Accepted byTopology Proceedings

# On sncc-inheritance of pointwise almost periodicity in flows

Xiongping Dai  Department of Mathematics; Nanjing University; Nanjing 210093;
Peoples Republic of China<EMAIL_ADDRESS>
###### Abstract.

Let $H$ be a subnormal co-compact closed subgroup of a Hausdorff topological
group $T$ and $X$ a compact Hausdorff space. We prove the inheritance theorem:
A point of $X$ is almost periodic (a.p.) for $T\curvearrowright X$ iff it is
a.p. for $H\curvearrowright X$. Moreover, if $T\curvearrowright X$ is minimal
with $H\lhd T$, then $\mathscr{O}_{H}\colon X\rightarrow 2^{X}$,
${x\mapsto\overline{Hx}}$ is a continuous mapping, and, $T\curvearrowright
X/H$ is an a.p. nontrivial factor of $T\curvearrowright X$ iff
$T\curvearrowright X\times T/H$ is not minimal.

###### Key words and phrases:

Flow, almost periodic point, inheritance

###### 2020 Mathematics Subject Classification:

Primary 37B05; Secondary 54H15

The author was supported in part by NSFC Grant #12271245.

##  0\. Introduction

Let $X$ be a compact Hausdorff space as our phase space and $T$ a Hausdorff
topological group as our phase group, unless otherwise specified. We say that
$(T,X)$ is a flow, denoted $\mathscr{X}$ or $T\curvearrowright X$, if there
exists a continuous map $T\times X\xrightarrow{(t,x)\mapsto tx}X$, called the
phase mapping, such that $ex=x$ and $(st)x=s(tx)$ for all $x\in X$ and $s,t\in
T$, where $e\in T$ is the identity element. If $\mathscr{X}$ has some
dynamical properties and $H$ is a closed subgroup of $T$, it is natural to ask
whether or not the induced subflow $H\curvearrowright X$ necessarily inherits
these dynamical properties.

We shall consider the pointwise almost periodic inheritance of subnormal co-
compact closed subgroup action (see Thm. 3.1 and its applications Thm. 3.4B,
Thm. 6.5, and Thm. 6.8). In literature 3.4B is a classical theorem that gives
us the first example of minimal distal non-equicontinuous metric flows. Its
proofs available in the literature (e.g., [3, Chap. IV] and [7, Exa. 6.19.2])
are cumbersome and lengthy. Ours is however very concise using the inheritance
theorem.

##  1\. Basic notions and lemmas

Let $\mathscr{X}$ be a flow. By $2^{T}$ and $2^{X}$ we denote respectively the
collections of nonempty compact subsets of $T$ and $X$, where $2^{X}$ is
endowed with the Vietoris topology. By $\mathfrak{N}_{e}(T)$ and
$\mathfrak{N}_{x}(X)$ we denote the systems of neighborhoods of $e$ in $T$ and
of $x$ in $X$, respectively. Here $\\{pt\\}$ stands for a singleton.

######  1.1.

Let $H$ be any subgroup of $T$. Then the quotient map $\rho\colon T\rightarrow
T/H$ is surjective, where $T/H=\\{tH\,|\,t\in T\\}$ is the left-coset space.
By

  1. $\mathscr{O}_{H}\colon X\rightarrow 2^{X}$, $x\mapsto\overline{Hx}$

it stands for the lower semi-continuous $H$-orbit-closure mapping.

######  1.2.

A set $A$ in $T$ is syndetic [10, Def. 2.02] if there exists $K\in 2^{T}$ such
that $T=K^{-1}A$, or equivalently, $Kt\cap A\not=\emptyset\ \forall t\in T$.

######  1.3.

A point $x$ in $X$ is called almost periodic (a.p.) for $\mathscr{X}$ if for
every $U\in\mathfrak{N}_{x}(X)$

  1. $N_{T}(x,U)=\\{t\,|\,t\in T,tx\in U\\}$

is syndetic in $T$. If every point of $X$ is a.p., then $\mathscr{X}$ is said
to be pointwise a.p. See [13, 10, 8, 4, 2, 6].

Let $\mathscr{U}$ be the uniformity on $X$. If, for each
$\alpha\in\mathscr{U}$, there exists a syndetic set $A$ in $T$ such that
$Ax\subseteq\alpha[x]$ for all $x\in X$, then $\mathscr{X}$ is called an a.p.
flow. Note that $\mathscr{X}$ is a.p. iff it is equicontinuous; i.e., given
$\varepsilon\in\mathscr{U}$ there exists $\delta\in\mathscr{U}$ such that
$T\delta\subseteq\varepsilon$ (cf. [10, 7, 4, 2, 6]).

######  1.4.

$\mathscr{X}$ is minimal iff $\overline{Tx}=X$ for all $x\in X$. In fact, a
point $x$ is a.p. for $\mathscr{X}$ iff $\overline{Tx}$ is a minimal set of
$\mathscr{X}$ (see, e.g., [13, 9, 10]).

######  1.5.

By $\mathcal{S}^{\textsl{ns}}(T)$ we denote the family of the normal syndetic
closed subgroups of $T$.

######  1.6.

A subgroup $H$ of $T$ is said to be co-compact iff the quotient space $T/H$ is
compact. In that case, $T$ is also called a compact extension of $H$. We
denote by $\mathcal{S}^{\textsl{ncc}}(T)$ the family of the normal co-compact
closed subgroups of $T$.

######  1.7 Lemma.

Let $T$ be an “wF-group” (i.e., $T$ is a group with a topology under which
$(s,t)\mapsto st^{-1}$ is separately continuous).111A group $K$ that is
compact with multiplication separately continuous and inversion continuous is
refereed to as an “F-group” in [14, Rem. 2.5]. An locally compact (LC)
Hausdorff wF-group is a topological group according to Ellis’ Joint
Continuity Theorem. Let $H_{1}<H_{2}$ be closed subgroups of $T$; then:

  1. $(\mathrm{a})$

If $H_{1}$ is syndetic in $T$, then it is syndetic in $H_{2}$ and $H_{2}$ is
also syndetic in $T$ (cf., e.g., [10, 4]).

  2. $(\mathrm{b})$

If $H_{1}$ is co-compact in $T$, then it is co-compact in $H_{2}$ and $H_{2}$
is also co-compact in $T$.

  3. $(\mathrm{c})$

If $T$ is a semi-topological group and $H_{1}$ is syndetic in $H_{2}$ and
$H_{2}$ is syndetic in $T$, then $H_{1}$ is syndetic in $T$ (cf., e.g., [10,
4]).

###### Proof.

(a): Obvious for a closed subset of a compact set is compact.

(b): This follows from the following canonically determined CD of continuous
mappings: and $H_{2}/H_{1}$ is closed in $T/H_{1}$.

(c): Obvious. The proof is completed. ∎

Note that if $H$ is a syndetic closed subgroup of $T$, then there exists $K\in
2^{T}$ which is mapped onto $T/H$ by $\rho\colon T\rightarrow T/H$, and so $H$
is co-compact. When $T$ is LC, then the converse is true (cf., e.g., [10,
2.01] or [4, 2.8.18]). In fact, this easily follows from the openness of
$\rho$ as follows:

######  1.8 Lemma.

Let $H$ be a subgroup of a topological group $T$, where $T$ need not be
Hausdorff. Then:

  1. $(\mathrm{a})$

The quotient map $\rho\colon T\rightarrow T/H$ is an open surjective map (cf.
[11, Thm. 5.17]).

  2. $(\mathrm{b})$

If $H$ is closed, then $T/H$ is Hausdorff. (Thus, if $s,t\in T$ cannot be
separated by open sets, then $sH=tH$ in $T/H$. See [11, Thm. 5.21].)222Recall
that a topological space $Y$ is Hausdorff iff the diagonal
$\Delta_{Y}=\\{(y,y)\,|\,y\in Y\\}$ is closed in $Y\times Y$.

  3. $(\mathrm{c})$

If $\\{t_{i}H\colon i\in I\\}$ converges in $T/H$ to $tH$, then there exists a
subnet $\xi\colon I^{\prime}\to I$ and a net $\\{k_{i^{\prime}}\colon
i^{\prime}\in I^{\prime}\\}$ in $T$ which converges in $T$ to $t$ with
$k_{i^{\prime}}H=t_{\xi(i^{\prime})}H$ for all $i^{\prime}\in I^{\prime}$. In
particular, there exists a net $\\{h_{i^{\prime}}\colon i^{\prime}\in
I^{\prime}\\}$ in $H$ such that
$t_{\xi(i^{\prime})}=k_{i^{\prime}}h_{i^{\prime}}$ for all $i^{\prime}\in
I^{\prime}$.

###### Proof.

(a): If $U$ is open in $T$, then $UH=\bigcup_{s\in H}Us$ is open in $T$ such
that $UH=\rho^{-1}\rho[U]$. Since $\rho$ is the quotient map, $\rho[U]$ is
open in $T/H$.

(b): Since $\rho$ is an open map, $\rho\times\rho\colon T\times T\rightarrow
T/H\times T/H$ is open and so is a quotient map. As

  1. $(\rho\times\rho)^{-1}[\Delta_{T/H}]=\\{(s,t)\in T\times T\,|\,s^{-1}t\in H\\}$

which is $\rho\times\rho$-saturated and which closed because $H$ is closed and
$T$ is a topological group, it follows that $\Delta_{T/H}$ is closed.

(c): Let $I^{\prime}=I\times\mathfrak{N}_{t}(T)$, where $I^{\prime}$ is
directed by $(i_{1},U_{1})\leq(i_{2},U_{2})$ iff $i_{1}\leq i_{2}$ and
$U_{1}\supseteq U_{2}$ in $I^{\prime}$. For $i^{\prime}=(i,U)\in I^{\prime}$,
by (a) we can choose $\xi(i^{\prime})>i$ in $I$ such that
$t_{\xi(i^{\prime})}H\in\rho[U]$. Let $k_{i^{\prime}}\in U$ with $\rho
k_{i^{\prime}}=\rho t_{\xi(i^{\prime})}$. The proof is completed. ∎

Now if $T$ is LC and $H<T$ is co-compact, then for any compact
$U\in\mathfrak{N}_{e}(T)$ we have that $\rho[U]\in\mathfrak{N}_{\rho(e)}(T/H)$
and there is a finite set $F\subset T$ with $T/H=\\{suH\,|\,s\in F,u\in U\\}$.
Let $K=FU$. Then $K\in 2^{T}$, $KH=T$ and $H$ is syndetic in $T$. However, if
$[T:H]$ is not finite and $T$ is not LC, the converse need not be true and so
$\mathcal{S}^{\textsl{ns}}(T)\subsetneq\mathcal{S}^{\textsl{ncc}}(T)$ in
general. For example, let $T$ be connected non-LC and $\chi\in\hat{T}$ (a
continuous character of $T$) with $\chi\not\equiv 1$. Then $\ker\chi$ is a
normal closed subgroup of $T$ such that $T/\ker\chi\cong\mathbb{S}$, where
$\mathbb{S}$ is the unit circle in $\mathbb{C}$. So $T/\ker\chi$ is compact
($\ker\chi$ is co-compact in $T$) but we cannot guarantee $\ker\chi$ syndetic
in $T$. If $\mathscr{X}$ is a minimal flow and $H=\ker\chi$, then it is of
interest to know whether or not $H\curvearrowright X$ is a pointwise a.p. flow
(see Thm. 3.1).

######  1.9.

A subgroup $H$ is called subnormal in $T$ ([7, Def. 6.16]), if there exists a
sequence of subgroups $H_{0}=H<H_{1}<\dotsm<H_{n}=T$ such that $H_{i}$ is
normal in $H_{i+1}$, denoted $H_{i}\lhd H_{i+1}$, for $0\leq i<n$. By
$\mathcal{S}^{\textsl{sncc}}(T)$ we denote the family of the subnormal co-
compact closed subgroups of $T$.

######  1.10.

As usual, $P(\mathscr{X})$ stands for the proximal relation of $\mathscr{X}$;
that is, for every pair $(x,y)\in X\times X$, $(x,y)\in{P}(\mathscr{X})$ iff
$\overline{T(x,y)}\cap\Delta_{X}\not=\emptyset$. If ${P}(\mathscr{X})=X\times
X$, then $\mathscr{X}$ is called a proximal flow. If
$P(\mathscr{X})=\Delta_{X}$, then $\mathscr{X}$ is called a distal flow. In
this case, $\mathscr{X}$ is pointwise a.p. [7, 8, 4, 2, 6].

######  1.11 Example (cf. [7, Ex. 6.19.2]).

Write $[x,y,z]$ for $\left[\begin{smallmatrix}1&x&y\\\ 0&1&z\\\
0&0&1\end{smallmatrix}\right]$ and let

  1. $G=\\{[x,y,z]\,|\,x,y,z\in\mathbb{R}\\}$,

  2. $K=\\{[x,y,z]\,|\,x,z\in\mathbb{Z},y\in\mathbb{R}\\}$,

  3. $H=\\{[x,y,z]\,|\,x,y,z\in\mathbb{Z}\\}$.

Then, under matrix multiplication $\cdot$, $H$ is a closed syndetic non-normal
discrete subgroup of the simply connected nilpotent Lie group $G$ such that
$H\lhd K\lhd G$. So $H\in\mathcal{S}^{\textsl{sns}}(G)$; but
$H\notin\mathcal{S}^{\textsl{ns}}(G)$ for

  1. $H\cdot[0,0,z]=\\{[i,iz+j,z+k]\,|\,i,j,k\in\mathbb{Z}\\}$ and

  2. $[0,0,z]\cdot H=\\{[i,j,z+k]\,|\,i,j,k\in\mathbb{Z}\\}$

such that $H\cdot[0,0,z]\not=[0,0,z]\cdot H$ for all
$z\in\mathbb{R}\setminus\mathbb{Q}$. The coset flow $G\curvearrowright G/H$ is
of course minimal. Moreover, it is known that $G\curvearrowright G/H$ is a
distal non-equicontinuous flow (see Thm. 3.4B for a self-contained proof).

######  1.12 Lemma.

Let $\mathscr{X}$ be any flow and $H$ a co-compact closed subgroup of $T$,
where $X$ need not be compact. If $X_{0}$ is an $H$-invariant closed subset of
$X$, then $TX_{0}$ is a $T$-invariant closed subset of $X$ such that
$t_{i}X_{0}\to tX_{0}$ whenever $t_{i}H\to tH$ in $T/H$.

###### Proof.

(If $H$ is syndetic, then $T=KH$ for some $K\in 2^{T}$, and so that
$TX_{0}=KHX_{0}=KX_{0}$ is closed, for $K$ is compact.)

Step 1. Assume $t_{i}\in T$ is any net with $t_{i}H\to tH$ in $T/H$ for some
$t\in T$. By going to a subnet (Lem. 1.8c) if necessary, we can select a net
$s_{i}$ in $H$ such that $t_{i}s_{i}\to t$ in $T$ so $(t_{i}s_{i})^{-1}\to
t^{-1}$. Let $t_{i}\in T$ and $y_{i}\in X_{0}$ such that $t_{i}y_{i}\to x\in
X$. To show $t_{i}X_{0}\to tX_{0}$, we need only prove $x\in tX_{0}$. As
$(t_{i}s_{i})s_{i}^{-1}y_{i}\to x$ and
$(t_{i}s_{i})^{-1}(t_{i}s_{i})(s_{i}^{-1}y_{i})=s_{i}^{-1}y_{i}\to t^{-1}x$,
it follows that $t^{-1}x\in X_{0}$ and $x\in tX_{0}$.

Step 2. Assume $H$ is co-compact. Let $t_{i}\in T$ and $y_{i}\in X_{0}$ such
that $t_{i}y_{i}\to x\in X$. Since $T/H$ is compact, we can assume $t_{i}H\to
tH$ for some $t\in T$. Then by Step 1, $x\in tX_{0}$ and thus, $TX_{0}$ is
closed. The proof is completed. ∎

######  1.13 Remark.

Condition “co-compact” is crucial for Lemma 1.12. Otherwise, we can easily
point out counterexamples; for instance, if $H=\\{e\\}$ and $x\in X$, then
$Hx=\\{x\\}$ is $H$-invariant closed but $Tx$ is generally not closed whenever
$T$ is not compact.

##  2\. Motivations

Now it is time to formulate our motivations.

######  2.1 Theorem (inheritance by ns-subgroup; [10, Thm. 4.04, 4.19], [7, 4,
2]).

Let $\mathscr{X}$ be a flow and $H\in\mathcal{S}^{\textsl{ns}}(T)$, then:

  1. $(1)$

$\mathscr{X}$ is pointwise a.p. iff $H\curvearrowright X$ is a pointwise a.p.
subflow of $\mathscr{X}$.

  2. $(2)$

If $\mathscr{X}$ is minimal, then $\mathscr{O}_{H}\colon X\rightarrow 2^{X}$
is continuous.

In fact, if $T$ is only a para-topological group—namely, $T$ is Hausdorff and
$(s,t)\mapsto st$ is continuous but $t\mapsto t^{-1}$ need not be continuous,
this theorem is still true (cf. [5, Lem. 1.5e and Thm. 2.7]). For
“$H\in\mathcal{S}^{\textsl{ncc}}(T)$” instead of condition
“$H\in\mathcal{S}^{\textsl{ns}}(T)$” there is a useful partial generalization
of Theorem 2.1 under an additional condition—proximal—as follows:

######  2.2 Theorem ([8, Lem. II.3.2]).

Let $\mathscr{X}$ be a minimal ‘proximal’ flow. If
$H\in\mathcal{S}^{\textsl{ncc}}(T)$, then $H\curvearrowright X$ is minimal
proximal as a subflow of $\mathscr{X}$.

See Theorem 4.2 and Corollary 4.4 below for generalizations of Theorem 2.2.
Under another kind of additional condition—disjointness and co-compactness—in
place of “ns”, there is a result as follows:

######  2.3 Theorem (cf. [8, Prop. IV.5.1]).

Let $H$ be a co-compact closed subgroup of $T$. Then

  1. $T\curvearrowright X\times T/H$, defined by $(t,(x,sH))\mapsto(tx,tsH)$,

is minimal iff $\mathscr{X}$ is minimal and $H\curvearrowright X$ is a minimal
subflow of $\mathscr{X}$.

###### Proof.

Necessity: Clearly, $\mathscr{X}$ is minimal. For proving that
$H\curvearrowright X$ is a minimal subflow of $\mathscr{X}$, let $X_{0}$ be a
minimal subset of $H\curvearrowright X$ and simply write $[e]=H\in T/H$. Then
as $H(X_{0}\times\\{[e]\\})=X_{0}\times\\{[e]\\}$, it follows by Lemma 1.12
that $T(X_{0}\times\\{[e]\\})$ is a closed invariant subset of $X\times T/H$
by $T$. So

$T(X_{0}\times\\{[e]\\})=X\times T/H\supset X\times\\{[e]\\}.$

As $t(x,[e])\in X\times\\{[e]\\}\Rightarrow t\in H$ and by
$X\times\\{[e]\\}=\bigcup_{t\in H}tX_{0}\times\\{[e]\\}$, it follows easily
that $X=HX_{0}=X_{0}$. Hence $H\curvearrowright X$ is a minimal subflow of
$\mathscr{X}$.

Sufficiency: Let $W$ be a minimal subset of $X\times T/H$ under $T$. We can
define a closed subset of $X$, $Y=\\{x\in X\,|\,(x,[e])\in
W\\}\not=\emptyset$. By $HY=Y$ and minimality of $H\curvearrowright X$, we
have that $Y=X$. Thus, $X\times\\{[e]\\}\subseteq W$, and then for any $t\in
T$, $t(X\times\\{[e]\\})=X\times\\{tH\\}$. So $W=X\times T/H$. The proof is
completed. ∎

The normality of $H$ plays only a role in the “only if” part of Theorem
2.1-(1) (see Thm. 3.1-(2) below). If $H$ is not normal, then this part is
generally false by constructing a counterexample as follows:

######  2.4 Example.

Let $T=\textrm{GL}(2,\mathbb{R})$, the general linear group with the usual
topology, which is LC. Let

  1. $H=\left\\{\left[\begin{smallmatrix}x&0\\\ y&z\end{smallmatrix}\right]\,|\,x,y,z\in\mathbb{R}\textrm{ s.t. }xz\not=0\right\\}$.

Then $H$ is a non-normal closed subgroup of $T$. Moreover, $H$ is syndetic and
so co-compact in $T$. Indeed, if

  1. $K=\left\\{\left[\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right]\,|\,a,b,c,d\in[-1,1]\textrm{ s.t. }ad-bc=1\right\\}$,

then $K\in 2^{T}$ and $T=KH$ (see [12, pp.392–393]). Let
$T\curvearrowright\mathbb{R}^{2}$ be the noncompact flow with the phase
mapping

  1. $T\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{2},\quad\left(\left[\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right],\left[\begin{smallmatrix}x\\\ y\end{smallmatrix}\right]\right)\mapsto\left[\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right]\left[\begin{smallmatrix}x\\\ y\end{smallmatrix}\right]=\left[\begin{smallmatrix}ax+by\\\ cx+dy\end{smallmatrix}\right]$.

Let $\mathbb{P}^{1}$ denote the real projective line space; i.e., the space of
all $1$-dimensional linear subspaces of $\mathbb{R}^{2}$. Then the above
natural action of $T$ on $\mathbb{R}^{2}$ induces an action of $T$ on
$\mathbb{P}^{1}$. It is clear that

  1. (1)

$T\curvearrowright\mathbb{P}^{1}$, with phase mapping
$T\times\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$, is a homogeneous flow (i.e.,
$Tx=\mathbb{P}^{1}\ \forall x\in\mathbb{P}^{1}$).

Let $x\in\mathbb{P}^{1}$. Since $\left[\begin{smallmatrix}1&k\\\
0&1\end{smallmatrix}\right]x\rightarrow L$ as $k\to\infty$, where
$L\in\mathbb{P}^{1}$ is the line through the points $(-1,0)$ and $(1,0)$ of
$\mathbb{R}^{2}$. Thus:

  1. (2)

$T\curvearrowright\mathbb{P}^{1}$ is a minimal proximal flow.

We now consider the induced subflow $H\curvearrowright\mathbb{P}^{1}\colon
H\times\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$. We can assert the following:

  1. (3)

$H\curvearrowright\mathbb{P}^{1}$ is proximal but not pointwise a.p. (and
hence not minimal).

###### Proof.

Suppose to the contrary that $H\curvearrowright\mathbb{P}^{1}$ is pointwise
a.p. Since $H$ is a syndetic subgroup of $T$,
$H\curvearrowright\mathbb{P}^{1}$ is proximal by (2) so $\mathbb{P}^{1}$
contains a unique minimal set under $H$. Thus,
$H\curvearrowright\mathbb{P}^{1}$ is minimal. However, since the line through
the points $(0,1)$ and $(0,-1)$ of $\mathbb{R}^{2}$ is a fixed point under
$H$, $\mathbb{P}^{1}=\\{pt\\}$, contrary to $\dim\mathbb{P}^{1}=1$. The proof
is completed. ∎

  1. (4)

$T\curvearrowright\mathbb{P}^{1}\times T/H$ is not a minimal flow.

###### Proof.

By Theorem 2.3 and (3) above. ∎

In fact, $H$ is not subnormal in $T$ according to (2), (3) and Theorem 3.1
below.

Note that if $\mathscr{X}$ is not equicontinuous/a.p., then the minimality
condition of $\mathscr{X}$ is critical for Theorem 2.1-(2), as shown by the
following

######  2.5 Theorem.

There exists a flow $\mathbb{R}\curvearrowright X\colon\mathbb{R}\times
X\xrightarrow{(t,x)\mapsto t\cdot x}X$ such that
$\mathscr{O}_{\mathbb{R}}\colon x\mapsto\overline{\mathbb{R}\cdot x}$ is
continuous (so it is pointwise a.p.; see Thm. 3.5A); however,
$\mathscr{O}_{\mathbb{Z}}\colon x\mapsto\overline{\mathbb{Z}\cdot x}$ is not
continuous from $X$ into $2^{X}$. (Note here that
$\mathbb{Z}\in\mathcal{S}^{\textsl{ns}}(\mathbb{R})$.)

###### Proof.

Let $X=\\{z\in\mathbb{C}\colon|z|\leq 1\\}$. Then we can write
$X={\bigcup}_{0\leq r\leq 1}\mathbb{S}_{r}$, where
$\mathbb{S}_{r}=\\{z\in\mathbb{C}\colon|z|=r\\}$ for all $r\in[0,1]$. The
continuous-time flow $\mathbb{R}\times X\rightarrow X$ is defined by
$(t,x)\mapsto t\cdot x=e^{2\pi i|x|t}x$. Clearly, $\overline{\mathbb{R}\cdot
x}=\mathbb{S}_{|x|}$ for all $x\in X$. If $x_{i}\to x$ in $X$, then
$\overline{\mathbb{R}\cdot x_{i}}\to\overline{\mathbb{R}\cdot x}$. However, if
$r$ with $0<r<1$ is rational and $\\{r_{i}\\}$ is a sequence of irrational
numbers with $r_{i}\to r$, then for $x_{i}\in\mathbb{S}_{r_{i}}\to
x\in\mathbb{S}_{r}$, as $i\to\infty$,

$\overline{\mathbb{Z}\cdot
x_{i}}=\mathbb{S}_{r_{i}}\to\mathbb{S}_{r}\not=\overline{\mathbb{Z}\cdot x}.$

Thus, $\mathscr{O}_{\mathbb{Z}}\colon X\ni x\mapsto\overline{\mathbb{Z}\cdot
x}\in 2^{X}$ is not a continuous mapping. The proof is completed. ∎

##  3\. Inheritance theorems

Example 2.4 also shows that the disjointness condition in Theorem 2.3 is
crucial. The main purpose of this paper is to generalize Theorem 2.1 without
the proximality and disjointness as follows:

######  3.1 Theorem (inheritance by sncc-subgroup).

Let $\mathscr{X}$ be any flow and $H$ a closed subgroup of $T$, let
$H\curvearrowright X$ be a subflow of $\mathscr{X}$. Then:

  1. $(1)$

If $H\in\mathcal{S}^{\textsl{sncc}}(T)$ and $x\in X$ is a.p. for
$T\curvearrowright X$, then $x$ is a.p. for $H\curvearrowright X$.

  2. $(2)$

If $H$ is co-compact in $T$ and $x\in X$ is a.p. for $H\curvearrowright X$,
then $x$ is a.p. for $T\curvearrowright X$.

  3. $(3)$

If $\mathscr{X}$ is minimal and $H\in\mathcal{S}^{\textsl{ncc}}(T)$, then
$\\{\overline{Hx}\,|\,x\in X\\}$ is a $T$-invariant partition of $X$ such that
$\mathscr{O}_{H}\colon X\rightarrow 2^{X}$ is continuous.

Consequently, for every $H\in\mathcal{S}^{\textsl{sncc}}(T)$, a point $x$ in
$X$ is a.p. for $T\curvearrowright X$ iff it is a.p. for $H\curvearrowright
X$.

###### Proof.

(1): Suppose $x\in X$ is a.p. for $T$. First, assume
$H\in\mathcal{S}^{\textsl{ncc}}(T)$. By Zorn’s lemma, it follows that we can
select a point $y\in\overline{Hx}$ that is a.p. for $H$. Since $\overline{Tx}$
is a minimal set for $T$ and $y\in\overline{Tx}$, there is a net
$\\{t_{n}\,|\,n\in D\\}$ in $T$ such that $t_{n}y\to x$. Since $T/H$ is a
compact group, we can find an element $k\in T$ such that $t_{n}H\to kH$ in
$T/H$. By Lemma 1.8c, we can assume that there is a net $k_{n}\in T$ such that
$k_{n}\to k$ in $T$ and $t_{n}H=k_{n}H$ for all $n\in D$. By $t_{n}H=k_{n}H$,
it follows that $k_{n}^{-1}t_{n}\in H$ for all $n\in D$. Thus $t_{n}y\to x$
and $k_{n}^{-1}\to k^{-1}$ implies that $k_{n}^{-1}t_{n}y\to k^{-1}x$ and
$k^{-1}x\in\overline{Hy}$ is a.p. for $H$. Finally, by normality of $H$, $x$
is a.p. for $H$. Indeed, for each point $z\in\overline{Hx}$, we have that
$k^{-1}z\in\overline{Hk^{-1}x}$ and $k^{-1}\overline{Hz}=\overline{Hk^{-1}x}$,
and, $\overline{Hz}=\overline{Hx}$. Thus $\overline{Hx}$ is minimal for $H$.
Since $\overline{Hx}$ is compact Hausdorff, $x$ is a.p. for $H$.

Next, suppose $H\in\mathcal{S}^{\textsl{sncc}}(T)$. Then there exists a finite
sequence of closed subgroups of $T$: $H_{0}=H\lhd H_{1}\lhd\dotsm\lhd
H_{n}=T$, as in Def. 1.9. Clearly, we have that
$H_{i}\in\mathcal{S}^{\textsl{ncc}}(H_{i+1})$. So by induction, it follows
easily that $x$ is an a.p. point for $H$.

(2): Suppose $x\in X$ is an a.p. point for $H$. Without loss of generality we
may assume $X=\overline{Tx}$ and by Zorn’s lemma we can select an a.p. point
$y\in X$ under $T$. Then every point of $\overline{Hy}$ is a.p. under $T$ for
$\overline{Hy}\subseteq\overline{Ty}$. Take a net $t_{i}\in T$ with $t_{i}x\to
y$. Since $T/H$ is a compact coset space, by Lemma 1.8c we can take elements
$s_{i},k\in T$ such that (a subnet of) $s_{i}\to k$ (so $s_{i}^{-1}\to
k^{-1}$) in $T$ and $s_{i}H=t_{i}H$. Then $s_{i}^{-1}t_{i}x\to z:=k^{-1}y$ and
$s_{i}^{-1}t_{i}\in H$. Thus, $z\in\overline{Hx}$ and $z$ is a.p. for $T$.
Then $\overline{Hx}=\overline{Hz}\,(\subseteq\overline{Tz})$ consists of
$T$-a.p. points. Whence $x$ is an a.p. point for $T$.

(3): By (1) and normality of $H$, $\\{\overline{Hx}\,|\,x\in X\\}$ is a
$T$-invariant partition of $X$ into $H$-minimal sets. Now let $x_{n}\to x$ in
$X$. We need only prove that $\overline{Hx_{n}}\to\overline{Hx}$ in $2^{X}$.
Suppose this is not true; then we may assume (a subnet of)
$\overline{Hx_{n}}\to L\in 2^{X}$ such that $\overline{Hx}\subsetneq L$. Since
$\mathscr{O}_{H}\colon X\rightarrow 2^{X}$ is lower semi-continuous and
$X=\overline{\bigcup\\{t\overline{Hx}\,|\,t\in T\\}}$, hence for
$\varepsilon_{n}\in\mathscr{U}$ with $\varepsilon_{n}\to\Delta_{X}$, we can
choose $t_{n}\in T$, with $t_{n}x$ sufficiently close to $x_{n}$, such that
$\varepsilon_{n}[t_{n}\overline{Hx}]\supseteq\overline{Hx_{n}}$ for all $n$.
By compactness of $T/H$ and using Lemma 1.8c, we can assume $t_{n}\to k$ in
$T$ and $t_{n}H\to kH$ in $T/H$ so that (a subnet of) $t_{n}\overline{Hx}\to
k\overline{Hx}$ in $2^{X}$. Then $\overline{Hx}\subsetneq L\subseteq
k\overline{Hx}=\overline{Hkx}$, contrary to that $\\{\overline{Hz}\,|\,z\in
X\\}$ is a partition of $X$ into $H$-minimal sets. The proof is completed. ∎

######  3.2 Corollary ([10, Thm. 4.04] and [7, Prop. 2.8]).

Let $\mathscr{X}$ be a flow and $H\in\mathcal{S}^{\textsl{ns}}(T)$, then $x\in
X$ is a.p. for $T\curvearrowright X$ iff $x$ is a.p. for $H\curvearrowright
X$.

Corollary 3.2 is still true if $X$ is only LC instead of $X$ being compact;
see [7, Prop. 2.8]. In fact, if $X$ is LC and $x$ is a.p. for
$H\in\mathcal{S}^{\textsl{ns}}(T)$, then $\overline{Hx}$ is compact and
$\overline{Tx}=K\overline{Hx}$ for some $K\in 2^{T}$ so that $\overline{Tx}$
is a compact set in $X$. Although the compactness of $X$ plays a role in our
proof of Theorem 3.1-(2), it turns out that we can conclude the following in
our sncc-setting.

######  3.3 Proposition.

Let $\mathscr{X}$ be a flow, with $X$ an LC Hausdorff space not necessarily
compact. Then:

  1. $(1)$

If $H$ is a co-compact closed subgroup of $T$ and $x\in X$ is a.p. for
$H\curvearrowright X$, then $\overline{Tx}$ is compact minimal under $T$.

  2. $(2)$

If $H\in\mathcal{S}^{\textsl{sncc}}(T)$, then $x\in X$ is a.p. for
$T\curvearrowright X$ iff $x$ is a.p. for $H\curvearrowright X$.

###### Proof.

(1): Let $x\in X$ be a.p. for $H\curvearrowright X$. Since $X$ is LC, hence by
a standard argument it follows that $\overline{Hx}$ is a compact set in $X$.

Step 1. If $\\{t_{n}x\\}$ is a net in $Tx$, then there exists a subnet
$\\{s_{i}\\}$ of $\\{t_{n}\\}$ such that $s_{i}x\to y$ for some $y\in X$.
Indeed, we can select a subnet $\\{s_{i}\\}$ of $\\{t_{n}\\}$ and some $k\in
T$ such that $s_{i}H\to kH$ in $T/H$, for $T/H$ is compact. Further by Lemma
1.8c, we can select a net $\\{k_{i}\\}$ in $T$ such that $k_{i}\to k$ and
$k_{i}H=s_{i}H$. Then $Hx\ni k_{i}^{-1}s_{i}x\to y^{\prime}\in\overline{Hx}$
and $s_{i}x\to y:=ky^{\prime}\in X$.

Step 2. To prove that $\overline{Tx}$ is compact, it suffices to show that
there exists a convergent subnet from every net $\\{y_{i}\colon i\in I\\}$ in
$\overline{Tx}$. Indeed, given $\\{y_{i}\colon i\in I\\}$ in $\overline{Tx}$,
let $\mathcal{U}$ be a uniformity on $X$ (any will do and there are usually
many). Let $I^{\prime}=I\times\mathcal{U}$. For $i^{\prime}=(i,\alpha)\in
I^{\prime}$ let $t_{i^{\prime}}\in I$ such that
$(y_{i},t_{i^{\prime}}x)\in\alpha$. By Step 1, there is a subnet
$\\{t_{i^{\prime\prime}}\\}$ of $\\{t_{i^{\prime}}\\}$ such that
$t_{i^{\prime\prime}}x\to y$ for some $y\in\overline{Tx}$. For any
$\varepsilon\in\mathcal{U}$ choose $\alpha\in\mathcal{U}$ with
$\alpha\circ\alpha\subseteq\varepsilon$. Eventually
$t_{i^{\prime\prime}}x\in\alpha[y]$ and eventually
$(y_{i^{\prime\prime}},t_{i^{\prime\prime}}x)\in\alpha$. Thus,
$y_{i^{\prime\prime}}\to y$ and $\overline{Tx}$ is compact.

Step 3. By Theorem 3.1, it follows that $x$ is a.p. and so $\overline{Tx}$ is
compact minimal for $T$. This proves (1).

(2): Obvious by (1) and Theorem 3.1. The proof is completed. ∎

######  3.4 Question.

If $\mathscr{X}$ is a minimal flow and $H\in\mathcal{S}^{\textsl{sncc}}(T)$
not normal in $T$, is $\mathscr{O}_{H}\colon X\rightarrow 2^{X}$ a continuous
map? The general answer is NO!

######  3.4A. Example.

Let $G$ and $H$ be as in Example 1.11. So $H\in\mathcal{S}^{\textsl{sncc}}(G)$
and $H\notin\mathcal{S}^{\textsl{ncc}}(G)$. Let $X=G/H$. Recall that
$G\curvearrowright X$ is a minimal flow. Then by Theorem 3.1,
$H\curvearrowright X$ is pointwise a.p. Given $g\in G$, write $[g]=gH\in X$.
Clearly, if $g\in K$, then $\overline{H[g]}=\\{[g]\\}$ because $Hg=gH$. As for
all $z\in\mathbb{R}$ and $i,j,k,i^{\prime},j^{\prime},k^{\prime}\in\mathbb{Z}$

$[i,j,k]\cdot[0,0,z]\cdot[i^{\prime},j^{\prime},k^{\prime}]=\begin{pmatrix}1&i^{\prime}+i&j^{\prime}+k^{\prime}i+iz+j\\\
0&1&k^{\prime}+z+k\\\ 0&0&1\end{pmatrix},$

it follows that if we choose a sequence $g_{n}=[0,0,1/2n]\in G$ with
$n\in\mathbb{N}$, then $g_{n}\to[0,0,0]$ the unit matrix and
$\overline{H[g_{n}]}\not\to\overline{H[e]}=\\{[e]\\}$ in $2^{X}$ as
$n\to\infty$. Thus, $\mathscr{O}_{H}\colon X\rightarrow 2^{X}$ is not a
continuous map and then by Theorem 3.1, $H\curvearrowright X$ is pointwise
a.p. non-minimal.

######  3.4B. Theorem (due to L. Auslander, F. Hahn and L. Markus 1963).

Let $G$, $H$ be as in Example 1.11. Then $G\curvearrowright G/H$ is a non-
equicontinuous distal coset flow.

###### Proof.

First $G\curvearrowright G/H$ is not equicontinuous; otherwise,
$\mathscr{O}_{H}\colon X\rightarrow 2^{X}$ would be continuous, contrary to
Example 3.4A. As $H\lhd K\lhd G$, it follows that
$G/H\xrightarrow{\pi_{1}}G/K\xrightarrow{\pi_{2}}\\{pt\\}$ are group
extensions.333In fact, $\pi_{2}$ is a group extension of $\\{pt\\}$ for $K\lhd
G$ and $G/K$ is a compact Hausdorff topological group. On the other hand,
$K/H$ is a compact Hausdorff topological group such that $G\curvearrowright
G/H\curvearrowleft K/H$, given by $(t,gH,kH)\mapsto tgkH$ for all $t\in G$,
$gH\in G/H$ and $kH\in K/H$, is a bi-flow. Now for $gH\in G/H$,
$\pi_{1}^{-1}(gH)=gKH=\textrm{Aut}_{\pi_{1}}(G\curvearrowright
G/H)(gH),\quad\textrm{Aut}_{\pi_{1}}(G\curvearrowright G/H)\cong
K/H\cong\mathbb{R}/\mathbb{Z}.$ Thus, $\pi_{1}$ is a group extension. So
$G\curvearrowright G/H$ is a distal coset flow. ∎

######  3.5Weakly a.p. in the sense of W. H. Gottschalk.

A flow $\mathscr{X}$ is called weakly a.p. [9], if given
$\alpha\in\mathscr{U}$ there exists a compact set $K$ in $T$ such that
$Tx\subseteq K^{-1}\alpha[x]$ for all $x\in X$.

######  3.5A. Theorem (cf. [9, Thm. 5] or [10, Thm. 4.24]).

A flow $\mathscr{X}$ is weakly a.p. iff $\mathscr{O}_{T}\colon X\rightarrow
2^{X}$ is continuous.

######  3.5B. Corollary.

A minimal flow is weakly almost periodic.

######  3.5C. Corollary.

Let $\mathscr{X}$ be a minimal flow and $H\in\mathcal{S}^{\textsl{ncc}}(T)$.
Then $H\curvearrowright X$, as a subflow of $\mathscr{X}$, is weakly a.p. of
Gottschalk.

###### Proof.

By Theorem 3.1-(3) and Theorem 3.5A. ∎

Thus, to prove that $\mathscr{O}_{H}\colon X\rightarrow 2^{X}$ continuous in
Question 3.4, it is enough to show that $H\curvearrowright X$ is weakly a.p.
in the sense of Gottschalk.

######  3.6 Corollary.

Let $\mathscr{X}$ be a minimal flow and $H\in\mathcal{S}^{\textsl{sncc}}(T)$
with a subnormal sequence $H_{0}=H\lhd H_{1}\lhd\dotsm\lhd H_{n}=T$ such that
$[H_{i+1}:H_{i}]$ is finite for $1\leq i<n$. Then $\mathscr{O}_{H}\colon
X\rightarrow 2^{X}$ is continuous.

###### Proof.

By induction and Theorem 3.1, it follows that $\\{\overline{H_{1}x}\,|\,x\in
X\\}$ is an $H$-invariant finite partition of $X$. By Theorem 3.1 again, it
follows that $\mathscr{O}_{H}\colon\overline{H_{1}x}\rightarrow
2^{\overline{H_{1}x}}\subseteq 2^{X}$ is continuous. Thus,
$\mathscr{O}_{H}\colon X\rightarrow 2^{X}$ is continuous for every
$\overline{H_{1}x}$ is clopen in $X$. The proof is completed. ∎

If $\mathscr{X}$ is a minimal flow and $H\in\mathcal{S}^{\textsl{ncc}}(T)$,
then Theorem 3.1 tells us that $\mathcal{P}=\\{\overline{Hx}\,|\,x\in X\\}$ is
a $T$-invariant closed partition of $X$. In fact, we shall prove that
$T\curvearrowright X/\mathcal{P}$ is an a.p. factor of $\mathscr{X}$ (see Thm.
6.5).

##  4\. Homomorphism of flows

######  4.1.

Let $\phi\colon\mathscr{X}\rightarrow\mathscr{Z}$ be a homomorphism of flows;
i.e., $\phi\colon X\rightarrow Z$ is a continuous mapping such that $\phi
tx=t\phi x$ for all $x\in X$ and all $t\in T$. If in addition $\phi X=Z$, then
$\mathscr{X}$ is called an extension of $\mathscr{Z}$ and $\mathscr{Z}$ is
called a factor of $\mathscr{X}$.

Write ${R}_{\phi}=\\{(x_{1},x_{2})\,|\,x_{1},x_{2}\in X,\phi x_{1}=\phi
x_{2}\\}$.

We say that $(x_{1},x_{2})\in{P}_{\\!\phi}(\mathscr{X})$ iff
$(x_{1},x_{2})\in{R}_{\phi}\cap{P}(\mathscr{X})$. Then
${P}(\mathscr{X})={P}_{\\!\phi}(\mathscr{X})$ if $Z=\\{pt\\}$. Thus,
$\mathscr{X}$ is $\phi$-proximal, or simply, $\phi$ is proximal, iff
${P}_{\\!\phi}(\mathscr{X})={R}_{\phi}$.

We say that $\mathscr{X}$ is $\phi$-distal, or simply $\phi$ is distal, iff
${P}_{\\!\phi}(\mathscr{X})=\Delta_{X}$ iff
$\overline{T(x_{1},x_{2})}\cap\Delta_{X}=\emptyset$ for all
$(x_{1},x_{2})\in{R}_{\phi}\setminus\Delta_{X}$.

Let $H\in\mathcal{S}^{\textsl{sncc}}(T)$ with $H=H_{0}\lhd H_{1}\lhd\dotsm\lhd
H_{n-1}\lhd H_{n}=T$. Then

  1. $T/H\rightarrow T/H_{1}\rightarrow\dotsm\rightarrow T/H_{n-1}\rightarrow T/H_{n}=\\{pt\\}$

is a sequence of a.p. extension. Thus, $T\curvearrowright T/H$ is a distal
coset flow, not necessarily a.p. (see Thm. 3.4B).

Then the following result together with Theorem 3.1 may give us a
relativization of Theorem 2.2.

######  4.2 Theorem (inheritance of relativized proximality; cf. [10, Thm.
10.03] for $Z=\\{pt\\}$, $H$ syndetic).

Let $\phi\colon\mathscr{X}\rightarrow\mathscr{Z}$ be a homomorphism of flows
and $H$ a co-compact closed subgroup of $T$. Then
$(x_{1},x_{2})\in{P}_{\\!\phi}(T\curvearrowright X)$ iff
$(x_{1},x_{2})\in{P}_{\\!\phi}(H\curvearrowright X)$.

###### Proof.

Let $(x_{1},x_{2})\in{P}_{\\!\phi}$ under $T$, then there is a net
$\\{t_{n}\,|\,n\in D\\}$ in $T$ and a point $x\in X$ such that
$t_{n}(x_{1},x_{2})\to(x,x)$. Since $T/H$ is compact, we can assume (a subnet
of) $t_{n}H\to kH$ for some element $k\in T$. Further by Lemma 1.8c, we can
assume that there is a net $k_{n}\in T$ with $k_{n}\to k$ in $T$ such that
$t_{n}H=k_{n}H$ for all $n\in D$. Then $k_{n}^{-1}t_{n}\in H$ and
$k_{n}^{-1}\to k^{-1}$. Thus $k_{n}^{-1}t_{n}(x_{1},x_{2})\to
k^{-1}(x,x)\in\Delta_{X}$ and $(x_{1},x_{2})\in{P}_{\\!\phi}$ under $H$. On
the other hand, obviously, ${P}_{\\!\phi}(H\curvearrowright
X)\subseteq{P}_{\\!\phi}(T\curvearrowright X)$. The proof is completed. ∎

######  4.3 Theorem (inheritance of relativized distality; cf. [7, Prop. 5.14]
for $H$ syndetic, $Z=\\{pt\\}$).

Let $\phi\colon\mathscr{X}\rightarrow\mathscr{Z}$ be a homomorphism of flows
and $H$ a co-compact closed subgroup of $T$. Then $\mathscr{X}$ is
$\phi$-distal iff $H\curvearrowright X$ is $\phi$-distal.

###### Proof.

By Theorem 4.2. ∎

######  4.4 Corollary.

Let $\mathscr{X}$ be a minimal proximal flow. If
$H\in\mathcal{S}^{\textsl{sncc}}(T)$, then $H\curvearrowright X$ is minimal
proximal as a subflow of $\mathscr{X}$.

###### Proof.

By Theorem 3.1, $H\curvearrowright X$ is pointwise a.p.; and by Theorem 4.2,
$H\curvearrowright X$ is proximal. Thus, $H\curvearrowright X$ is minimal
proximal. ∎

##  5\. Point-envelopes of subgroups

######  5.0.

Let $\mathscr{X}$ be a flow, $x\in X$, and $H$ a subset of $T$, as in [10,
Def. 2.08], the $x$-envelope of $H$ in $T$ is defined by $E_{x}[H]=\\{t\in
T\,|\,tx\in\overline{Hx}\\}$.

Clearly, $H\subseteq E_{x}[H]$ and then
$\overline{Hx}\subseteq\overline{E_{x}[H]x}\subseteq\overline{Hx}$ from which
it follows that

  1. 5.0a.

$E_{x}[H]=E_{x}[E_{x}[H]]\ \forall x\in X$.

If $H\lhd T$, then $tx\in\overline{Hx}$ implies
$tHx=Htx\subseteq\overline{Hx}$; thus,

  1. 5.0b.

$t\in E_{x}[H]$ iff $t\overline{Hx}\subseteq\overline{Hx}$.

Next we will be concerned with the case $H\in\mathcal{S}^{\textsl{ncc}}(T)$.
Clearly, $E_{x}[H]$ is a closed subsemigroup of $T$ for this case. However, is
it a subgroup of $T$? First, using Theorem 3.1 we can easily obtain the
following result.

######  5.1 Proposition.

If $\mathscr{X}$ is a minimal flow with $T$ abelian and $H<T$ is co-compact,
then $E_{x}[H]=E_{y}[H]$ for all $x,y\in X$.

###### Proof.

Let $x,y\in X$ and $H\in\mathcal{S}^{\textsl{ncc}}(T)$. Let $t\in E_{x}[H]$
and let $s_{n}\in T$ such that $s_{n}x\to y$. Then by Theorem 3.1-(3),

$ty=\lim s_{n}tx\in\lim
s_{n}\overline{Hx}=\lim\overline{Hs_{n}x}=\overline{Hy}.$

So $t\in E_{y}[H]$. Symmetrically, $E_{y}[H]\subseteq E_{x}[H]$. The proof is
complete. ∎

######  5.2 Proposition.

Let $\mathscr{X}$ be a flow and $x\in X$ an a.p. point. Let
$H\in\mathcal{S}^{\textsl{ncc}}(T)$. Then $E_{x}[H]=E_{y}[H]$ for all
$y\in\overline{Hx}$.

###### Proof.

Let $x\in X$ be a.p. for $T$. Then $x$ is a.p. for $H$ by Theorem 3.1 and so
$\overline{Hx}=\overline{Hy}$ for all $y\in\overline{Hx}$. It then follows
from 5.0b that $E_{x}[H]=E_{y}[H]$ for all $y\in\overline{Hx}$. The proof is
completed. ∎

Now we will prove that $E_{x}[H]$ is a closed subgroup of $T$. However, it is
“non-normal” in general. The following theorem has the flavor of [7, Cor.
2.12] that asserts that a closed subsemigroup that contains a normal syndetic
subgroup is a subgroup of $T$.

######  5.3 Theorem.

Let $T$ be an wF-group (cf. Lem. 1.7). If $S$ be a closed subsemigroup of $T$
such that for all $V\in\mathfrak{N}_{e}(T)$ there is a finite set $F$ in $T$
with $FVS=T$, then $S$ is a subgroup of $T$.

######  Note.

If $T$ is a topological group, then ‘a finite set $F$’ may be relaxed as ‘a
compact set $F$’ here.

###### Proof.

Let $s\in S$. We need only prove $s^{-1}\in S$. For this, take
$V\in\mathfrak{N}_{e}(T)$. Then there is a finite subset $F$ of $T$ such that
$FVS=T$. Let $k_{0}\in FV$; then $k_{0}s^{-1}=k_{1}s_{1}$ for some $k_{1}\in
FV$ and $s_{1}\in S$. Further, $k_{1}s^{-1}=k_{2}s_{2}$ for some $k_{2}\in FV$
and $s_{2}\in S$. By induction, there are two sequences
$k_{0},k_{1},k_{2},\dotsc$ in $FV$ and $s_{1},s_{2},s_{3},\dotsc$ in $S$ with
$k_{i}s^{-1}=k_{i+1}s_{i+1}$ for all $i=0,1,\dotsc$. Write $k_{i}=f_{i}v_{i}$
with $f_{i}\in F$ and $v_{i}\in V$ for all $i$. Since $F$ is finite, we can
find some $0\leq m<n<\infty$ such that $f_{m}=f_{n}=f$. Then

$\begin{split}k_{n}^{-1}k_{m}s^{-1}&=k_{n}^{-1}k_{m+1}s_{m+1}=k_{n}^{-1}(k_{m+1}s^{-1})ss_{m+1}\\\
&=k_{n}^{-1}k_{m+2}s_{m+2}ss_{m+1}\\\
&=k_{n}^{-1}(k_{m+2}s^{-1})ss_{m+2}ss_{m+1}=\dotsm\\\
&=k_{n}^{-1}k_{n}s_{n}ss_{n-1}\dotsm ss_{m+1}\\\ &=s_{n}ss_{n-1}\dotsm
ss_{m+1}.\end{split}$

Hence $k_{n}^{-1}k_{m}s^{-1}\in S$ and $v_{n}^{-1}v_{m}s^{-1}\in S$. Thus
$v_{m}s^{-1}\in v_{n}S\subseteq VS$. Taking $v_{m}\to e$ in $T$, this implies
that $s^{-1}\in VS$ for all $V\in\mathfrak{N}_{e}(T)$. Next we can select two
nets $v_{i}\to e$ in $T$ and $s_{i}\in S$ such that $s^{-1}=v_{i}s_{i}$ and
then $v_{i}^{-1}s^{-1}=s_{i}\to s^{-1}\in S$, for $S$ is closed. The proof is
completed. ∎

######  5.3A. Proposition.

Let $S$ be a closed subsemigroup of an wF-group $T$, which includes a co-
compact closed subgroup of $T$. Then $S$ is a co-compact closed subgroup of
$T$.

###### Proof.

Let $H\leq S$ be a co-compact closed subgroup of $T$. Let
$V\in\mathfrak{N}_{e}(T)$. Then there is a finite set $F$ in $T$ with
$FVH\,(\subseteq FVS)=T$. Thus, $S$ is a subgroup of $T$ by Theorem 5.3.
Moreover, by Lemma 1.7, $S$ is co-compact. ∎

######  5.3B. Proposition (cf. [10, Lem. 2.06] or [4, Lem. 2..8.17] for $T$ a
Hausdorff topological group).

If $S$ is a syndetic closed subsemigroup of an wF-group $T$, then $S$ is a
syndetic closed subgroup of $T$.

###### Proof.

Let $V\in\mathfrak{N}_{e}(T)$. Since $S$ is syndetic, there is a set $K\in
2^{T}$ with $KS=T$. Further, we can find a finite set $F\subseteq T$ with
$FV\supseteq K$. Thus, $FVS=T$. Then $S$ is a syndetic closed subgroup of $T$
by Theorem 5.3. ∎

######  5.4 Corollary (comparable with [7, Cor. 2.11]).

Let $E$ be an F-group and $M$ a closed subsemigroup of $E$. Then $M$ is a
subgroup of $E$.

######  Note.

In [7, Cor. 2.11], $E$ is a compact $T_{1}$ left(or right)-topological group.
Then $M$ is a compact $T_{1}$ left(or right)-topological semigroup, and so
that $M$ contains the unique idempotent $e$ ([7, Lem. 2.9]). In view of this,
Ellis’ argument is invalid for our setting.

###### Proof.

As $E$ is compact, it follows that for every $V\in\mathfrak{N}_{e}(E)$, there
is a finite set $F\subseteq E$ with $FV=E$, and so that $FVM=EM=E$. Thus, $M$
is a subgroup of $E$ by Theorem 5.3. ∎

######  5.5 Corollary (cf. [10, Lem. 2.10-(1)] or [4, Lem. 2.8.19]).

Let $\mathscr{X}$ be a flow, $x\in X$, and $A\in\mathcal{S}^{\textsl{ns}}(T)$.
Then $E_{x}[A]$ is closed syndetic subgroup of $T$ with $A\subseteq E_{x}[A]$
and $\overline{Ax}=\overline{E_{x}[A]x}$.

######  5.6 Corollary (cf. [10, Lem. 2.10-(3)] for $T$ an LC group).

Let $\mathscr{X}$ be any flow, $x\in X$ and
$S\in\mathcal{S}^{\textsl{ns}}(T)$. If $U\in\mathfrak{N}_{x}(X)$, then there
exists $K\in 2^{T}$ such that $Kx\subseteq U$ and $E_{x}[S]=K^{-1}S$.

###### Proof.

We may assume $U\in\mathfrak{N}_{x}(X)$ is open. Let $L\in 2^{T}$ with $T=LS$.
Set $H=L\cap E_{x}[S]$. Since $E_{x}[S]$ is a closed subgroup of $T$ and
$S\subseteq E_{x}[S]$ by Corollary 5.5, $H\not=\emptyset$ is a compact set
with $E_{x}[S]=HS\,(=SH)$.

Let $t\in H$. Then $tx\in\overline{Sx}$ and $x\in\overline{t^{-1}Sx}$. So
there exists an element $s_{t}\in S$ with $t^{-1}s_{t}x\in U$. Further, there
exists a compact set $V_{t}\in\mathfrak{N}_{t}(H)$ such that
$V_{t}^{-1}s_{t}x\subseteq U$. Therefore we can find a finite set
$\\{t_{1},\dotsc,t_{m}\\}$ in $H$, which is such that
$H=\bigcup_{i=1}^{m}V_{t_{i}}$ and $V_{t_{i}}^{-1}s_{t_{i}}x\subseteq U$ for
$i=1,\dotsc,m$. Now we put $K=\bigcup_{i=1}^{m}V_{t_{i}}^{-1}s_{t_{i}}$. Then
$K$ is compact in $T$ with $Kx\subseteq U$. Moreover,
$E_{x}[S]=S\left({\bigcup}_{i=1}^{m}V_{t_{i}}\right)=S\left({\bigcup}_{i=1}^{m}s_{t_{i}}^{-1}V_{t_{i}}\right)=SK^{-1}=K^{-1}S$.
The proof is completed. ∎

######  5.7 Corollary.

Let $\mathscr{X}$ be any flow, $x\in X$, and
$H\in\mathcal{S}^{\textsl{ncc}}(T)$. Then $E_{x}[H]$ is co-compact closed
subgroup of $T$ and $T\curvearrowright T/E_{x}[H]$ is an a.p. minimal flow.

###### Proof.

Clearly $E_{x}[H]$ is a co-compact closed subgroup of $T$ such that
$H\subseteq E_{x}[H]$ by Theorem 5.3. Since $T/E_{x}[H]$ is a factor of $T/H$,
hence $T\curvearrowright T/E_{x}[H]$ is an a.p. coset flow. The proof is
completed. ∎

##  6\. Relation of orbital closure of ncc-subgroups

Let $\mathscr{X}$ be a flow with compact Hausdorff phase space $X$ and with
phase mapping $T\times X\rightarrow X$, $(t,x)\mapsto tx$ in the sequel of
this section.

######  6.1.

Given a subgroup $H$ of $T$, let $H\curvearrowright X$ be the induced subflow
of $\mathscr{X}$. We define

  1. $R_{H}=\\{(x,x^{\prime})\,|\,x,x^{\prime}\in X\textrm{ s.t. }x^{\prime}\in\overline{Hx}\\}$,

which is called the $H$-orbital closure relation of $\mathscr{X}$. Write $X/H$
for $X/R_{H}$, where $X/H$ is equipped with the quotient topology.

If $R_{H}$ is a closed $T$-invariant equivalence relation on $X$, then $X/H$
is a compact Hausdorff space (by Lem. 6.3 below) so that $T\curvearrowright
X/H$ is a flow. If $H\curvearrowright X$ is pointwise a.p., then $R_{H}$ is an
equivalence relation. However, $R_{H}$ is generally not an invariant closed
relation.

######  6.2 Lemma.

If $\mathscr{X}$ is a minimal flow and $H\in\mathcal{S}^{\textsl{ncc}}(T)$,
then $R_{H}$ is a closed invariant equivalence relation of $\mathscr{X}$.

###### Proof.

By Theorem 3.1-(1) and (2), it follows that $R_{H}$ is a $T$-invariant
equivalence relation on $X$. Further by Theorem 3.1-(3), $R_{H}$ is closed in
$X\times X$. The proof is completed. ∎

It turns out that, for $H\in\mathcal{S}^{\textsl{ncc}}(T)$, $X/R_{H}$ is a
compact Hausdorff space by the following folklore lemma in general topology,
where the “compact Hausdorff” of $X$ plays a role in its proof. We shall give
a proof here for reader’s convenience, which is comparable with Lemma 1.8b.

######  6.3 Lemma (cf. [7, Lem. 4.9] without proof).

Let $R$ be an equivalence relation on $X$. Then $R$ is closed in $X\times X$
iff $X/R$ is a Hausdorff space.

###### Proof.

Let $\rho\colon X\rightarrow X/R$ be the quotient map. If $X/R$ is Hausdorff,
then $\Delta_{X/R}$ is closed in $X/R\times X/R$ so that
$R=(\rho\times\rho)^{-1}[\Delta_{X/R}]$ is closed in $X\times X$. (This still
holds for $X$ a topological space not necessarily compact and Hausdorff.)

Conversely, suppose $R$ is closed in $X\times X$. Let $A\not=\emptyset$ be a
closed subset of $X$ and set $R[A]=\bigcup_{a\in A}R(a)\subseteq X$. Clearly,
$R[A]=\rho^{-1}\rho[A]$ is closed in $X$, for $R[A]$ is exactly equal to the
image of $A\times X\cap R$ under the canonical map $(x_{1},x_{2})\mapsto
x_{2}$. So $\rho$ is closed. If $U\subseteq X$ is open, then

  1. $R^{*}(U):=\\{x\in X\,|\,R(x)\subseteq U\\}=X\setminus R[X\setminus U]=X\setminus\rho^{-1}\rho[X\setminus U]$

is open. So $\rho[R^{*}(U)]$ is open in $X\setminus R$. Now if
$\rho(x)\not=\rho(y)$ (or equivalently, $R(x)\cap R(y)=\emptyset$), then we
can choose disjoint open sets $U_{x}$, $U_{y}$ with $R(x)\subseteq U_{x}$ and
$R(y)\subseteq U_{y}$. Thus,
$\rho[R^{*}(U_{x})]\in\mathfrak{N}_{\rho(x)}(X/R)$ and
$\rho[R^{*}(U_{y})]\in\mathfrak{N}_{\rho(y)}(X/R)$ are disjoint, and, $X/R$ is
a Hausdorff space. The proof is completed. ∎

######  6.4 Remark.

For the necessity proof above, “compact Hausdorff” of $X$ has played a role in
proving that $R[A]$ is a closed subset of $X$.

If $R$ is closed in $X\times X$, then the cells $R(x)$, for $x\in X$, are
closed sets in $X$. However, the converse is false (so that this negates [6,
A.2.3]). Let’s see a counterexample.

######  Example.

Let $X=[0,1]\times[0,1]=\\{x=(x_{1},x_{2})\,|\,0\leq x_{1},x_{2}\leq 1\\}$.
Define a relation $R$ on $X$ by the way: for $x=(x_{1},x_{2})$,
$y=(y_{1},y_{2})\in X$,

  1. $x\sim y$ iff $x_{1}=y_{1}$ if $0\leq x_{1},y_{1}<1$; $x\sim y$ iff $x=y$ if $x_{1}=y_{1}=1$.

Clearly, $R$ is an equivalence relation on $X$ such that $R(x)$, for $x\in X$,
is a closed set in $X$. However, $R$ is not closed in $X\times X$. Thus, $X/R$
is compact and not of Hausdorff.

######  6.5 Theorem.

Let $\mathscr{X}$ be a minimal flow and $H\in\mathcal{S}^{\textsl{ncc}}(T)$.
Then for all $x\in X$, the evaluation mapping

  1. $\mathfrak{e}\colon T/E_{x}[H]\rightarrow X/H,\quad tE_{x}[H]\mapsto t\overline{Hx}$

is an isomorphism from $T\curvearrowright T/E_{x}[H]$ onto $T\curvearrowright
X/H$, and moreover, $T\curvearrowright X/H$ is an a.p. factor of
$\mathscr{X}$.

######  Note.

Consequently, $T\curvearrowright X/H$ may be thought of as an a.p. coset flow
of $T$ if $H\in\mathcal{S}^{\textsl{ncc}}(T)$. On the other hand, if
$H\in\mathcal{S}^{\textsl{sncc}}(T)\setminus\mathcal{S}^{\textsl{ncc}}(T)$,
then the statement is generally false. For example, let $H\lhd K\lhd G$ be as
in Example 1.11; let $X=G/H$ and $x=[e]=H\in X$. Then for $T=G$, $E_{x}[H]=H$
and by Example 3.4A, $R_{H}$ is not closed so that $X/H$ is not Hausdorff by
Lemma 6.3. Since $G/E_{x}[H]=G/H$ is Hausdorff, hence $G/E_{x}[H]\not\cong
X/H$.

###### Proof.

At first, since $\\{\overline{Hy}\,|\,y\in X\\}$ is a closed $T$-invariant
partition of $X$ into $H$-minimal sets by Theorem 3.1. So $X/H$ is a compact
Hausdorff space and $T\curvearrowright X/H$ is a minimal flow by Lemmas 6.2
and 6.3. By Corollary 5.7, $T\curvearrowright T/E_{x}[H]$ is an a.p. minimal
flow. Obviously $\mathfrak{e}$ is a well-defined 1-1 onto map such that
$\mathfrak{e}t=t\mathfrak{e}$ for all $t\in T$. It remains to show that
$\mathfrak{e}$ is a continuous mapping. For this, consider the CD:

$\begin{diagram}\qquad\textrm{where}\begin{cases}f\colon t\mapsto
t\overline{Hx},\\\ \rho\colon t\mapsto tE_{x}[H].\end{cases}$

Notice that $t_{n}\to t$ in $T$ implies that $t_{n}x\to tx$; and
$H\in\mathcal{S}^{\textsl{ncc}}(T)$ and $t_{n}\to t$ in $T$ implies that
$f(t_{n})\to f(t)$ in $X/H$. Thus, $f$ and $\rho$ both are continuous maps.
Moreover, if $tE_{x}[H]=sE_{x}[H]$ (or equivalently, if $t^{-1}s\in
E_{x}[H]$), then $f(t)=f(s)$ so that $\mathfrak{e}$ is continuous. Thus,
$T\curvearrowright X/H$ is equicontinuous. The proof is completed. ∎

######  6.6 Corollary (cf. [10, Thm. 2.25]).

Let $\mathscr{X}$ be a minimal flow, $x\in X$, and
$S\in\mathcal{S}^{\textsl{ns}}(T)$. Then $\\{\overline{Sz}\,|\,z\in X\\}$ is a
$T$-invariant partition of $X$; and moreover, $\\{\overline{Sz}\,|\,z\in X\\}$
and $T/E_{x}[S]$ have the same cardinality.

######  6.7 Corollary (cf. [4, Lem. 2.8.26] for $T$ abelian).

Let $\mathscr{X}$ be a minimal flow, $x\in X$, and
$A\in\mathcal{S}^{\textsl{ns}}(T)$. Then
$\mathfrak{e}\colon(T,T/E_{x}[A])\rightarrow(T,X/A)$ is an isomorphism.

We now will conclude our arguments with an application of Theorem 2.3 and 6.5.

######  6.8 Theorem.

Let $\mathscr{X}$ be a minimal flow, $H\in\mathcal{S}^{\textsl{ncc}}(T)$. Then
$T\curvearrowright X/H$ is an a.p. nontrivial factor of $\mathscr{X}$ iff
$T\curvearrowright X\times T/H$ is not a minimal flow.

###### Proof.

If $T\curvearrowright X/H$ is nontrivial, then by Theorem 6.5,
$T\curvearrowright X/H$ is a common factor of $\mathscr{X}$ and
$T\curvearrowright T/H$ so that $T\curvearrowright X\times T/H$ is not
minimal. Now conversely, suppose $T\curvearrowright X\times T/H$ is not
minimal. Then $H\curvearrowright X$ is not minimal by Theorem 2.3. So $X/H$ is
not a singleton. Now by Theorem 6.5, $T\curvearrowright X/H$ is an a.p.
nontrivial factor of $\mathscr{X}$. The proof is completed. ∎

### Acknowledgements

The author is grateful to the anonymous referee for her/his constructive
comments. The work was supported by National Natural Science Foundation of
China (No. 12271245).

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