On the virtual automorphism group of a minimal flow

TL;DR

This paper introduces the concepts of virtual automorphism groups and semi-regular flows in minimal dynamical systems, exploring their properties and relationships to actual automorphism groups.

Contribution

It defines new notions in the study of minimal flows and investigates their connections to classical automorphism groups.

Findings

Virtual automorphism groups are distinct from actual automorphism groups.

Semi-regular flows exhibit specific automorphism properties.

The relationship between virtual and actual automorphism groups is characterized.

Abstract

We introduce the notions "virtual automorphism group" of a minimal flow and "semi-regular flow" and investigate the relationship between the virtual and actual group of automorphisms.

Equations

$$\mathfrak{G}(X,x_0)=\{g\in G:g{x}_0=x_0\}.$$

$$\mathfrak{G}(X,g{x}_0)=g\mathfrak{G}(X,x_0)g^{-1}.$$

$${\varphi }_h(p{x}_0)=ph{x}_0.$$

$$\psi (p{x}_0)=p\psi (x_0)=ph{x}_0.$$

$$h{x}_0=\psi (x_0)=\psi (a{x}_0)=a\psi (x_0)=ah{x}_0,$$

$$h{\psi }^{-1}(x_0)={\psi }^{-1}(h{x}_0)={\psi }^{-1}(\psi (x_0))=x_0,$$

$${\varphi }_h(p{x}_0)=ph{x}_0,$$

$$L=(\pi \times \pi )(\mathrm{graph}(R_h))=(\pi \times \pi )\{(p,ph):p\in M\}=\{(p{x}_0,ph{x}_0):p\in M\}.$$

$${\varphi }_h(p{x}_0)=ph{x}_0,(p\in M).$$

$${\varphi }_h(q{x}_0)=qh{x}_0=pah{x}_0=ph(h^{-1}ah)x_0=ph{x}_0={\varphi }_h(p{x}_0).$$

$$ph{x}_0=ph{a}^{\prime }{x}_0=ph(h^{-1}ah)x_0=pah{x}_0=p{p}^{-1}qh{x}_0=vqh{x}_0,$$

$$\pi (ph{x}_0)=ph{y}_0={\tilde{\varphi }}_h(p{y}_0)$$

$$\pi (qh{x}_0)=qh{y}_0={\tilde{\varphi }}_h(q{y}_0),$$

$$X\stackrel{\sigma }{\to }Y\stackrel{\theta }{\to }Z,$$

$$W[a]:=\{c\in X:(a,c)\in W\}\subseteq \{a\}\times {\pi }^{-1}(\pi (b)).$$

$$(\pi \times \pi )(x_0,c)=(\pi \times \pi )(p(x_0,x))=(p{z}_0,p\pi (x))=(z_0,pz).$$

$$W[x_0]=\{c\in X:(x_0,c)\in W\}\subseteq \{x_0\}\times {\pi }^{-1}(z).$$

$$\text{if (a,b)∈W then (a,κ(b))∈W}.$$

$$\begin{array}{r}u(b,b^{\prime },\bar{b},{\bar{b}}^{\prime })=(b,b^{\prime },b,b^{\prime })\\ \tilde{u}(b,b^{\prime },\bar{b},{\bar{b}}^{\prime })=(b,b^{\prime },b^{\prime },b)\end{array}$$

$$\mu =\breve{Q}Q,{\mu }^{\prime }={\breve{Q}}^{\prime }{Q}^{\prime },$$

$$\nu =\breve{Q^{\prime }}Q,{\nu }^{\prime }={\breve{Q}}^{\prime }{Q}^{\prime }.$$

$$(Y,y_0)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$z_0(h{x}_0)=h{x}_0,(h\in \Gamma ).$$

$$(b{z}_0)(x_0)=b{z}_0(x_0)=b{x}_0=z_0(x_0)=x_0,$$

$${\varphi }_h:Z\to Z,{\varphi }_h(p{z}_0)=ph{z}_0,(p\in M),$$

$$(ph{z}_0)(h^{\prime }{x}_0)=ph(z_0(h^{\prime }{x}_0))=ph{h}^{\prime }{x}_0=p{z}_0(h{h}^{\prime }{x}_0)$$

$$(qh{z}_0)(h^{\prime }{x}_0)=qh(z_0(h^{\prime }{x}_0))=qh{h}^{\prime }{x}_0=q{z}_0(h{h}^{\prime }{x}_0).$$

$$(\eta \circ {\varphi }_h)(y_0)=\eta (h{y}_0)=h{x}_0,$$

$$\lambda :(Y,y_0,T)\to \underset{h\in \Gamma }{\bigvee }(X,h{x}_0,T)\cong Z_{\Gamma }$$

$$\Pi (A)=\{p\circ A:p\in M\}.$$

$$ph\circ A=p\circ hA=(p\circ A)h.$$

$$ph\circ A=p\circ h\circ A\supseteq p\circ hA$$

Full text

Introduction

The notion of a virtual subgroup was introduced in ergodic theory by W.A. Mackey (see e.g. [M]). Here is a brief description of the basic idea behind this highly technical notion, as described by R. Zimmer [Zi].

If $X$ is an ergodic $G$-space, one of two mutually exclusive statements holds:

(i) There is an orbit whose complement is a nullset. In this case, $X$ is called essentially transitive.

(ii) Every orbit is a nullset. $X$ is then called properly ergodic.

In the first case, the action of $G$ on $X$ is essentially equivalent to the action defined by translation on $G/H$, where $H$ is a closed subgroup of $G$; furthermore, this action is determined up to equivalence by the conjugacy class of $H$ in $G$. In the second case, no such simple description of the action is available, but it is often useful to think of the action as being defined by a “virtual subgroup” of $G$. Many concepts defined for a subgroup $H$, can be expressed in terms of the action of $G$ on $G/H$; frequently, this leads to a natural extension of the concept to the case of an arbitrary virtual subgroup, i.e., to the case of an ergodic $G$-action that is not necessarily essentially transitive. Perhaps the most fundamental notions that can be extended in this way are those of a homomorphism, and the concomitant ideas of kernel and range. These and other related matters are discussed in [M].

In some sense the concept of the “Ellis group associated to a pointed minimal flow” (see Section 1) is an analogue of Mackey’s virtual group in topological dynamics. It became a keystone object in the abstract theory of topological dynamics, which was developed by R. Ellis and collaborators in the 60’s and 70’s of the last century (see e.g. [E69], [EGS] and [V]).

If one carries this idea a bit further, and one thinks of $A$, the Ellis group of a minimal flow $(X,T)$, as a virtual subgroup, then the group $N_{G}(A)/A$, where $N_{G}(A)$ is the normalizer of $A$ in the ambient group $G$ ( a subgroup of the enveloping semigroup of $(X,T)$), can be thought of as the “virtual automorphism group” of the flow $(X,T)$.

In the present work we make this notion precise (Section 2) and investigate the question of realization of the virtual automorphism group as an actual group of flow automorphisms.

We thank, Ethan Akin and Andrew Zucker for their helpful comments.

1. Some notations and basic facts concerning minimal flows

In this work $T$ denotes an arbitrary (discrete) group. A $T$-flow $(X,T)$ on a compact Hausdorff space $X$ is given by a homomorphism $\rho:T\to{\mathrm{Homeo}}\,(X)$ of $T$ into the group of self homeomorphisms of $X$. We usually suppress the homomorphism $\rho$ from our notation of a flow (even when $\rho$ is not an injection) and we write $tx$ for the image of the point $x\in X$ under the homeomorphism $\rho(t)$ ($t\in T$).

In the next few paragraphs we will survey some of the basic definitions and facts from the theory of abstract topological dynamics which will be repeatedly used in this work. This theory started with the classical monograph by Gottschalk and Hedlund [GH] and was then greatly developed by R. Ellis. For more details we refer to the following monographs: [E69], [Gl76], [A88] and [dV93].

The flow $(X,T)$ is minimal if every point in $X$ has a dense orbit. A pair of points $x,x^{\prime}\in X$ is proximal if there is a net $t_{i}\in T$ and a point $z\in X$ such that $\lim t_{i}x=\lim t_{i}x^{\prime}=z$. We write $P[x]$ for the proximal cell of $x$ (i.e. the set of points proximal to $x$). A point $x\in X$ is a distal point if it is proximal only to itself : $P[x]=\{x\}$. A minimal flow is point distal if there is at least one distal point in $X$, and it is distal if every point is distal. Ellis has shown that in a metric minimal flow the existence of one distal point implies that the set $X_{0}\subseteq X$ of distal points is a dense $G_{\delta}$ set. A continuous map $\pi:(X,T)\to(Y,T)$ between two minimal flows is a homomrphism (or an extension) if it intertwines the $T$-actions ($t\pi(x)=\pi(tx),\ \forall x\in X,t\in T$). We say that the homomorphism is proximal if for every $y\in Y$ every pair of points in $\pi^{-1}(y)$ is proximal, and that it is distal if for every $y\in Y$ we have $P[x]\cap\pi^{-1}(y)=\{x\},\ \forall x\in\pi^{-1}(y)$.

The enveloping semigroup of the flow $(X,T)$, denoted by $E(X,T)$, is the closure of the set $\{\rho(t):t\in T\}$ in the compact space $X^{X}$. This is indeed a compact subsemigroup of the semigroup (under composition of maps) $X^{X}$, and thus for any fixed $p\in E(X,T)$, right multiplication by $p$, $R_{p}:q\mapsto qp,\ (q\in E(X,T))$ is continuous on $E(X,T)$. Left multiplication $L_{p}:q\mapsto pq,\ q\in E(X,T)$ is however often highly non-continuous (usually not even measurable) unless $p$ is a continuous map. As the elements of $T$ are continuous maps the homomorphism $t\mapsto L_{t}\ (t\in T)$ makes $E(X,T)$ a $T$-flow.

It is well known that the semigroup $\beta T$, the Stone-Čech compactification of the discrete $T$, is the universal point transitive $T$-flow and therefore also a universal enveloping semigroup. We will use this universality and often consider elements of $\beta T$ as maps in the enveloping semigroup of each and every $T$-flow under consideration. The semigroup $\beta T$ admits many (for infinite $T$) minimal left ideals (which coincide with the minimal subflows). All these ideals are isomorphic to each other both as compact right topological semigroups and as minimal flows. As usual, we will fix a minimal ideal $M$ of $\beta T$. The universality of $\beta T$ implies that $(M,T)$ is a universal minimal flow. Ellis has shown that as a flow $(M,T)$ is coalescent; i.e every endomorphism of $(M,T)$ is an automorphism, and thus up to an automorphism $(M,T)$ is the unique universal minimal flow. Each minimal ideal contains (usually many) idempotents and for convenience we usually fix one such idempotent $u=u^{2}\in M$. We denote the collection of idempotents in $M$ by the letter $J$.

It turns out that the set $G=uM\subset M$ is actually a group and moreover, via the representation $g\mapsto R^{-1}_{g},\ g\in G$, this group is isomorphic to the group ${\mathrm{A}ut}\,(M,T)$ of automorphisms of the flow $(M,T)$. $M$ is the disjoint union of the collection of groups $\{vM:v\in J\}$ and each member $p$ of $M$ has a unique representation $p=vg$ where $v=v^{2}$ is an idempotent in $M$ and $g$ is in $G$. We sometimes write $p^{-1}$ for $vg^{-1}$; this is indeed the inverse element of $p$ in the group $vG$.

If $(X,T)$ is minimal then for every $x\in X$ there is an idempotent $v\in M$ such that $vx=x$. In other words $X=\bigcup\{vX:v\in J\}$. However, whereas $M=\bigcup\{vM:v\in J\}$ is a disjoint union, usually the sets $vX$ are not necessarily disjoint. For example, a point $x$ in a minimal flow $(X,T)$ is distal iff $vx=x$ for every $v\in J$. In particular a minimal flow $(X,T)$ is distal iff $X=uX=vX$ for all the idempotents $v\in J$. Thus, in a minimal distal flow $E(X,T)$ is a group.

A minimal flow with a distinguished point $x_{0}\in X$ is called a pointed flow and we usually assume that $ux_{0}=x_{0}$; i.e. $x_{0}\in uX$. We write

[TABLE]

This subgroup of $G$ is called the Ellis group of the pointed flow $(X,x_{0},T)$. It is easy to check that for $g\in G$ we have

[TABLE]

In the sequel we will often use the following fact. A homomorphism $\pi:(X,x_{0},T)\to(Y,y_{0},T)$ of pointed minimal flows is a proximal homomorphism iff ${\mathfrak{G}}(X,x_{0})={\mathfrak{G}}(Y,y_{0})$.

Note that when the group $T$ is abelian we have $tp=pt$ for every $t\in T$ and $p\in E(X,T)$. In particular it follows that the subgroup $\{tu:t\in T\}\subset G$ is dense in $M$, and for every minimal flow $(X,T)$ the subset $uX=Gx_{0}\cong G/A$ is dense in $X$. More generally, following [Gl75], a minimal flow $(X,T)$ is called incontractible when for every $k\in{\mathbb{N}}$ the set of almost periodic points is dense in the product flow $(X^{k},T)$ (equivalently when $u\circ uX=X$, see Section 6 below). The group $T$ is called strongly amenable [Gl75], when $(M,T)$ is incontractible (equivalently, when $T$ admits no non-trivial minimal proximal flows). The interpretation of the group $A={\mathfrak{G}}(X,x_{0})$ as a “virtual group” is more meaningful for strongly amenable groups $T$, as then, for every minimal flow $(X,T)$ the homogeneous space $uX=Gx_{0}\cong G/A$ “approximates” the flow $X$. (See [FTVF] for a characterisation of strongly amenable groups.) However, if we consider the relative theory (where one studies the collection of all the minimal extensions of a fixed minimal flow) the corresponding analogy will make sense for every acting group $T$.

2. The group of virtual automorphisms of a minimal flow

Given a flow $(X,T)$ we let ${\mathrm{A}ut}\,(X,T)$ denote its group of automorphisms and ${\mathrm{E}nd}\,(X,T)$ its semigroup of endomorphisms. Given a subgroup $A<G$ we write $N_{G}(A)=\{h\in G:h^{-1}Ah=A\}$ and $N^{\prime}_{G}(A)=\{h\in G:h^{-1}Ah\subseteq A\}$

Proposition 2.1**.**

Let $(X,x_{0},T)$ be a pointed minimal flow, $A=\mathfrak{G}(X,x_{0})$, and let $\psi$ be an endomorphism of the flow $(X,T)$. Then there is an element $h\in N^{\prime}_{G}(A)$ such that $\psi=\phi_{h}$, where for every $p\in M$,

[TABLE]

If $\psi$ is an automorphism then $h\in N_{G}(A)$ and $\psi^{-1}=\phi_{h^{-1}}$.

Proof.

Because $\psi$ is an endomorphism it commutes with $u$, hence $u\psi(x_{0})=\psi(ux_{0})=\psi(x_{0})\in uX=Gx_{0}$, and there exists an element $h\in G$ such that $\psi(x_{0})=hx_{0}$. Now for every $p\in\beta T$ (or $p\in E(X,T)$) we have

[TABLE]

If $a\in A$ then

[TABLE]

hence $h^{-1}ah\in A$, so that $h\in N^{\prime}_{G}(A)$.

If $\psi$ is an automorphism then

[TABLE]

hence $\psi^{-1}(x_{0})=h^{-1}x_{0}$, so that, as above, $\psi^{-1}=\phi_{h^{-1}}$, and also $hAh^{-1}\subseteq A$, hence $h^{-1}Ah=A$. ∎

Proposition 2.2**.**

Let $(X,x_{0},T)$ be a pointed minimal flow, $A=\mathfrak{G}(X,x_{0})$ its Ellis group.

1. (1)

Let $h\in G$ be such that the map $\phi_{h}$, where for every $p\in M$

[TABLE]

is well defined. Then $h\in N^{\prime}_{G}(A)$ and $\phi_{h}$ is a continuous endomorphism; i.e. $\phi_{h}\in End(X,T)$. 2. (2)

If $h\in G$ is such that both maps $\phi_{h}$ and $\phi_{h^{-1}}$ are well defined, then $h\in N_{G}(A)$ and $\phi_{h}\in{\mathrm{A}ut}\,(X,T)$.

Proof.

(1) We have for $a\in A$, $hx_{0}=\phi_{h}(x_{0})=\phi_{h}(ax_{0})=ahx_{0}$, whence $h^{-1}ah\in A$; i.e. $h\in N_{G}^{\prime}(A)$. Let $\pi:M\to X$ be the evaluation map $p\mapsto px_{0},\ (p\in M)$. Let $R_{h}:M\to M$ denote right multiplication by $h$ and let

[TABLE]

By our assumption $L$ is a graph of a map $\phi_{h}:X\to X,\ px_{0}\mapsto phx_{0},\ (p\in M)$. Since the graph of $R_{h}$ is $T$-invariant ($R_{h}$ commutes with the elements of $T$), we deduce that also $\phi_{h}$ commutes with the $T$-action. Since $L$ is a closed subset of $M\times M$, it follows that the map $\phi_{h}$ is continuous. Thus $\phi_{h}\in{\mathrm{E}nd}\,(X,T)$.

(2) If $\phi_{h^{-1}}$ is also well defined then, as above, $hAh^{-1}\subseteq A$, hence $h\in N_{G}(A)$, and we have $\phi_{h^{-1}}=(\phi_{h})^{-1}$, so that $\phi_{h}\in{\mathrm{A}ut}\,(X,T)$. ∎

Example 2.3*.*

A minimal flow $(X,T)$ is said to be coalescent if every endomorphism of $(X,T)$ is an automorphism. This notion was introduced in [A63] where an example of a non-coalescent minimal ${\mathbb{Z}}$-flow was described. This example, constructed as a factor of Ellis’ “two circles” flow, is not metrizable. However a slight modification of the argument there will produce a non-coalescent minimal subshift. Later, in [D] Downarowicz constructed a Toeplitz flow $(X,T)$ which is not coalescent; i.e. it admits an endomorphism which is not an automorphism. These flows are both almost one-to-one extension of their maximal equicontinuous factor $\pi:X\to Y$ (which is an irrational rotation in the modified Auslander example and an adding machine in the Toeplitz flow). This implies the following facts: (i) For any choice of a base point $x_{0}\in uX$, $\mathfrak{G}(X,x_{0})=A$ is a normal subgroup of $G$ (i.e. $N_{G}(A)=G$). (ii) The extension $\pi$ is proximal, whence $\mathfrak{G}(X,x_{0})=\mathfrak{G}(Y,y_{0})=A$ (with $y_{0}=\pi(x_{0})$). Now let $\phi$ be an endomorphism of $(X,T)$ which is not one-to-one. As in Proposition 2.1 we have $\phi=\phi_{h}$ for some $h\in G$ and, although here $h^{-1}Ah=A$, yet $\phi_{h}$ is not an automorphism. This phenomenon however can not occur for distal flows, as we can see in the following proposition.

Proposition 2.4**.**

Let $(X,T)$ be a minimal distal flow. If $h\in N_{G}(A)$ is such that the map $\phi_{h}(px_{0})=phx_{0}$ is well defined then it is an automorphism.

Proof.

For distal $(X,x_{0},T)$ we have $Gx_{0}=X$. Suppose $gx_{0}\not=g^{\prime}x_{0}$ for some $g,g^{\prime}\in G$ but $\phi_{h}(gx_{0})=\phi_{h}(g^{\prime}x_{0})$. Then $gx_{0}\not=g^{\prime}x_{0}$ implies that $b=g^{-1}g^{\prime}\not\in A$, and $ghx_{0}=g^{\prime}hx_{0}$, hence $x_{0}=h^{-1}g^{-1}g^{\prime}hx_{0}$, implies that $h^{-1}bh\in A$. Thus $b\in hAh^{-1}\setminus A$, contradicting our assumption that $h\in N_{G}(A)$. ∎

We call $N_{G}(A)/A$ the group of virtual automorphisms of the minimal flow $(X,T)$.

3. Semi-regular flows

The notion of regular minimal flows was introduced in [A66]. A minimal flow $(X,T)$ is regular if for any pair of points $x,y\in X$ there is an automorphism $\psi\in{\mathrm{A}ut}\,(X,T)$ such that the pair $(x,\psi(y))$ is proximal; iff for every almost periodic point $(x,y)\in X^{2}$, there is an automorphism $\psi$ of $X$ such that $y=\psi(x)$. Equivalently we can say that $(X,T)$ is regular iff (for $x_{0}\in uX=Gx_{0}$) for every $g\in G$, there is an automorphism $\psi\in{\mathrm{A}ut}\,(X,T)$ such that $\psi(x_{0})=gx_{0}$ (recall that $\psi\circ u=u\circ\psi$ and that for every $x\in X$ the point $x$ is proximal to $ux$).

Definition 3.1**.**

We say that a pointed minimal flow $(X,x_{0},T)$ (with $\mathfrak{G}(X,x_{0})=A$), is semi-regular (SR for short) if for every $h\in N_{G}(A)$ there is an automorphism $\psi\in{\mathrm{A}ut}\,(X,T)$ such that $\psi(x_{0})=hx_{0}$. More generally, given a subgroup $\Gamma<N_{G}(A)$, we say that $(X,x_{0},T)$ is $\Gamma$-semi-regular if for every $h\in\Gamma$ there is an automorphism $\psi\in{\mathrm{A}ut}\,(X,T)$ such that $\psi(x_{0})=hx_{0}$. (See also [H].) Clearly then a minimal flow $(X,T)$ is regular iff it is SR with $A\triangleleft G$.

Thus a minimal flow is SR iff every virtual automorphism of $(X,T)$ is realized, so that $N_{G}(A)/A\cong{\mathrm{A}ut}\,(X,T)$.

Remark 3.2*.*

The dependence on the various choices we made in order to formulate the definition of the SR property; namely the choice of $M$ in $\beta T$, the choice of $u$ in $J$ and finally the choice of $x_{0}$ in $uX$, are either immaterial or have the effect of replacing a subgroup of $G$ by some conjugate.

Examples 3.3*.*

1. (1)

Every regular flow is SR (clear). 2. (2)

Every minimal distal flow is SR (see Section 4). 3. (3)

Every minimal proximal flow is SR (a minimal proximal flow is regular). 4. (4)

If $(X,T)$ is SR then so is its maximal highly proximal extension (see Section 6). 5. (5)

The Morse minimal set is SR (see Section 5). 6. (6)

The Sturmian minimal set is not SR (see Section 4). 7. (7)

Toeplitz flows are not SR (see Section 4). 8. (8)

More generally a metrizable almost automorphic flow which is not equicontinuous is never SR (see Section 4). 9. (9)

A regular point distal minimal flow is distal (see Section 4). 10. (10)

Every minimal flow admits a proximal extension which is SR (see Section 6).

Remark 3.4*.*

We note that an element $g\in G$ belongs to $N_{G}(A)$ iff ${\mathfrak{G}}(X,gx_{0})={\mathfrak{G}}(X,x_{0})=A$. For a commutative $T$ we have ${\mathfrak{G}}(X,tx_{0})=A$ for every $t\in T$ (hence every $tu\in G$). Thus for commutative $T$ the set $\{gx_{0}:g\in G\ \&\ {\mathfrak{G}}(X,gx_{0})=A\}\supset Tx_{0}$, is always dense in $X$.

One can easily relativize the notion of SR.

Definition 3.5**.**

Let $\pi:(X,x_{0},T)\to(Y,y_{0},T)$ be a homomorphism of pointed minimal flows. Let ${\mathfrak{G}}(X,x_{0})=A<{\mathfrak{G}}(Y,y_{0})=F$ (where $y_{0}=\pi(x_{0})$). We say that $\pi$ is an SR extension if for every $h\in N_{F}(A)=F\cap N_{G}(A)$ the map $\phi_{h}:X\to X$ defined by $\phi_{h}(px_{0})=phx_{0}$ is a well defined automorphism of $(X,T)$.

4. Distal minimal flows are semi-regular

Theorem 4.1**.**

Every minimal distal flow is SR.

Proof.

Let $(X,x_{0},T)$ be a minimal distal flow with $A=\mathfrak{G}(X,x_{0})$. We have $X=Gx_{0}$ and for $h\in N_{G}(A)$ we let $\phi_{h}:X\to X$ be defined by

[TABLE]

To see that this is well defined suppose $px_{0}=qx_{0}$ for some $p,q\in M$. We have to show that $phx_{0}=qhx_{0}$. Now in a distal flow the enveloping semigroup $E(X,T)$ is a group and the the image of $M$ under the canonical map from $\beta T$ onto $E(X,T)$ is surjective. Thus we can consider $p,q$ as elements of the group $G=E(X,T)$. As $px_{0}=qx_{0}$, it follows that $p^{-1}qx_{0}=x_{0}$, hence $a=p^{-1}q\in A$. Thus $q=pa$ and we get

[TABLE]

The continuity of $\phi_{h}$ follows from the continuity of right multiplication $R_{h}$ on $M$ and it follows that $\phi_{h}$ is an endomorphism of $(X,T)$. Finally, as the same argument applies for $\phi_{h^{-1}}$, we conclude that $\phi_{h}\in{\mathrm{A}ut}\,(X,T)$. ∎

Example 4.2*.*

In [PW] the authors construct an example of a minimal metric cascade $(X,T)$ which is not coalescent. If $\phi$ is an endomorphism of $(X,T)$ which is not an automorphism, then $\phi=\phi_{h}$ for some $h\in G$ for which $h^{-1}Ah\subsetneq A$, where $A={\mathfrak{G}}(X,x_{0})$ for some choice of $x_{0}\in X$. In fact, as we have seen above (Proposition 2.4), in a minimal distal flow $h^{-1}Ah=A$, would imply that $\phi=\phi_{h}$ is an automorphism.

Proposition 4.3**.**

A minimal point distal regular flow is distal.

Proof.

Let $x_{0}$ be a distal point. Given $x\in X$ there is, by regularity, an automorphism $\psi\in{\mathrm{A}ut}\,(X)$ such that the points $\psi(x)$ and $x_{0}$ are proximal. But $x_{0}$ being a distal point, we have $\psi(x)=x_{0}$, and it follows that $x$ is also a distal point. Thus $X$ is a distal flow.

∎

Corollary 4.4**.**

For abelian $T$ a minimal metric almost automorphic flow which is SR is actually equicontinuous. Thus Toeplitz flows and Sturmian like flows are not SR.

Proof.

By definition an almost automorphic flow is a metric minimal flow $(X,T)$ such that the homomorphism $\pi:(X,T)\to(Y,T)$ from $(X,T)$ onto its maximal equicontinuous factor $(Y,T)$, is an almost one-to-one extension. Such a flow is point distal and it satisfies ${\mathfrak{G}}(X,x_{0})={\mathfrak{G}}(Y,y_{0})=A\triangleleft G$ (here we use the assumption that $T$ is abelian). Thus $N_{G}(A)=G$ and semi-regularity for $(X,T)$ is the same as regularity. Now the previous proposition applies. ∎

It is natural to ask whether a minimal, point distal, flow which is not distal can be SR. We will see in Section 5 below that the Morse minimal cascade (i.e a ${\mathbb{Z}}$-flow), which is point distal, metric and not distal, is in fact semi-regular.

We also have an analogous statement concerning distal extensions.

Proposition 4.5**.**

A distal extension $\pi:(X,x_{0},T)\to(Y,y_{0},T)$ of pointed minimal flows is an SR extension.

Proof.

Let ${\mathfrak{G}}(X,x_{0})=A<{\mathfrak{G}}(Y,x_{0})=F$, and let $h\in N_{F}(A)$. We have to show that the map $\phi_{h}$ is well defined. Suppose then that $px_{0}=qx_{0}$ for $p,q\in M$. Then $a=up^{-1}q\in A$, hence $h^{-1}ah=a^{\prime}\in A$ and we have

[TABLE]

where $v=v^{2}$ is the unique idempotent in $J$ such that $vp=p$. Thus the points $phx_{0}$ and $qhx_{0}$ are proximal. On the other hand we have $\pi(phx_{0})=ph\pi(x_{0})=phy_{0}=py_{0}$ and also $\pi(qhx_{0})=qh\pi(x_{0})=qhy_{0}=qy_{0}$. Since by assumption $px_{0}=qx_{0}$ we also have $py_{0}=qy_{0}$, so that $phx_{0}$ and $qhx_{0}$ are in the same $\pi$ fiber. Since $\pi$ is a distal extension we conclude that the points $phx_{0}$ and $qhx_{0}$ are both proximal and distal, whence equal. Thus $\phi_{h}$ is a well defined element of ${\mathrm{E}nd}\,(X,T)$ and, as the same argument applies to $\phi_{h^{-1}}$, we see that $\phi_{h}\in{\mathrm{A}ut}\,(X,T)$. ∎

A similar argument yields the following proposition.

Proposition 4.6**.**

Let $\pi:(X,x_{0},T)\to(Y,y_{0},T)$ be a distal extension with $Y$ being SR. If, in addition, we assume that $N_{G}(A)\subset N_{G}(F)$, then $X$ is also SR. In particular, this is the case when $Y$ is regular.

Proof.

Let $h\in N_{G}(A)$, we show that $\phi_{h}:X\to X$, defined by $\phi_{h}(px_{0})=phx_{0},\ (p\in M)$ is well defined. Assuming $px_{0}=qx_{0}$ we have

[TABLE]

where $\tilde{\phi}_{h}$ is the element of ${\mathrm{A}ut}\,(Y)$ defined by $h$ (recalling that $Y$ is SR and that $N_{G}(A)\subset N_{G}(F)$). Since $px_{0}=qx_{0}$ it follows that $py_{0}=qy_{0}$, whence also $\tilde{\phi}_{h}(py_{0})=\tilde{\phi}_{h}(qy_{0})$. Thus the points $phx_{0}$ and $qhx_{0}$ lie in the same $\pi$-fiber, and therefore are distal. As in the proof of Proposition 4.5 (4.1) they are also proximal points hence equal.

The last assertion follows since by regularity $F\triangleleft G$ and $N_{G}(A)\subseteq N_{G}(F)=G$. ∎

5. The Morse minimal set is semi-regular

We have already raised the question whether a minimal point distal, non-distal flow can be SR. In this section we will show that the classical Morse minimal set provides such an example. This example will also show that there is a minimal metric point distal SR ${\mathbb{Z}}$-flow which is not equicontinuous. This should be contrasted with the fact that a regular Proximal-Isometric (PI) ${\mathbb{Z}}$-flow which is not equicontinuous is necessarily non-metrizabe (see [Gl92]). For a description and detailed analysis of the Morse minimal set we refer to [GH, Chapter 12].

Theorem 5.1**.**

The Morse minimal set is semi-regular.

Proof.

Let $(X,S)$ denote the Morse minimal flow. Here we deviate from our usual notation and use the letter $S$ to denote the shift homeomorphism that generates the ${\mathbb{Z}}$-flow on $X$, a subshift $X\subset\{0,1\}^{\mathbb{Z}}$. We know that $(X,S)$ has the following structure:

[TABLE]

where (i) $\pi=\theta\circ\sigma$ is the homomorphism of $(X,T)$ onto its maximal equicontinuous factor (a dyadic adding machine), (ii) $\theta$ is an almost one-to-one extension, and $\sigma$ is a ${\mathbb{Z}}_{2}$ group extension. More precisely, there is a point $z_{1}\in Z$ such that $\theta^{-1}(z_{1})=\{y_{1},\bar{y}_{1}\}$ and for every point $z\in Z$ which is not in the orbit of $z_{1}$, we have that $\theta^{-1}(z)$ is a singleton. Finally on $X$ there is an involution $\kappa$ (a self homeomorphism satisfying $\kappa^{2}={\rm{id\,}}$) which commutes with the shift. (It sends the sequence $x\in X\subset\{0,1\}^{\mathbb{Z}}$ into the “fliped” sequence $\kappa(x)=x^{\prime}$, where $x^{\prime}(n)={x(n)^{\prime}}$, and $0^{\prime}=1,\ 1^{\prime}=0$.) The map $\sigma:X\to Y$ is then the quotient map under the action of ${\mathbb{Z}}_{2}=\{{\rm{id\,}},\kappa\}$. We let $\pi^{-1}(z_{1})=\sigma^{-1}\{y_{1},\bar{y}_{1}\}=\{x_{1},x^{\prime}_{1},\bar{x}_{1},\bar{x}_{1}^{\prime}\}$.

Next fix a point $x_{0}\in uX$ which is not on the orbits of the four points $\{x_{1},x^{\prime}_{1},\bar{x}_{1},\bar{x}_{1}^{\prime}\}$. We let $y_{0}=\sigma(x_{0})$ and $z_{0}=\theta(y_{0})$. Also set ${\mathfrak{G}}(X,x_{0})=A$ and ${\mathfrak{G}}(Y,y_{0})={\mathfrak{G}}(Z,z_{0})=F$. We then have $F\triangleleft G$ and $Z\cong G/F$, and $A\triangleleft F$ and ${\mathbb{Z}}_{2}\cong F/A$. It is also shown in [GH] that the group ${\mathrm{A}ut}\,(X,S)$ is the group $\{S^{n}:n\in{\mathbb{Z}}\}\oplus\{{\rm{id\,}},\kappa\}\cong{\mathbb{Z}}\oplus{\mathbb{Z}}_{2}$.

Since $\theta^{-1}(z_{0})=\{y_{0}\}$ it follows that $\pi^{-1}(z_{0})=\sigma^{-1}(y_{0})=\{x_{0},x^{\prime}_{0}\}$ and both $x_{0}$ and $x^{\prime}_{0}$ are distal points.

Let now $x$ be an arbitrary point in $uX$ and set $W=\overline{{\mathcal{O}}(x_{0},x)}$, the orbit closure of the point $(x_{0},x)\in X\times X$. Because $x\in uX$ the point $(x_{0},x)$ is an almost periodic point of the product flow $X\times X$, so that $W$ is a minimal flow. Also, there exists $g\in G$ with $x=gx_{0}$. Let $y=\sigma(x)$ and $z=\pi(x)=\theta(y)$.

We claim that for every $(a,b)\in W$ we have

[TABLE]

To see this note that if $(x_{0},c)\in W$ then $(x_{0},c)=p(x_{0},x)$ for some $p\in M$ and

[TABLE]

Since $Z$ is equicontinuous, $pz_{0}=z_{0}$ implies that $pz=z$, so that $\pi(c)=z$. Thus

[TABLE]

and therefore $W[px_{0}]\subseteq\pi^{-1}(\pi(px))$ for every $p\in M$, as claimed.

We now consider two possible cases:

Case 1: Suppose that $(x_{0},x^{\prime})=(x_{0},\kappa(x))\in W$. (Note that this implies that $({\rm{id\,}}\times\kappa)(W)\cap W\not=\emptyset$, whence $({\rm{id\,}}\times\kappa)(W)=W$.) Since $(x_{0},x^{\prime})\in uW=G(x_{0},x)$, there exists $h\in G$ such that $h(x_{0},x)=(x_{0},x^{\prime})$. In particular $hx_{0}=x_{0}$, whence $h\in A$. On the other hand $x^{\prime}=\kappa(x)=\kappa(gx_{0})=gx^{\prime}_{0}=hx=hgx_{0}$, hence $x^{\prime}_{0}=g^{-1}hgx_{0}$, hence $g^{-1}hg\not\in A$; i.e. $g\not\in N_{G}(A)$.

Case 2: Now assume that $(x_{0},x^{\prime})\not\in W$. As in Case 1, we deduce that

[TABLE]

Let $P$ denote the projection map from $W$ onto $X$ (as its first coordinate). We have $P^{-1}(a)=W[a]\subseteq\{a\}\times\sigma^{-1}(\theta^{-1}(z))$.

Claim: For every $a\in X$ we have $P^{-1}(a)=\{(a,\phi(a))\}$ for a surjection $\phi:X\to X$.

We recall here that the enveloping semigroup $E(X,S)$ of the Morse flow has exactly two minimal ideals, say $I_{1},I_{2}$, each containing exactly two idempotents, say $J_{1}=\{u,v\}\subset I_{1}$ and $J_{2}=\{\tilde{u},\tilde{v}\}\subset I_{2}$ (see [HJ] and [S]). Now for the point $a$ we have at least one of the possibilities $ua=a$ or $va=a$. We will assume that $ua=a$ and the other case is treated in the same way with $v$ replacing our usual $u$.

Fix some $b\in X$ with $(a,b)\in W$ and $ub=b$. We write $\eta=\sigma(b)$ and $\zeta=\pi(b)=\theta(\eta)$.

To prove our claim we again consider two cases:

Case $2_{a}$: The point $\eta$ is a non-split point; i.e. $\theta^{-1}(\zeta)=\{\eta\}$.

In this case we have $P^{-1}(a)\subseteq\{a\}\times\sigma^{-1}(\eta)=\{a\}\times\{b,b^{\prime}\}$, and, since by assumption $(a,b^{\prime})\not\in W$, we indeed have $P^{-1}(a)=\{(a,b)\}$, as claimed (putting $b=\phi(a)$).

Case $2_{b}$: The point $\eta$ is a split point; i.e. $\theta^{-1}(\zeta)=\{\eta,\bar{\eta}\}$, with $\eta\not=\bar{\eta}$.

In this case we have $\pi^{-1}(\zeta)=\{b,b^{\prime},\bar{b},\bar{b}^{\prime}\}$. Suppose $(a,\bar{b})\in W$ (the case $(a,\bar{b}^{\prime})\in W$ is symmetric). By the general theory of enveloping semigroups there is in $I_{2}$ a unique idempotent $\tilde{u}$ equivalent to $u$; i.e. $u\tilde{u}=\tilde{u}$ and $\tilde{u}u=u$ (see e.g. [Gl76, I, Proposition 2.5]). Both $u$ and $\tilde{u}$ act as the identity on $uX$, however, as they are distinct elements of $E(X,S)$, we must have

[TABLE]

(or vice versa). (In fact, if $u\bar{b}=\tilde{u}\bar{b}$ then also $uS^{n}\bar{b}=\tilde{u}S^{n}\bar{b}$ and $uS^{n}\bar{b}^{\prime}=\tilde{u}S^{n}\bar{b}^{\prime}$ for every $n\in{\mathbb{Z}}$, whence $u=\tilde{u}$, which is impossible.) Thus we have $\tilde{u}(a,\bar{b})=(a,b^{\prime})\in W$ (or $u(a,\bar{b})=(a,b^{\prime})\in W$) contradicting our assumption that $(a,b^{\prime})\not\in W$. We have shown, in view of (5.1), that indeed $P^{-1}(a)=\{(a,b)\}$, and we let $\phi(a)=b$.

Given $c\in X$, there is a point $a\in X$ such that $(a,c)\in W$ and as also $(a,\phi(a))\in W$, we have $\phi(a)=c$. This shows that $\phi$ is surjective and our claim is proven.

We now have $W=\{(a,\phi(a)):a\in X\}$ and it follows that $\phi\in{\mathrm{E}nd}\,(X,S)$. Since $g^{-1}(x_{0},x)=g^{-1}(x_{0},gx_{0})=(g^{-1}x_{0},x_{0})\in W$ we conclude, by symmetry, that $\phi\in{\mathrm{A}ut}\,(X)$.

To sum up, we have shown that for every $h\in G$ either $h\not\in N_{G}(A)$ or $\phi_{h}\in{\mathrm{A}ut}\,(X,S)$; in other words, we have shown that $(X,S)$ is semi-regular. Thus the group of virtual automorphisms $N_{G}(A)/A$ is realized as ${\mathrm{A}ut}\,(X,S)=\{S^{n}:n\in{\mathbb{Z}}\}\oplus\{{\rm{id\,}},\kappa\}\cong{\mathbb{Z}}\oplus{\mathbb{Z}}_{2}$. ∎

Remark 5.2*.*

Using [GH]’s notation we can write $\{x_{1},x^{\prime}_{1},\bar{x}_{1},\bar{x}_{1}^{\prime}\}=\{\mu,\mu^{\prime},\nu,\nu^{\prime}\}$, where

[TABLE]

From this description it follows immediately that the pairs $\{\nu,\mu\}=\{\bar{x}_{1},x_{1}\}$ and $\{\nu,\mu^{\prime}\}=\{\bar{x}_{1},x^{\prime}_{1}\}$ are positively asymptotic and negatively asymptotic pairs, respectively, hence proximal. This directly implies (5.2). In fact, this argument can be used to prove that indeed $E(X,T)$ has exactly two minimal ideals, each having exactly two idempotents.

Remark 5.3*.*

We note that the proof of Theorem 5.1 shows also that the Morse flow is coalescent.

6. Every minimal flow admits a proximal extension which is SR

Theorem 6.1**.**

Let $(X,x_{0},T)$ be a minimal flow with ${\mathfrak{G}}(X,x_{0})=A$, and let $\Gamma\leq N_{G}(A)$ be a subgroup. Then there exists a minimal pointed flow $Z_{\Gamma}=(Z,z_{0},T)$ and a homomorphism $\pi:Z\to X$ such that:

1. (1)

${\mathfrak{G}}(Z,z_{0})=A$* (so that $\pi$ is a proximal extension).* 2. (2)

$\Gamma A/A\leq{\mathrm{A}ut}\,(Z)$. 3. (3)

For every minimal flow $(Y,y_{0},T)$ which is a proximal extension $\eta:(Y,y_{0},T)\to(X,x_{0},T)$ such that $\Gamma A/A<{\mathrm{A}ut}\,(Y)$, there is a commutative diagram:

[TABLE]

Proof.

Let $C=\{hx_{0}:h\in\Gamma\}\subset X$. Let $z_{0}\in X^{C}$ be the point

[TABLE]

Let $Z=\overline{{\mathcal{O}}_{T}(z_{0})}\subset X^{C}$ (alternatively $Z=\bigvee_{h\in\Gamma}(X,hx_{0})$). Because $C\subseteq uX$, it follows that the point $z_{0}$ is an almost periodic point of the product flow $X^{C}$, hence $Z$ is a minimal flow. The projection on the $x_{0}\in C$ coordinate is a homomorphism $\pi:(Z,z_{0},T)\to(X,x_{0},T)$.

Clearly $az_{0}=z_{0}$ for every $a\in A$, so that $A\subseteq{\mathfrak{G}}(Z,z_{0})$. Conversely, if $bz_{0}=z_{0}$ for some $b\in G$ then

[TABLE]

hence $b\in A$, ${\mathfrak{G}}(Z,z_{0})\subseteq A$, hence ${\mathfrak{G}}(Z,z_{0})=A$.

For $h\in\Gamma$ the map

[TABLE]

is well defined. In fact if $pz_{0}=qz_{0}$ for $p,q\in M$, then for all $h^{\prime}\in\Gamma$

[TABLE]

As $\Gamma$ is a group, $hh^{\prime}\in\Gamma$ and $hh^{\prime}x_{0}\in C$. Therefore, by our assumption, $phz_{0}(hh^{\prime}x_{0})=qhz_{0}(hh^{\prime}x_{0})$, whence $phz_{0}=qhz_{0}$. We conclude, as in Section 2, that $\phi_{h}\in{\mathrm{A}ut}\,(Z)$. We thus get $\Gamma A/A\leq{\mathrm{A}ut}\,(Z)$.

Next assume that $(Y,y_{0},T)$ is as in (3). By assumption $\Gamma A/A\leq{\mathrm{A}ut}\,(Y)$ and it follows that for each $h\in\Gamma$ the function $\phi_{h}(py_{0})=phy_{0}$ is a well defined element of ${\mathrm{A}ut}\,(Y,T)$. Thus the map $\eta\circ\phi_{h}:Y\to X$ satisfies

[TABLE]

and $\eta\circ\phi_{h}:(Y,y_{0},T)\to(X,hx_{0},T)$ is a homomorphism of pointed flows. It then follows that there is a homomorphism

[TABLE]

which satisfies $\eta=\pi\circ\lambda$. ∎

Remark 6.2*.*

Note that when $(X,T)$ is metrizable and the group $\Gamma A/A$ is countable then the SR flow $Z_{\Gamma}$ is also metrizable. In particular this is the case for $\Gamma=\langle\gamma_{0}\rangle=\{\gamma_{0}^{n}:n\in{\mathbb{Z}}\}$ for some $\gamma_{0}\in N_{G}(A)$.

Corollary 6.3**.**

Every minimal flow admits a proximal extension which is SR

Proof.

Take $\Gamma=N_{G}(A)$. ∎

Definition 6.4**.**

We write $X_{SR}$ for the flow $Z_{N_{G}(A)}$. With this notation it is easy to check that $(X,T)$ is SR iff $X=X_{SR}$.

Example 6.5*.*

Let $(Z,R_{\alpha})$ denote the rotation by $\alpha\in{\mathbb{R}}\setminus{\mathbb{Q}}$ on the circle $Z={\mathbb{R}}/{\mathbb{Z}}$,  $R_{\alpha}(x)=z+\alpha\pmod{1}$. Let $(X,S)$ be the Sturmian flow with $\pi:X\to Z$ being its maximal equicontinuous factor. Then the flow $Z$ is regular and the extension $\pi$ is almost one-to-one (so that $X$ is almost automorphic). The regularizer of $(X,S)$, which is the same as $X_{SR}$, is the Ellis’ two circles flow $\tilde{X}$, a nonmetrizable flow (see e.g. [GMe, Example 14.10] for more details on this). We have ${\mathfrak{G}}(Z,z_{0})={\mathfrak{G}}(X,x_{0})={\mathfrak{G}}(\tilde{X},\tilde{x}_{0})=A$, and we observe that the virtual automorphism group $N_{G}(A)/A$ is realized on $Z$ as the compact group ${\mathrm{A}ut}\,(Z)\cong{\mathbb{T}}={\mathbb{R}}/{\mathbb{Z}}$, on $\tilde{X}$ again as ${\mathrm{A}ut}\,(\tilde{X})\cong{\mathbb{R}}/{\mathbb{Z}}$, but with the discrete topology, and it is mostly nonrealizable on $X$, where ${\mathrm{A}ut}\,(X)=\{S^{n}:n\in{\mathbb{Z}}\}$.

Given any minimal flow $(X,T)$, we will next describe another natural construction that yields an SR flow $X^{SR}$ which is a proximal extension of $X$. In the collection of all the minimal flows which are SR and are proximal extensions of $X$, the flow $X_{SR}$ is the minimum and the flow $X^{SR}$ is the maximum (with respect to being a factor).

Let $(X,x_{0},T)$ be a minimal flow with ${\mathfrak{G}}(X,x_{0})=A$. Let $X^{SR}=\Pi(A)$ denote the minimal flow which is the maximal proximal extension of $X$. As described in [Gl76] this flow can be presented as a quasifactor of $M$, as follows:

[TABLE]

Moreover the map $\pi:M\to\Pi(A)$ is such that $\pi^{-1}(p\circ(u\circ A))=p\circ A,\ \forall p\in M$. In particular the elements of $\Pi(A)$ form a partition of $M$. (For more details on the quasifactor $\Pi(A)$ and the the circle operation see [Gl76, Chapter IX].)

Lemma 6.6**.**

For $h\in N_{G}(A)$ and $p=\lim t_{i}\in M$,

[TABLE]

Proof.

We have

[TABLE]

However, as the elements of $\Pi(A)$ form a partition of $M$, we get $ph\circ A=p\circ hA=(p\circ A)h$. ∎

Proposition 6.7**.**

For every pointed minimal flow $(X,x_{0},T)$ with ${\mathfrak{G}}(X,x_{0})=A$, the universal minimal proximal extension $\Pi(A)$ of $X$ (which depends only on $A$) is semi-regular.

Proof.

Given $h\in N_{G}(A)$ and $p\in M$ we set $\phi_{h}(p\circ A)=ph\circ A$. This is well defined since if $p\circ A=q\circ A$ then, by Lemma 6.6,

[TABLE]

∎

Remark 6.8*.*

Since every $\tau$-closed subgroup $A$ of $G$ (see e.g. [Gl76, Chapter IX]) is the Ellis group of a minimal flow, namely $A={\mathfrak{G}}(\Pi(A),u\circ A)$, we get, in view of Proposition 6.7, that $N_{G}(A)/A\cong{\mathrm{A}ut}\,(\Pi(A))$, hence we conclude that for every $\tau$-closed subgroup $A<G$, the group $N_{G}(A)/A$ is realized as an actual automorphism group of some minimal flow.

We end this section with the following proposition (for information on the maximal highly proximal extension of a minimal flow see [AG]) :

Proposition 6.9**.**

If $(X,T)$ is SR then so is its maximal highly proximal extension.

Proof.

One way to describe the maximal highly proximal extension $X^{*}$ of $X$ is as the qusifactor of $M$ obtained from a homomorphism $\pi:M\to X$, as follows:

[TABLE]

where $x_{0}\in uX$ and $\pi(p)=px_{0},\ (p\in M)$. As with the quasifactor $\Pi(A)$, it can be shown that $\{p\circ\pi^{-1}(x_{0}):p\in M\}$ is a partition of $M$. As $X$ is SR we have $N_{G}(A)/A\cong{\mathrm{A}ut}\,(X)$ and it suffices to show that every $\psi\in{\mathrm{A}ut}\,(X)$ lifts to an automorphism $\psi^{*}$ of $X^{*}$. We define $\psi^{*}(p\circ\pi^{-1}(x_{0}))=p\circ\pi^{-1}(\psi(x_{0})),\ (p\in M)$. If $p\circ\pi^{-1}(x_{0})=q\circ\pi^{-1}(x_{0})$, for $p,q\in M$, then also $px_{0}=qx_{0}$, whence

[TABLE]

hence $p\circ\pi^{-1}(\psi x_{0})=q\circ\pi^{-1}(\psi x_{0})$. (It is not hard to check that $p\circ\pi^{-1}(x)\in X^{*}$ for every $p\in M$ and every $x\in X$.)

An alternative proof is as follows:

If $(X,T)$ is a minimal flow then, as a topological space, the maximal HPI extension of $X$, say $X^{*}$, is the Stone space of the Boolean algebra of regular open sets in $X$ (see e.g. [Zu]). It thus follows that every self-homeomorphism of $X$ lifts to $X^{*}$, and clearly an automorphism of $(X,T)$ lifts to an automorphism of $(X^{*},T)$. ∎

7. A Koopman representation of the virtual automorphism group

When a minimal flow $(X,T)$ is strictly ergodic (i.e. it admits a $T$-invariant probability measure and this measure is unique) then the group ${\mathrm{A}ut}\,(X,T)$ also preserves this measure. In fact, if $\mu$ is the $T$-invariant measure on $X$ and $\psi\in{\mathrm{A}ut}\,(X,T)$, then clearly the pushforward measure $\psi_{*}(\mu)$ is $T$-invariant as well, and thus $\psi_{*}(\mu)=\mu$ by uniqueness.

The Koopman representation of $T$ associated to the measure preserving system $(X,{\mathcal{X}},\mu,T)$ is the representation on the Hilbert space $L_{2}(\mu)$ given by $t\mapsto U_{t}$, where for $t\in T$ the unitary operator $U_{t}$ is defined by $U_{t}(f)=f\circ t^{-1}$.

Now in a special case we are able to show that also the virtual automorphism group $N_{G}(A)/A$ admits such a faithful representation.

Theorem 7.1**.**

Let $(X,T)$ be a minimal, metrizable, point distal, uniquely ergodic flow with $T$-invariant probability measure $\mu$, and suppose that $X$ is measure-regular, where the latter property means that the $T$-invariant $G_{\delta}$ set $X_{0}\subseteq X$ consisting of distal points, has measure one. Then each element of the virtual group automorphisms of the flow defines an automorphism of the measure space $(X,{\mathcal{X}},\mu)$ and this correspondence defines a unitary representation of the virtual group of automorphisms as a group of unitary operators on the separable Hilbert space $L_{2}(\mu)$.

Proof.

As usual we pick a point $x_{0}\in X_{0}$, so necessarily $ux_{0}=x_{0}$, then let $A={\mathfrak{G}}(X,x_{0})$. Our virtual automorphism group is the the group $N_{G}(A)/A$. Clearly $uX=Gx_{0}\supseteq X_{0}$. Next define, for $h\in N_{G}(A)$, $\phi_{h}:uX\to uX$ by

[TABLE]

Then the map $\phi_{h}$ is well defined and it is a homeomorphism of the (usually not even measurable) set $uX$. In fact, if $gx_{0}=g^{\prime}x_{0}$ then $g^{-1}g^{\prime}x_{0}=x_{0}$, hence $a=g^{-1}g^{\prime}\in A$ and

[TABLE]

The continuity of $\phi_{h}$ follows from the continuity of right multiplication on $G$. Moreover, we have $t\phi_{h}=\phi_{h}t,\ \forall t\in T$.

Now $\phi_{h}:X_{0}\to\phi_{h}(X_{0})$ is a homeomorphism and it pushes the measure $\mu$ on $X_{0}$ to a measure $(\phi_{h})_{*}(\mu)$ on $\phi_{h}(X_{0})$. By uniqueness $(\phi_{h})_{*}(\mu)=\mu$.

In particular $\mu(X_{0}\cap\phi_{h}(X_{0}))=1$. Similarly $\mu(X_{0}\cap\phi_{h}^{n}(X_{0}))=1$ for every $n\in{\mathbb{Z}}$ and we conclude that the $T$-invariant dense $G_{\delta}$ set $X_{\infty}=\bigcap_{n\in{\mathbb{Z}}}\phi_{h}^{n}(X_{0})$ has measure $1$. Since it is also $\phi_{h}$-invariant this shows that $\phi_{h}$ is an automorphism of the measure space $(X,{\mathcal{X}},\mu)$.

Now the composition map $U_{h}:f\mapsto f\circ\phi^{-1}_{h}$ defines a unitary operator on $L_{2}(\mu)$ and the map $h\mapsto U_{h}$ from $N_{G}(A)/A\to{\mathcal{U}}(L_{2}(\mu))$ is the desired unitary representation. ∎

8. Questions

We conclude with the following list of questions:

1. (1)

Given a $\tau$-closed subgroup $A$ of $G$ (see e.g. [Gl76, Chapter IX]), when is there a minimal metric pointed flow $(X,x_{0},T)$ with ${\mathfrak{G}}(X,x_{0})=A$ ? 2. (2)

Given a minimal metric pointed flow $(X,x_{0},T)$ when is there a metric SR proximal extension of $X$ ? 3. (3)

(A. Zucker) For a minimal metric flow $(X,T)$ is there always a metrizable extension which is coalescent ? (Of course $(M,T)$ is a non-metrizable one, when $T$ is infinite.) 4. (4)

When is there a unitary representation of $N_{G}(A)/A$ on a separable Hilbert space ? 5. (5)

In [GTWZ] the authors show that for any infinite countable group $T$: (i) There is a minimal metric flow $(X,T)$ such that ${\mathrm{A}ut}\,(X,T)$ embeds every compact metrizable group, and (ii) The group ${\mathrm{A}ut}\,(M(T),T)\cong G$ has cardinality $2^{\mathfrak{c}}$, the largest possible one. Is there an algebraic obstruction on an abstract group $\Gamma$ of cardinality $\leq 2^{\mathfrak{c}}$ to be isomorphic to a (virtual) automorphism group of a minimal $T$-flow ? (Regarding this question see the work [CP] and remark 11.6 in [GTWZ].)