiplication with q,
$\unicode{x3bb} _q:{\mathcal H}_p\to {\mathcal H}_q$
, and if
$p,q$
belong to the same right ideal, then the isomorphism is given by right multiplication with q,
$\rho _q:{\mathcal H}_p\to {\mathcal H}_q$
. Given any two
$p,q\in J_{\min }$
, let
$r\in J_{\min }$
be such that the intersection of the left ideal containing
${\mathcal H}_p$
with the right ideal containing
${\mathcal H}_q$
is the group
${\mathcal H}_{r}$
. Then,
$\rho _q\circ \unicode{x3bb} _r$
is an isomorphism from
${\mathcal H}_p$
to
${\mathcal H}_q$
. The situation can be summarized by saying that
$\ker E$
is a disjoint union of isomorphic groups, for each minimal idempotent one, and the way how elements of different groups multiply is encoded in the matrix
$(a_{\unicode{x3bb} \,i})$
mentioned in §1.
Let
$\langle J_{\min }\rangle $
be the subsemigroup of
$\ker E$
generated by its idempotents. The following definition was introduced in [Reference Kellendonk and Yassawi14].
Definition 3.1. [Reference Kellendonk and Yassawi14]
Let e be a minimal idempotent. The structure group is the group
${\mathcal H}_e=eEe$
. The little structure group
$\Gamma _e$
is its subgroup
$e\langle J_{\min }\rangle e$
.
In the general theory of semigroups,
${\mathcal H}_e$
is called the Rees-structure group of
$\ker E$
. We recall that
${\mathcal H}_e$
depends on the choice of e only up to isomorphism. As the isomorphism
$\rho _q\circ \unicode{x3bb} _r : {\mathcal H}_p \to {\mathcal H}_q$
is given by multiplication with idempotents, it maps
$\Gamma _p$
bijectively onto
$\Gamma _q$
. All statements below hold for any choice of e.
3.1 The proximal relation and the little structure group
As is well known, the proximal relation can be studied with the help of the Ellis semigroup. Indeed,
$x,y\in X$
are proximal if and only if there exists
$f\in E$
such that
$f(x) = f(y)$
. By multiplying this equation with an element of
$\ker E$
if necessary, we see that f can be taken from
$\ker E$
and hence belongs to some minimal left ideal L. It then follows that
$f(x)=f(y)$
for any element of L. This has the following simple consequence.
Lemma 3.2. Let e be a minimal idempotent of
$E(X)$
. All elements of
$e\pi ^{-1}(\xi )$
are pairwise distal. In particular, if
$(X,T)$
is minimal, then
$e\pi ^{-1}(\xi )$
has
$\mathrm {cr}$
elements.
Proof. Suppose
$x,y\in e\pi ^{-1}(\xi )$
are proximal. Then, there is
$f\in E(X)$
such that
$f(x)=f(y)$
. As
$x=e(x)$
and
$y=e(y)$
, we also have
$efe(x) = efe(y)$
. Let
$g\in {\mathcal H}_e$
be the inverse of
$efe$
in
${\mathcal H}_e$
. Then,
$x=e(x)=gefe(x)=gefe(y) = y$
.
Lemma 3.3. [Reference Auslander2, Theorem 13(iii), p. 89]
Let
$(X,T)$
be minimal. Then,
$x,y\in X$
are proximal if and only if there exists a minimal idempotent p such that
$p(y)=x$
.
Proof. We first show that for any
$x\in X$
, one can find a minimal idempotent
$p\in E$
such that
$p(x) = x$
. Let
$x\in X$
. Let q be any minimal idempotent and
$y=q(x)$
. By minimality, its T-orbit is dense in X and hence
$E(y)=X$
. Thus, there is
$f\in E$
such that
$x=f(y)= fq(x)$
. As q is minimal,
$fq\in \ker E$
. Upon replacing f by
$fq$
, we may thus assume
$x=f(x)$
for some
$f\in \ker E$
. By the Rees structure theorem, f belongs to one of the groups
${\mathcal H}_{p}$
,
$p\in J_{\min }$
. Let
$f^{-1}$
be its inverse in this group. Then
$p = f f^{-1}$
is a minimal idempotent and
$p(x) = ff^{-1}(x) = ff^{-1}f(x) = f(x) = x$
.
Let
$x,y\in X$
be proximal. Then, there exists a minimal idempotent q such that
$q(x)=q(y)$
. Let p be a minimal idempotent such that
$p(x) = x$
. Let r be the (unique) minimal idempotent in the intersection of the minimal right ideal to which p belongs with the minimal left ideal to which q belongs. Then,
$rp=p$
and
$rq=r$
. Hence,
$$ \begin{align*}x = p(x) = rp(x) = rqp(x) = rq(x)=rq(y) = r(y).\end{align*} $$
As for the converse,
$p(y) = x$
implies
$p(y)=p(x)$
, and hence x and y are proximal.
It is well known that the proximal relation is transitive if and only if E has a unique minimal left ideal [Reference Auslander2]. We can relate this also to the triviality of the little structure group.
Lemma 3.4. Let
$\xi \in X_{\max }$
. The following are equivalent.
-
(1)
$\Gamma _e$
acts trivially on
$e\pi ^{-1}(\xi )$
. -
(2) For all
$p\in J_{\min }$
,
$\Gamma _p$
acts trivially on
$p\pi ^{-1}(\xi )$
. -
(3) The product of two minimal idempotents acts as an idempotent on
$\pi ^{-1}(\xi )$
.
Proof. ‘
$1\Rightarrow 2$
’: Suppose that
$\Gamma _e$
acts trivially on
$e\pi ^{-1}(\xi )$
and let
$p\in J_{\min }$
. Recall that
$\rho _p\circ \unicode{x3bb} _r : \Gamma _e\to \Gamma _p$
is an isomorphism where r is the unique idempotent in the intersection of the left ideal containing e and with the right ideal containing p. Let
$g\in \Gamma _e$
act trivially on
$e\pi ^{-1}(\xi )$
and
$x\in p\pi ^{-1}(\xi )$
. Then,
This shows that
$\Gamma _p$
acts trivially on
$p\pi ^{-1}(\xi )$
.
‘
$2\Rightarrow 3$
’: Let
$p,q\in J_{\min }$
. Let r be the unique idempotent such that
$pq\in {\mathcal H}_r$
. Then,
$pq = pqr = rpq r$
and thus
$pq\in \Gamma _r$
. As
$\Gamma _r$
acts trivially on
$r\pi ^{-1}(\xi )$
, we have
$pq(x) = pqr(x) = r(x)$
and so the restriction of
$pq$
to
$\pi ^{-1}(\xi )$
coincides with the restriction of the idempotent r to that fibre.
‘
$3\Rightarrow 1$
’: Suppose that
$pq(x)=x$
for all
$x\in \pi ^{-1}(\xi )$
and
$p,q\in J_{\min }$
. Then, also
$epe(x)=x$
showing that
$\Gamma _e$
acts trivially on
$e\pi ^{-1}(\xi )$
.
The above shows in particular that the little structure group is trivial if and only if the product of two minimal idempotents is an idempotent. Semigroups whose idempotents form a subsemigroup are called orthodox [Reference Howie12]. So
$\Gamma _e$
is trivial if and only if
$\ker E$
is orthodox.
As before, we choose for the following a minimal idempotent e. The results hold for any choice.
Lemma 3.5. Let
$(X,T)$
be minimal with maximal equicontinuous factor
$\pi :X\to X_{\max }$
. Let
$\xi \in X_{\max }$
. The proximal relation restricted to
$\pi ^{-1}(\xi )$
is transitive if and only if
$\Gamma _e$
acts trivially on
$e\pi ^{-1}(\xi )$
. In particular, the proximal relation is transitive if and only if the little structure group
$\Gamma _e$
is trivial.
Proof. Suppose that the proximal relation restricted to
$\pi ^{-1}(\xi )$
is transitive. Let
$x\in \pi ^{-1}(\xi )$
and
$p,q\in J_{\min }$
. Let L be the left ideal to which p belongs and R be the right ideal to which q belongs. Let
$q'$
be another idempotent of R. We have
$q(x)\sim _p x\sim _p q'(x)$
so
$q(x)\sim _p q'(x)$
by transitivity. Therefore, there is a left ideal
$L'$
such that
$rq(x)=rq'(x)$
for all idempotents
$r\in L'$
. Choose
$r\in L'\cap R$
. Then,
$q(x)=rq(x)=rq'(x)=q'(x)$
. Now choose
$q'\in L\cap R$
. Then,
$pq(x) = pq'(x) = p(x)$
. Hence,
$pq$
acts as an idempotent on
$\pi ^{-1}(\xi )$
. By Lemma 3.4,
$\Gamma _e$
acts trivially on
$e\pi ^{-1}(\xi )$
.
Suppose that
$\Gamma _e$
acts trivially on
$e\pi ^{-1}(\xi )$
. Let
$(x,y)$
and
$(y,z)$
be proximal pairs in
$\pi ^{-1}(\xi )$
. As the system is minimal, there exist minimal idempotents
$p,q$
such that
$p(x)=y$
and
$q(y)=z$
. Hence,
$qp(x) = z$
. By Lemma 3.4,
$qp$
acts as an idempotent on
$\pi ^{-1}(\xi )$
. Hence,
$qp(z) = qp(x)$
showing that
$(x,z)$
is a proximal pair.
The last statement follows as
$\Gamma _e$
acts trivially on
$e\pi ^{-1}(\xi )$
for all
$\xi $
if and only if
$\Gamma _e=\{e\}$
.
3.2 When the structure group is small
Like any factor map,
$\pi :X\to X_{\max }$
induces an epimorphism of semigroups
$\pi _{*}:E(X)\to E(X_{\max })$
, namely
$\pi _{*}(f)(\xi ) = \pi (f(x))$
, where x is any element of
$\pi ^{-1}(\xi )$
. Note that idempotents
$p\in E$
preserve the fibres of
$\pi $
and hence
${\pi }_{*}(p)(\xi ) = \pi (p(x))=\xi $
. It follows that
${\pi }_{*}:E(X)\to E(X_{\max })$
restricts to a (continuous) epimorphism of groups
$$ \begin{align*} {\pi}_{*}|_{{\mathcal H}_e} : {\mathcal H}_e\twoheadrightarrow E(X_{\max}), \end{align*} $$
whose kernel contains
$\Gamma _e$
. If no confusion is possible, we write
$\pi _{*}$
for
${\pi }_{*}|_{{\mathcal H}_e}$
.
Lemma 3.6. [Reference Aujogue, Barge, Kellendonk and Lenz1]
Let
$(X,T)$
be minimal. If the proximal relation agrees with the equicontinuous structure relation (
$\mathrm {cr}=1$
), then
$\pi _{*}: {\mathcal H}_e\to E(X_{\max })$
is an isomorphism. If
$E(X_{\max })$
acts freely on
$X_{\max }$
, then the converse is true as well,
$\pi _{*}: {\mathcal H}_e\to E(X_{\max })$
is an isomorphism if and only if
$\mathrm {cr}=1$
.
While the above isomorphism is a continuous bijection, it is not bi-continuous, as
$eEe$
is not compact if E contains more than one minimal idempotent.
Proof. Suppose that the proximal relation agrees with the equicontinuous structure relation and therefore is transitive. Let
$f\in eEe$
with
${\pi }_{*}(f) = {\mathrm {id}}$
. Then, f preserves the fibres of
$\pi $
. As
$\mathrm {cr}=1$
, all elements of a fibre are proximal. Therefore,
$x\sim _p f(x)$
for all x. Hence, there is a minimal idempotent p (which a priori may depend on x) such that
$p(x) = pf(x)$
. As the proximal relation is transitive, E has a unique minimal left ideal. This left ideal contains e and p, and hence,
$ep = e$
. This implies
$e(x) = f(x)$
, and hence,
$f=e$
.
To prove the converse statement, let
$\mathrm {cr}(\xi )>1$
. Then, there are
$y,y'\in \pi ^{-1}(\xi )$
that are not proximal. Therefore,
$x:=e(y)\neq x':=e(y')$
. By minimality, there exists
$f\in E(X)$
such that
$f(x') = x$
. Therefore,
$efe(x')=e(x)=x$
. Hence,
$\pi _{*}(efe)(\xi ) =\pi (efe(x')) = \pi (x)=\xi $
. As
$E(X_{\max })$
acts fixed point freely on
$X_{\max }$
, we conclude
$\pi _{*}(efe)={\mathrm {id}}$
. However,
$efe\neq e$
, because
$x'\neq x$
. So
$\pi _{*}$
is not injective.
Theorem 3.7. Let
$(X,T)$
be minimal. Suppose that X is an infinite set. If
$\mathrm {cr}=1$
, then the structure group
${\mathcal H}_e$
has cardinality
${\mathfrak c}$
.
Proof. By Lemma 3.6,
${\mathcal H}_e \cong E(X_{\max })$
. As
$(X_{\max },T)$
is equicontinuous,
$E(X_{\max })$
is metrizable. As X is infinite and
$\mathrm {cr}=1$
, the set of regular points in
$X_{\max }$
is infinite. It follows that
$E(X_{\max })$
is an infinite compact metrizable space. As T acts minimally on
$E(X_{\max })$
, the latter does not have isolated points. Therefore,
$E(X_{\max })$
is a non-empty perfect compact metrizable space. As such, it has cardinality
${\mathfrak c}$
[Reference Kechris13, Corollary 6.3].
3.3 When the structure group is huge
In this section, we provide a partial converse of the result of the last section, that
$\mathrm {cr}=1$
implies that the structure group is small.
Let
$K_e\subset {\mathcal H}_e$
be the subgroup of elements that preserve the fibres
$\pi ^{-1}(\xi )$
,
$\xi \in X_{\max }$
. Clearly,
$\Gamma _e$
is a subgroup of
$K_e$
. Here,
$E(X_{\max })$
acts transitively on
$X_{\max }$
. We assume for the rest of this section that this action is free so that, given any point
$\xi _0\in X_{\max }$
, the evaluation map
$ev_{\xi _0}:E(X_{\max })\to X_{\max }$
,
$ev_{\xi _0}(f) = f(\xi _0)$
is a bijection. We use this bijection to identify
$E(X_{\max })$
with
$X_{\max }$
and thus define a group structure on
$X_{\max }$
with neutral element
$\xi _0$
. We then have an exact sequence of groups
$$ \begin{align*} K_e\hookrightarrow {\mathcal H}_e \stackrel{ev_{\xi_0} \circ \pi_{*}}\twoheadrightarrow X_{\max}, \end{align*} $$
where
$\pi _{*}:E(X)\to E(X_{\max })$
is the epimorphism induced by
$\pi $
. Choose a right inverse
$s:X_{\max }\to {\mathcal H}_e$
to
$ev_{\xi _0} \circ \pi _{*}$
that intertwines the T actions, that is,
$ev_{\xi _0}\circ \pi _{*}\circ s = {\mathrm {id}}$
and
$s\circ \delta ^t = \alpha ^t\circ s$
(s need not be a group homomorphism).
Let
$f\in {\mathcal H}_e$
. We say that f acts trivially at
$\xi \in X_{\max }$
if all points of
$e\pi ^{-1}(\xi )$
are fixed points of f. We define the support of f to be the set of points of
$X_{\max }$
at which f acts non-trivially. Recall that a point
$\xi \in X_{\max }$
is called regular if
$\pi ^{-1}(\xi )$
contains only distal points and denote by
$X_{\max }^{\text {sing}}$
the complement of the regular points.
Lemma 3.8. Let
$e\neq f\in \Gamma _e$
. Then,
$\emptyset \neq \mathrm {supp}(f) \subset X_{\max }^{\mathrm {sing}}$
.
Proof.
$\emptyset \neq \mathrm {supp}(f)$
is clear by definition of the support and the inclusion follows from Lemma 3.5 as proximality restricted to regular fibres is the identity relation.
Lemma 3.9. Suppose that
$E(X_{\max })$
acts freely on
$X_{\max }$
. Let
$f\in K_e$
and
$a\in X_{\max }$
. The support of
$s(a) f s(a)^{-1}$
is
$a\,\mathrm {supp}(f)$
(product in
$X_{\max }$
).
Proof.
${\mathcal H}_e$
is a group acting on
$eX$
. Thus,
$s(a)$
restricts to a bijection from
$e\pi ^{-1}(\xi )$
to
$e\pi ^{-1}(a\xi )$
. This implies the statement.
Theorem 3.10. Let
$(X,T)$
be a minimal system with a finite coincidence rank. Suppose that
$E(X_{\max })$
acts freely on
$X_{\max }$
. If there is
$e\neq f\in K_e$
and an uncountable set
${A\subset X_{\max }}$
such that for all
$a\neq b\in A$
, we have
$$ \begin{align*}(a\,\mathrm{supp}(f))\cap (b\,\mathrm{supp}(f)) = \emptyset,\end{align*} $$
then the structure group
${\mathcal H}_e$
has cardinality
$2^{\mathfrak {c}}$
.
Proof. Let f and A be as in the theorem. As
$f\neq e$
, its support is not empty. Given
$a\in A$
, set
$f_a=s(a)f s(a)^{-1}$
. By Lemma 3.9, we have
$\mathrm {supp}(f_a) = a\,\mathrm {supp}(f)$
. Let
$a_1,\ldots ,a_k$
be k distinct points of A and
$f_{\{a_1,\ldots ,a_k\}} = f_{a_1} \cdots f_{a_k}$
, the product. As
$(a_i\,\mathrm {supp}(f))\cap (a_j\,\mathrm {supp}(f)) = \emptyset $
for
$i\neq j$
, we have
$$ \begin{align*}\mathrm{supp}(f_{\{a_1,\ldots,a_k\}}) = \bigcup_{i=1}^k (a_i\,\mathrm{supp}(f)).\end{align*} $$
Let
$B\subset A$
and consider the directed system of all finite subsets
$F_\nu \subset B$
of B, ordered by inclusion. The net
$f_{F_\nu }$
converges in the pointwise topology to the element
$f_B$
which, at
$\xi \in b\,\mathrm {supp}(f)$
,
$b\in B$
acts as
$f_b$
, while it acts trivially at all points in the complement of
$B\,\mathrm {supp}(f)$
. As
$Ee$
is a minimal left ideal, it is closed and therefore
$f_B\in Ee$
. Moreover,
$e\pi ^{-1}(\xi )$
is finite for all
$\xi \in X_{\max }$
, by Lemma 3.2. Therefore, the restriction of the net
$f_{F_\nu }$
to
$e\pi ^{-1}(\xi )$
becomes eventually constant. It follows that the limit
$f_B$
preserves all
$e\pi ^{-1}(\xi )$
so that
$ef_B=f_B$
, and hence,
$f_B\in {\mathcal H}_e$
. By construction,
$f_B\neq f_{B'}$
if B and
$B'$
are different subsets of A. Therefore,
$B\mapsto f_B$
is an injective map from the power set of A to
${\mathcal H}_e$
.
Given
$f\in K_e$
, let
${\mathcal L}^f$
be the subgroup of
$X_{\max }$
generated by products
$a b^{-1}$
with
$a,b\in \mathrm {supp}(f)$
(we let
${\mathcal L}^f$
be the trivial group if
$\mathrm {supp}(f)=\emptyset $
).
Corollary 3.11. Let
$(X,T)$
be a minimal system with finite coincidence rank and such that
$E(X_{\max })$
acts freely on
$X_{\max }$
. If there is
$f\in K_e$
such that
${\mathcal L}^f$
is not trivial and
$X_{\max }/{\mathcal L}^f$
is uncountable, then the structure group has cardinality
$2^{\mathfrak {c}}$
.
Proof. Let
$\tilde s: X_{\max }/{\mathcal L}^f \to X_{\max }$
be a right inverse of the quotient map, that is,
$\tilde s(\tilde a)$
is a choice of representatives for
$\tilde a$
. Set
$A=\tilde s(X_{\max }/{\mathcal L}^f)$
and let
$a, b\in A$
. If
$(a\,\mathrm {supp}(f)\cap (b\,\mathrm {supp}(f)) \neq \emptyset $
, then there are
$c,d\in \mathrm {supp}(f)$
such that
$ac=bd$
, and hence
$a=bdc^{-1}$
. As
$dc^{-1}\in {\mathcal L}^f$
, this implies
$a=b$
and so A satisfies the condition of the last theorem.
Finally, we formulate a criterion that is based on the non-transitivity of the proximal relation. In this case, X must be infinite. Let
${\mathcal L}^{\text {sing}}$
be the subgroup of
$X_{\max }$
generated by products
$a b^{-1}$
with
$a,b\in X_{\max }^{\text {sing}}$
.
Corollary 3.12. Let
$(X,T)$
be a minimal system with finite coincidence rank and such that
$E(X_{\max })$
acts freely on
$X_{\max }$
. If the proximal relation is not transitive and
$X_{\max }/{\mathcal L}^{\mathrm {sing}}$
uncountable, then the structure group has cardinality
$2^{\mathfrak {c}}$
.
Proof. Suppose that the proximal relation is not transitive. Then,
$\Gamma _e$
is non-trivial and so, by Lemma 3.8, there is
$f\in \Gamma _e$
with
$\emptyset \neq \mathrm {supp} f \subset X_{\max }^{\text {sing}}$
and, hence,
$\emptyset \neq {\mathcal L}^f\subset {\mathcal L}^{\text {sing}}$
. Thus, if
$X_{\max }/{\mathcal L}^{\text {sing}}$
is uncountable, then also
$X_{\max }/{\mathcal L}^{f}$
is uncountable and we conclude with Corollary 3.11.
Note that if
${\mathcal L}^{\text {sing}}$
is open, then
$X_{\max }/{\mathcal L}^{\text {sing}}$
is a single point by minimality of the T-action.
Theorem 3.13. Let
$(X,T)$
be a minimal system with finite coincidence rank and such that
$E(X_{\max })$
acts freely on
$X_{\max }$
. If the proximal relation is not transitive and
${\mathcal L}^{\mathrm {sing}}$
is not open, then the structure group has cardinality
$2^{\mathfrak {c}}$
.
Proof. Suppose that the proximal relation is not transitive and that
${\mathcal L}^{\text {sing}}$
is not open. Then,
$\Gamma _e$
is non-trivial and so, by Lemma 3.8, there is
$f\in \Gamma _e$
with
$\emptyset \neq \mathrm {supp} f \subset X_{\max }^{\text {sing}}$
and, hence,
$\emptyset \neq {\mathcal L}^f\subset {\mathcal L}^{\text {sing}}$
. Furthermore,
${\mathcal L}^{\text {sing}}\neq X_{\max }$
and, hence, X contains a finite distal fibre and, therefore, a dense subset on which
$\pi $
is
$\mathrm {cr}$
to one. As X is infinite and
$\mathrm {cr}$
finite,
$X_{\max }$
must be infinite. By minimality of the T-action, it has no isolated points. Thus,
$X_{\max }$
is uncountable [Reference Kechris13, Corollary 6.6]. Suppose that
$X_{\max }/{\mathcal L}^{\text {sing}}$
is countable. Then,
${\mathcal L}^{\text {sing}}$
cannot be meager. As the system is point distal, regular points are residual [Reference Ellis4]. Therefore,
$X_{\max }^{\text {sing}}$
and also
${\mathcal L}^{\text {sing}}$
are Borel sets, and hence, analytic. It follows that
${\mathcal L}^{\text {sing}}$
has the Baire property [Reference Kechris13, Theorem 29.5]. By Pettis’ theorem [Reference Kechris13, Theorem 9.9],
${\mathcal L}^{\text {sing}}$
must be open. As this contradicts our assumption,
$X_{\max }/{\mathcal L}^{\text {sing}}$
must be uncountable. Hence, also
$X_{\max }/{\mathcal L}^f$
is uncountable and the result follows from Corollary 3.11.
We recall that the condition that
$E(X_{\max })$
acts freely on
$X_{\max }$
is always satisfied for abelian T. In this case,
${\mathcal L}^{\text {sing}}$
is the group of differences of singular points. If there are only countably many singular points in
$X_{\max }$
, then
${\mathcal L}^{\text {sing}}$
is, of course, not open and thus the structure group has cardinality
$2^{\mathfrak {c}}$
provided the proximal relation is not transitive. If there is only one orbit of singular points, then more can be said about
${\mathcal H}_e$
, see [Reference Kellendonk and Yassawi14]. For instance, if
$(X_\theta ,{\mathbb Z})$
is the shift dynamical system associated to a bijective substitution
$\theta $
of length
$\ell $
with trivial generalized height, then the maximal equicontinuous factor is the
$\ell $
-adic odometer
$\mathbb Z_\ell $
, and the structure group
${\mathcal H}$
topologically and algebraically isomorphic to the group of all functions
$g:\mathbb Z_\ell /{\mathbb Z}\to G_\theta $
from the space of orbits of
$\mathbb Z_\ell $
to the group
$G_\theta $
which generated by the column maps of the substitution. Here, the group multiplication on the group of functions is point-wise and the topology is that of point-wise convergence. The fact that any function is allowed leads to the large cardinality.
Acknowledgements
The author is grateful to Todor Tsankov for explaining the equivalence between
$X_{\max }/{\mathcal L}^{\text {sing}}$
being countable and
${\mathcal L}^{\text {sing}}$
being open which underlies the proof of Theorem 3.13. He is also grateful to the referee, whose numerous suggestions and questions led to an improvement of the exposition of some of the results.
