2.1 Topological transformation groups

A flow or a topological dynamical system is a triple $\mathcal {X}=(X, T, \Pi )$ , where X is a compact Hausdorff space, T is a Hausdorff topological group, and $\Pi : T\times X\rightarrow X$ is a continuous map such that $\Pi (e,x)=x$ and $\Pi (s,\Pi (t,x))=\Pi (st,x)$ , where e is the unit of T, $s,t\in T$ , and $x\in X$ . We shall fix T and suppress the action symbol.

In this paper, we always assume that T is infinite countable and discrete, unless we state it explicitly in some places. Moreover, we always assume that X is a compact metric space with metric $d(\cdot , \cdot )$ .

When $T=\mathbb Z$ , $(X,T)$ is determined by a homeomorphism f, that is, f is the transformation corresponding to $1$ of $\mathbb Z$ . In this case, we usually denote $(X,\mathbb Z)$ by $(X,f)$ , and also call it a discrete flow.

Let $(X,T)$ be a flow and $x\in X$ . Let $\mathcal{O} (x,T)=\{tx: t\in T\}$ be the orbit of x, which is also denoted by $T x$ . We usually denote the closure of $\mathcal{O} (x,T)$ by $\overline {\mathcal{O} }(x,T)$ or $\overline {Tx}$ . A subset $A\subseteq X$ is called invariant if $t a\subseteq A$ for all $a\in A$ and $t\in T$ . When $Y\subseteq X$ is a closed and invariant subset of the flow $(X, T)$ , we say that the flow $(Y, T)$ is a subflow of $(X, T)$ . If $(X, T)$ and $(Y,T)$ are two flows, their product flow is the flow $(X \times Y, T)$ , where $t(x, y) = (tx, ty)$ for any $t\in T$ and $x,y\in X$ . For $n \geq 2$ , we write $(X^n,T)$ for the n-fold product flow $(X\times \cdots \times X,T)$ .

A flow $(X,T)$ is called minimal if X contains no proper non-empty closed invariant subsets. A point $x\in X$ is called a minimal point or an almost periodic point if $(\overline {\mathcal{O} }(x,T), T)$ is a minimal flow.

A flow $(X,T)$ is called transitive if every invariant open subset of X is dense; and it is point transitive if there is a point with a dense orbit (such a point is called a transitive point). It is easy to verify that a flow is minimal if and only if every orbit is dense.

The flow $(X,T)$ is weakly mixing if the product flow $(X \times X,T)$ is transitive.

A factor map $\pi : X\rightarrow Y$ between the flow $(X,T)$ and $(Y, T)$ is a continuous onto map which intertwines the actions; we say that $(Y, T)$ is a factor of $(X,T)$ and that $(X,T)$ is an extension of $(Y,S)$ . The flows are said to be isomorphic if $\pi $ is bijective.

Let $(X,T)$ be a flow. Fix $(x,y)\in X^2$ . It is a proximal pair if $\inf _{t\in T} d(tx, ty)=0$ ; it is a distal pair if it is not proximal. Denote by $P(X,T)$ or $P(X)$ the set of proximal pairs of $(X,T)$ . Here, $P(X,T)$ is also called the proximal relation of $(X,T)$ . A flow $(X,T)$ is distal if $P(X,T)= \Delta _X$ , where $\Delta _X=\{(x,x)\in X^2: x\in X\}$ is the diagonal of $X\times X$ . A flow $(X,T)$ is equicontinuous if for any $\epsilon>0$ , there is a $\delta>0$ such that whenever $x,y\in X$ with $d(x,y)<{\delta }$ , then $d(tx,ty)<\epsilon $ for all $t\in T$ . Any equicontinuous flow is distal.

Let $(X,T)$ be a flow. There is a smallest invariant equivalence relation $S_{\mathrm {eq}}$ such that the quotient flow $(X/S_{\mathrm {eq}},T)$ is equicontinuous [Reference Ellis and Gottschalk12, Theorem 1]. The equivalence relation $S_{\mathrm {eq}}$ is called the equicontinuous structure relation and the factor $(X_{\mathrm {eq}}=X/S_{\mathrm {eq}}, T)$ is called the maximal equicontinuous factor of $(X,T)$ .

2.2 Enveloping semigroups

Given a flow $(X,T)$ , its enveloping semigroup or Ellis semigroup $E(X,T)$ is defined as the closure of the set $\{t: t\in T\}$ in $X^X$ (with its compact, usually non-metrizable, pointwise convergence topology). For an enveloping semigroup, $E(X,T)\rightarrow E(X,T): q\mapsto qp$ and $p\mapsto tp$ is continuous for all $p\in E(X,T)$ and $t\in T$ . Note that $(X^X,T)$ is a flow and $(E(X,T),T)$ is its subflow.

It is easy to see that for a flow $(X,T)$ , the enveloping semigroup $E(X,T)$ is an $\mathcal E$ -semigroup.

For a semigroup, the element u with $u^2=u$ is called idempotent. Ellis–Numakura theorem says that for any $\mathcal E$ -semigroup E, the set $J(E)$ of idempotents of E is not empty (see [Reference Ellis10, Corollary 2.10] or [Reference Glasner17, Ch. I, Lemma 2.2]).

Let E be an $\mathcal E$ -semigroup. A non-empty subset $I \subseteq E$ is a left ideal if $EI \subseteq I$ . A minimal left ideal is a left ideal that does not contain any proper left ideal of E. Every left ideal is a semigroup and every left ideal contains some minimal left ideal.

2.3 Universal point transitive flow and universal minimal flow

For a fixed T, there exists a universal point transitive flow $(S_T,T)$ such that T can densely and equivariantly be embedded in $S_T$ (see [Reference Auslander4, Ch. 8], [Reference Glasner17, Ch. I]). The multiplication on T can be extended to a multiplication on $S_T$ , and $S_T$ is an $\mathcal E$ -semigroup. The universal minimal flow $\boldsymbol{\mathfrak {M}}=(\mathbf {M},T)$ is isomorphic to any minimal left ideal in $S_T$ and $\mathbf {M}$ is also an $\mathcal E$ -semigroup. Hence $J=J(\mathbf {M})$ of idempotents in $\mathbf {M}$ is non-empty.

Since $v\mathbf {M}$ is a group ( $v\in J$ ), for $p\in v\mathbf {M}$ , we denote $p^{-1}$ as the inverse of p in $v\mathbf {M}$ , that is, $p^{-1}\in v\mathbf {M}$ such that $p^{-1}p=pp^{-1}=v$ . If we choose an arbitrary idempotent $u\in J$ and denote $G=u\mathbf {M}$ , then every element p of $\mathbf {M}$ has unique representation $p=v{\alpha }$ for $v\in J$ and ${\alpha }\in G$ . Moreover, $p^{-1}=v{\alpha }^{-1}$ (see [Reference Glasner17, Ch. I, Proposition 2.3, Corollary 2.4]).

The sets $S_T$ and $\mathbf {M}$ act on X as semigroups and $S_T x=\overline {T x}$ , while for a minimal flow $(X,T)$ , we have $\mathbf {M} x=\overline {T x}=X$ for every $x\in X$ (see [Reference Glasner17, Ch. I, Proposition 3.1] for details). For $x\in X$ , set $J_x=\{u\in J: ux=x\}$ .

Thus, for a closed invariant subset A of X, $JA=\{ua: u\in J, a\in A\}$ is the set of all minimal points contained in A.

2.4 Hyperspace flow and circle operation

Let X be a compact metric space. Let $2^X$ be the collection of non-empty closed subsets of X endowed with the Hausdorff topology. Let $(X,T)$ be a flow. We can induce a flow on $2^X$ . The action of T on $2^X$ is given by $tA=\{ta:a\in A\}$ for each $t\in T$ and $A \in 2^X$ . Then $(2^X,T)$ is a flow and it is called the hyperspace flow.

As $(2^X, T)$ is a flow, $S_T$ acts on $2^X$ too. To avoid ambiguity, we denote the action of $S_T$ on $2^X$ by the circle operation as follows. Let $p\in S_T$ and $D\in 2^X$ . Define $p{\circ } D=\lim _{2^X} t_i D$ for any net $\{t_i\}_i$ in T with $t_i\to p$ . Moreover,

$$ \begin{align*} p{\circ} D=\{x\in X: \text{there are }d_i\in D\text{ with }x=\lim_i t_id_i\} \end{align*} $$

for any net $t_i\to p$ in $S_T$ . We always have $pD\subseteq p{\circ } D$ .

Note that if $A\in 2^X$ is finite and $p\in S_T$ , then $pA=p{\circ } A$ .

2.5 Almost one-to-one extensions and O-diagram

Let $(X,T)$ and $(Y,T)$ be flows and let $\pi : X \to Y$ be a factor map. One says that:

1. (1) $\pi $ is an open extension if it is open as a map;

2. (2) $\pi $ is an almost one-to-one extension if there exists a dense $G_{\delta }$ set $X_0\subseteq X$ such that $\pi ^{-1}(\{\pi (x)\})=\{x\}$ for any $x\in X_0$ .

The following is a well-known fact about open mappings (see [Reference de Vries8, Appendix A.8], for example).

Every extension of minimal flows can be lifted to an open extension by almost one-to-one modifications ([Reference Veech27, Theorem 3.1], [Reference Auslander and Glasner5, Lemma III.6] or [Reference de Vries8, Ch. VI]). To be precise, we have the following theorem.

Theorem 2.6. For every extension $\pi :X\rightarrow Y$ of minimal flows, there exists a commutative diagram of extensions (called the O-diagram)

with the following properties:

1. (a) $\sigma $ and $\tau $ are almost one-to-one;

2. (b) $\pi ^*$ is an open extension;

3. (c) $X^*$ is the unique minimal set in $R_{\pi \tau }=\{(x,y)\in X\times Y^*: \pi (x)=\tau (y)\}$ and $\sigma $ and $\pi ^*$ are the restrictions to $X^*$ of the projections of $X\times Y^*$ onto X and $Y^*$ , respectively.

We sketch the construction of these factors. Let $x \in X$ , $u \in J_{x}$ and $y=\pi (x)$ . Let $y^*=u{\circ } {\pi }^{-1}(y).$ One has that $y^*$ is a minimal point of $(2^X, T)$ and define $Y^*=\{p{\circ } y^*: p \in \mathbf {M}\}$ as the orbit closure of $y^*$ in $2^X$ for the action of T. Finally, $X^*=\{(px, p{\circ } y^*)\in X\times Y^*: p\in \mathbf {M} \}$ , $\tau (p{\circ } y^*)= p y$ and $\sigma ((px,p{\circ } y^*)) =px$ . It can be proved that $X^*=\{(\tilde x,\tilde y)\in X \times Y^* : \tilde x \in \tilde y\}$ .

There is another equivalent way to get an O-diagram. Let $\pi ^{-1}: Y\rightarrow 2^X, y\mapsto \pi ^{-1}(y)$ . Then $\pi ^{-1}$ is a u.s.c. map, and the set $Y_c$ of continuous points of $\pi ^{-1}$ is a dense $G_{\delta }$ subset of Y. Let

$$ \begin{align*}\widetilde{Y}=\overline{\{\pi^{-1}(y): y\in Y\}}\quad \text{and}\quad Y^*=\overline{\{\pi^{-1}(y): y\in Y_c\}},\end{align*} $$

where the closure is taken in $2^X$ . It is obvious that $Y^*\subseteq \widetilde {Y}\subseteq 2^X$ . Note that for each $A\in \widetilde {Y}$ , there is some $y\in Y$ such that $A\subseteq \pi ^{-1}(y)$ , and hence $A\mapsto y$ define a map $\tau : \widetilde {Y}\rightarrow Y$ . It is easy to verify that $\tau : (\widetilde {Y},T)\rightarrow (Y,T)$ is a factor map. One can show that if $(Y,T)$ is minimal, then $(Y^*,T)$ is a minimal flow and it is the unique minimal subflow in $(\widetilde {Y},T)$ , and $\tau : Y^*\rightarrow Y$ is an almost one-to-one extension such that $\tau ^{-1}(y)=\{\pi ^{-1}(y)\}$ for all $y\in Y_c$ (see [Reference Veech28, §2.3]). When $Y^*$ and $\tau $ are defined, $X^*, \sigma $ , and $\pi ^*$ are defined as above.

2.6 Proximal extensions and RIC-diagram

Let $(X,T)$ and $(Y, T)$ be flows and let $\pi : X \to Y$ be a factor map. One says that:

1. (1) $\pi $ is a distal extension if $\pi (x_1)=\pi (x_2)$ and $x_1\neq x_2$ implies $(x_1,x_2) \not \in P(X,T)$ ;

2. (2) $\pi $ is an equicontinuous or isometric extension if for any $\epsilon>0$ , there exists ${\delta }>0$ such that $\pi (x_1)=\pi (x_2)$ and $d(x_1,x_2)<{\delta }$ imply $d(tx_1,tx_2)<\epsilon $ for any $t\in T$ ;

3. (3) $\pi $ is a weakly mixing extension if $(R_\pi , T)$ as a subflow of the product flow $(X\times X, T)$ is transitive, where $R_\pi =\{(x_1,x_2)\in X^2: \pi (x_1)=\pi (x_2)\}$ .

Let $\pi : (X,T )\rightarrow (Y,T)$ be a factor map of minimal flows, and $x_0\in X$ , $y_0=\pi (x_0)$ , and $u\in J_{x_0}$ . We say that $\pi $ is an RIC (relatively incontractible) extension if for every $y = py_0\in Y$ , $p\in \mathbf {M}$ ,

$$ \begin{align*}\pi^{-1}(y)=p{\circ} u\pi^{-1}(y_0).\end{align*} $$

One can show that $\pi : X \to Y$ is RIC if and only if it is open and for every $n \ge 1$ , the minimal points are dense in the relation

$$ \begin{align*} R^n_\pi = \{(x_1,\ldots,x_n) \in X^n : \pi(x_i)=\pi(x_j)\quad \text{for all } 1\le i \le j \le n\}. \end{align*} $$

Note that every distal extension is RIC.

Every factor map between minimal flows can be lifted to an RIC extension by proximal extensions (see [Reference Ellis, Glasner and Shapiro11, Theorem 5.13] or [Reference de Vries8, Ch. VI]).

Theorem 2.7. Given a factor map $\pi :X\rightarrow Y$ of minimal flows, there exists a commutative diagram of factor maps (called an RIC-diagram)

such that:

1. (a) $\tau '$ and $\sigma '$ are proximal extensions;

2. (b) $\pi '$ is an RIC extension;

3. (c) $X '$ is the unique minimal set in $R_{\pi \tau ' }=\{(x,y)\in X\times Y ': \pi (x)=\tau '(y)\}$ , and $\sigma '$ and $\pi '$ are the restrictions to $X '$ of the projections of $X\times Y '$ onto X and $Y '$ , respectively.

We sketch the construction of these factors. Let $x\in X$ , $u\in J_{x}$ , and $y=\pi (x)$ . Let $y'=u {\circ } u \pi ^{-1}(y)$ , then $y'$ is a minimal point in $2^X$ . Define $Y'=\{p{\circ } y': p\in \mathbf {M}\}$ to be the orbit closure of $y'$ and $X'=\{(px, p{\circ } y')\in X\times Y': p\in \mathbf {M}\}$ , and factor maps are given by $\tau ' (p{\circ } y')= p y$ and $\sigma '((px,p{\circ } y')) =px$ . It can be proved that $X'=\{(\tilde x,\tilde y)\in X \times Y' : \tilde x \in \tilde y\}$ .

3.1 Almost proximal extensions

An almost periodic set for $(X,T)$ is a subset A of X such that if $z\in X^{|A|}$ with $\mathrm {range} (z)=A$ , then z is a minimal point of the flow $(X^{|A|},T)$ . (Here, $|A|$ denotes the cardinality of A.) For example, a finite set $A=\{x_1,x_2,\ldots , x_n\}$ is an almost periodic set if and only if $(x_1,x_2,\ldots , x_n)$ is a minimal point of $(X^n,T)$ . Using the basis for the Tychonoff topology, we see that a set A is an almost periodic set if and only if every finite subset of A is an almost periodic set. The notion of almost periodic sets was introduced by Auslander, refer to [Reference Auslander4, Ch. 5] for more information.

Let A be an almost periodic set of $(X,T)$ . Then for each $z\in X^{|A|}$ with $\mathrm {range} (z)=A$ , z is a minimal point of $(X^{|A|},T)$ . By Proposition 2.3, there is some $u\in J$ such that $uz=z$ . It follows that $uA=\{ua: a\in A\}=A$ . Thus, for a subset Z of X, a subset $A\subseteq Z$ is an almost periodic subset of Z if and only if $A\subseteq uZ$ for some $u\in J$ .

Proof. We only show the case when $u\pi ^{-1}(y), v\pi ^{-1}(y')$ are finite. The same proof works for the general case. Let $u\pi ^{-1}(y)=\{x_1,x_2,\ldots , x_n\}$ and $v\pi ^{-1}(y')=\{x_1',x_2',\ldots , x_m'\}$ for some $n,m\in \mathbb N$ . Then, $(x_1,x_2,\ldots , x_n)\in X^n$ is a minimal point of $(X^n, T)$ as $u(x_1,x_2,\ldots , x_n)=(x_1,x_2,\ldots , x_n)$ . Since $(X,T)$ is minimal, there is $p\in \mathbf {M}$ such that $x_1'=px_1$ . Note that $x_1'=vx_1'=vpx_1$ and $x_1,x_2,\ldots , x_n\in \pi ^{-1}(y)$ . It follows that

$$ \begin{align*}y'=\pi(x_1')=\pi(vpx_1)=\cdots =\pi(vpx_n)=vp\pi(x_1)=vpy.\end{align*} $$

Since $x_1,x_2,\ldots ,x_n$ are distinct and $(x_1,x_2,\ldots , x_n)$ is minimal, $vpx_1, vpx_2, \ldots , vpx_n$ are also distinct. As

$$ \begin{align*}vpx_1, vpx_2, \ldots, vpx_n \in v\pi^{-1}(y')=\{x_1',x_2',\ldots, x_m'\},\end{align*} $$

we have $n\le m$ . Similarly, we have $m\le n$ . Thus, $|u\pi ^{-1}(y)|=|v\pi ^{-1}(y')|$ .

By Lemma 3.2 and the fact that A is an almost periodic subset of subset Z of $(X,T)$ if and only if $A\subseteq uZ$ for some $u\in J$ , we have the following proposition readily.

For $x\in X$ , $P[x]=\{x'\in X: (x,x')\in P(X, T)\}$ is called the proximal cell of x. It is clear that $\pi $ is proximal if and only if for any $y\in Y$ and any $x\in \pi ^{-1}(y)$ , $\pi ^{-1}(y)\subseteq P[x]$ .

Let $\pi :(X,T)\rightarrow (Y,T)$ be an extension of flows $(X,T)$ and $(Y,T)$ and $n\in \mathbb N$ . Let

$$ \begin{align*}2^X_\pi=\{A\in 2^X: A\subseteq \pi^{-1}(y) \text{ for some } y\in Y \},\end{align*} $$

and

$$ \begin{align*}2^X_{\pi, n}=\{A\in 2^X_\pi: |A|\le n\}, \quad \text{and}\quad 2^X_{\pi,*}=\{A\in 2^X_\pi: |A|< \infty\}=\bigcup_{n=1}^\infty 2^X_{\pi, n}.\end{align*} $$

It is clear that $(2^X_\pi , T)$ , $(2^X_{\pi ,n}, T)$ , and $(2^X_{\pi ,*}, T)$ are subflows of $(2^X,T)$ . Let

$$ \begin{align*}\widetilde{\pi}: (2^X_\pi, T) \rightarrow (Y,T), \quad A\mapsto \pi(A).\end{align*} $$

Then $\widetilde {\pi }$ is an extension. Note that $2^X_{\pi ,1}=\{\{x\}\in 2^X: x\in X\}$ is isomorphic to X and $\widetilde {\pi }|_{2^X_{\pi ,1}}$ is the same to $\pi $ . Thus, $\pi : (X,T) \rightarrow (Y,T)$ is proximal if and only if $\widetilde {\pi }|_{2^X_{\pi ,1}}\!: (2^X_{\pi ,1}, T)\rightarrow (Y,T)$ is proximal.

Proof. If $\pi $ is almost proximal, then there is some $n\in \mathbb N$ such that $|u\pi ^{-1}(y)|=n$ for all $y\in Y, u\in J_y$ . Let $A\in 2^X_{\pi , *} \setminus 2^X_{\pi , n}$ and $y=\pi (A)$ . Then $|A|\ge n+1$ . For each $u\in J_y$ , $|uA|\le n$ since $\pi $ is almost proximal. Thus, $u\circ A=uA\neq A$ , in particular, by Proposition 2.3, A is not minimal.

Conversely, assume that there is some $n\in \mathbb N$ such that there are no minimal points in $2^X_{\pi ,*}\setminus 2^X_{\pi , n}$ . If $\pi $ is not almost proximal, then there is some $y\in Y$ and $u\in J_y$ such that $|u\pi ^{-1}(y)|=\infty $ . We choose $A\subseteq u\pi ^{-1}(y)$ with $|A|= n+1$ . Then

$$ \begin{align*}u\circ A=uA=A,\end{align*} $$

which means A is a minimal point of $2^X_{\pi ,*}\setminus 2^X_{\pi , n}$ , which is a contradiction. The proof is complete.

3.2 Almost finite-to-one extensions

By Proposition 3.8, it is easy to check the following corollary.

3.3 Structure of almost proximal extensions

The following result is well known and it is easy to be verified (one may find a proof in [Reference Huang, Lian, Shao and Ye20, Lemma 2.20]).

Theorem 3.12. Let $\pi : (X,T)\rightarrow (Y,T)$ be an almost proximal extension of minimal flows. Then it has the following structure:

where $\sigma '$ and $\tau '$ are proximal, $\pi '$ is a finite-to-one equicontinuous extension.

Moreover, $\pi $ is almost finite-to-one if and only if $\tau ', \sigma '$ are almost one-to-one.

If, in addition, the extension is regular (see definition below), an almost proximal extension has a succinct structure. Let $\mathrm {Aut} (X,T)$ be the group of automorphisms of the flow $(X,T)$ , that is, the group of all self-homeomorphisms $\psi $ of X such that $\psi \circ t=t\circ \psi \text { for all } t\in T$ . For an extension $\pi : (X,T)\rightarrow (Y,T)$ , let

$$ \begin{align*}\mathrm{Aut}_\pi(X,T)=\{\chi\in \mathrm{Aut}(X,T): \pi\circ \chi=\pi\},\end{align*} $$

that is, elements of $\mathrm {Aut}(X,T)$ mapping every fiber of $\pi $ into itself.

The notion of regularity was introduced by Auslander [Reference Auslander3]. Examples of regular extensions are proximal extensions, group extensions. For more information about regularity, refer to [Reference Auslander3, Reference Auslander4, Reference Glasner15].

Theorem 3.14. Let $\pi : (X,T)\rightarrow (Y,T)$ be an extension of minimal flows. If $\pi $ is regular and almost proximal, then there exists a flow $(Y^\#,T)$ with the following commutative diagram:

where $\tau ^\#$ is proximal and $\pi ^\#$ is a finite-to-one equicontinuous extension. And $\pi $ is almost finite-to-one if and only if $\tau ^\#$ is almost one-to-one.

Proof. Let $y_0\in Y$ and $u\in J_{y_0}$ , that is, $uy_0=y_0$ . Since $\pi $ is almost proximal, by Proposition 3.3, there exits an $n\in \mathbb N$ such that $|v\pi ^{-1}(y)|=n$ for all $y\in Y, v\in J_y$ . Thus, $|v\pi ^{-1}(py_0)|=n$ for all $p\in \mathbf {M}$ and all $v\in J(\mathbf {M})$ with $vp=p$ .

Let $\mathbf {y}_0=u\pi ^{-1}(y_0)=\{x_1,x_2,\ldots , x_n\}$ . As $\{x_1,x_2,\ldots , x_n\}$ is an almost periodic set, $(x_1,\ldots , x_n)$ is a minimal point of $(X^n,T)$ , and it follows that for all $p\in \mathbf {M}$ , $|p\mathbf { y}_0|=|\{px_1,px_2,\ldots , px_n\}|=n$ . Let

$$ \begin{align*}Y^\#=\{p\mathbf{y}_0: p\in \mathbf{M}\}.\end{align*} $$

It is clear that $X=\bigcup _{\mathbf {y}\in Y^\#} \mathbf {y}$ . We show that $Y^\#$ is a partition of X, that is, for all $p,q\in \mathbf {M}$ , either $p\mathbf {y}_0=q\mathbf {y}_0$ or $p\mathbf {y}_0 \cap q\mathbf {y}_0=\emptyset $ .

Assume that there are $p,q\in \mathbf {M}$ such that $p\mathbf {y}_0=\{px_1,\ldots , px_n\}\not =\{qx_1, \ldots , qx_n\}=q\mathbf {y}_0$ and $p\mathbf {y}_0 \cap q\mathbf {y}_0\neq \emptyset $ . Without loss of generality, we assume that $qx_1=px_j\in p\mathbf {y}_0$ for some $j\in \{1,2,\ldots , n\}$ and $qx_2\not \in p\mathbf {y}_0$ .

Since $(x_1,\ldots , x_n)$ is a minimal point of $(X^n,T)$ , $(qx_1, \ldots , qx_n)=q(x_1,\ldots , x_n)$ is also a minimal point of $(X^n,T)$ . In particular, $(qx_1,qx_2)\in R_\pi $ is a minimal point. As $\pi $ is regular, there is some $\chi \in \mathrm {Aut}_\pi (X,T)$ such that $qx_2=\chi (qx_1)$ . Let $v\in J(\mathbf {M})$ such that $vp=p$ (Proposition 2.2). Then,

$$ \begin{align*}qx_2=\chi(qx_1)=\chi(px_j)=\chi(vpx_j)=v\chi(px_j)=v\chi(qx_1)=vqx_2.\end{align*} $$

Thus, $qx_2\in v\pi ^{-1}(py_0)$ and $v\{px_1,px_2,\ldots , px_n, qx_2\}=\{px_1,px_2,\ldots , px_n, qx_2\}$ . It follows that $\{px_1,px_2,\ldots , px_n, qx_2\}$ is an almost periodic subset of $\pi ^{-1}(py_0)$ , whose cardinality is $n+1$ . In particular, $|v\pi ^{-1}(py_0)|\ge n+1$ , which contradicts with the fact $|v\pi ^{-1}(py_0)|= n$ . Thus, $Y^\#$ is a partition of X.

Since $Y^\#$ is a partition of X, it induces a map

$$ \begin{align*}\pi^\#: X\rightarrow Y^\#, \ x\mapsto p\mathbf{y}_0, \, (x\in p\mathbf{y}_0), p\in \mathbf{M}.\end{align*} $$

It is easy to check that $\pi $ is open, and by Lemma 3.11, it is an n-to- $1$ equicontinuous extension.

Let

$$ \begin{align*}\tau^\# : Y^\#\rightarrow Y, p\mathbf{y_0}\mapsto py_0 \quad\text{for all } p\in \mathbf{M}.\end{align*} $$

We show that it is proximal. Let $\tau ^\#(p\mathbf {y}_0)=\tau ^\#(q\mathbf {y}_0)$ for some $p,q\in \mathbf {M}$ . Then, $py_0=qy_0$ . There are minimal idempotents $u_1,u_2\in J(\mathbf {M})$ such that

$$ \begin{align*}p=u_1 p,q=u_2q.\end{align*} $$

Since $|\mathbf {y}_0|<\infty ,$ we have

$$ \begin{align*}p\mathbf{y}_0=u_1p\mathbf{y}_0=u_1p u\pi^{-1}(y_0)=u_1\pi^{-1}(py_0).\end{align*} $$

Similarly, one has $q\mathbf {y}_0=u_2\pi ^{-1}(qy_0)$ . Then,

$$ \begin{align*}u_1q\mathbf{y}_0=u_1u_2\pi^{-1}(qy_0)=u_1\pi^{-1}(py_0)=p\mathbf{y}_0.\end{align*} $$

So $p\mathbf {y}_0$ and $q\mathbf {y}_0$ are proximal. That is, $\tau ^\#$ is a proximal extension.

3.4 Relations between almost proximal extensions and almost one-to-one extensions

It is clear that any almost finite-to-one extension is an almost proximal extension. In this subsection, we show that if, in addition, the extension is point distal, then the converse holds.

Let $\pi : (X,T)\rightarrow (Y,T)$ be an extension of minimal flows. A point $x\in X$ is called a $\pi $ -distal point whenever $P_\pi [x]\triangleq \{x'\in \pi ^{-1}(\pi (x)): (x,x')\in P(X,T)\}=\{x\}$ , that is, $(x,x')$ is a distal pair for every $x'\in \pi ^{-1}(\pi (x))\setminus \{x\}$ . The extension $\pi $ is said to be point distal when there exists a $\pi $ -distal point in X. Note that $\pi $ is distal if and only if every point is $\pi $ -distal. It is easy to see that a point x is $\pi $ -distal if and only if $ux=x$ for all $u\in J_{\pi (x)}$ [Reference de Vries8, Ch. VI(4.3)].

Proof. If $\pi $ is almost finite-to-one, it is almost proximal. By Proposition 3.8, there exist $N\in \mathbb N$ and $y_0\in Y$ such that $|\pi ^{-1}(y_0)|=N$ and $\pi ^{-1}(y_0)$ is a minimal point of $(2^X, T)$ . Each point in $\pi ^{-1}(y_0)$ is $\pi $ -distal.

Conversely, assume that $\pi $ is almost proximal and point distal. Let $x_0\in X$ be a $\pi $ -distal point and $y_0=\pi (x_0)$ . Since $x_0$ is $\pi $ -distal, $ux_0=x_0$ for all $u\in J_{y_0}$ . Thus, ${x_0\in \bigcap _{u\in J_{y_0}}u\circ u \pi ^{-1}(y_0)}$ . By Theorem 3.15, the RIC-diagram and O-diagram of $\pi $ coincide. That is, in the RIC-diagram,

Here, $\sigma '$ and $\tau '$ are almost one-to-one, and $\pi '$ is a finite-to-one extension. Thus, $\pi $ is almost finite-to-one.

4.1 Weakly mixing extensions

Let $(X,T)$ be a flow. A point $x\in X$ is recurrent if for each neighborhood U of x, the set of return times of x to U, $N(x,U)=\{t\in T: tx\in U\}$ is infinite. A point x is recurrent if and only if there exists $p\in S_T\setminus T$ such that $px=x$ , if and only if there exists an idempotent $u\in S_T\setminus T$ such that $ux=x$ [Reference Ellis, Ellis and Nerurkar9, Lemma 5.18].

The following result says that each non-trivial weakly mixing extension contains plenty of strongly scrambled subsets.

Theorem 4.2. [Reference Akin, Glasner, Huang, Shao and Ye2, Theorem 5.10]

Let $(X,T), (Y,T)$ be a flow and $\pi : (X,T)\rightarrow (Y, T)$ an open non-trivial weakly mixing extension. Then there is a residual subset $Y_0\subseteq Y$ such that for every point $y\in Y_0$ , the set $\pi ^{-1}(y)$ contains a dense uncountable strongly scrambled subset K such that

$$ \begin{align*}K\times K\setminus \Delta_X\subseteq \mathrm{Trans}(R_\pi),\end{align*} $$

where $ \mathrm {Trans}(R_\pi )$ denotes the set of all transitive points in $R_\pi $ , and in particular, $\pi ^{-1}(y)$ is perfect, and it has the cardinality of the continuum.

Note that a $2$ -weakly mixing extension is just weakly mixing. Totally weakly mixing extensions will present stronger chaotic properties than weakly mixing extensions, see [Reference Akin, Glasner, Huang, Shao and Ye2, Reference Shao and Ye23] for more details. For RIC extensions, weak mixing is equivalent to total weak mixing.

In general, the fact that $\pi : (X,T)\rightarrow (Y,T)$ is weakly mixing does not imply that $\pi $ is totally weakly mixing. In [Reference Glasner16, Theorem 4.1], for $(Y,T)$ being trivial, the author gave a weakly mixing flow $(X,T)$ such that $(X^3,T)$ is not transitive. By Theorem 4.4, such T is not abelian. In §6, we will show that there exists $\pi : (X,\mathbb Z)\rightarrow (Y,\mathbb Z)$ which is weakly mixing but not 3-weakly mixing. In fact, we have the following example.

4.2 Proximal but not almost one-to-one extensions

Let $P^{(n)}(X)$ be the subset of $X^n$ of all points whose orbit closure intersects the diagonal of $X^n$ . Thus, $P(X)=P^{(2)}(X)$ . Glasner showed that for a minimal flow $(X,T)$ , if $P^{(n)}(X)$ is dense in $X^n$ for every $n\ge 2$ , then X is weakly disjoint from all minimal flows, that is, $(X\times Y,T)$ is transitive for all minimal flow $(Y,T)$ [Reference Glasner17, Ch. II, Proposition 2.1]. In particular, any minimal proximal flow is weakly mixing. This was extended by van der Woude [Reference van der Woude26, Ch. VII, Corollary 2.14] as the following theorem (see also [Reference Glasner16, Corollary 6.4]).

Thus, by Theorem 4.2, we have the following theorem.

In fact, we have the following result which is slightly stronger than Theorem 4.7.

Theorem 4.8. Let $\pi :X\rightarrow Y$ be a proximal but not almost one-to-one extension of minimal flows. Then there is a residual subset $Y_0\subseteq Y$ such that for each $y\in Y_0$ , $\pi ^{-1}(y)$ contains an uncountable strongly scrambled subset K satisfying that for any $x_1 \neq x_2\in K$ ,

$$ \begin{align*} \pi^{-1}(y)\times \pi^{-1}(y)\subseteq \overline{\mathcal{O}}((x_1,x_2),T).\end{align*} $$

Proof. Let $\pi :X\rightarrow Y$ be a proximal but not almost one-to-one extension of minimal flows. We consider its O-diagram:

Recall the construction of an O-diagram (see Theorem 2.6):

$$ \begin{align*}Y^*=\overline{\{\pi^{-1}(y): y\in Y_c\}}\subseteq 2^X,\quad X^*=\{(x, \mathbf{y})\in X\times Y^*: x\in \mathbf{y}\},\end{align*} $$

where $Y_c$ is the set of continuous points of $\pi ^{-1}: Y\rightarrow 2^X$ and it is a residual subset of Y, $X^*$ is the unique minimal set in $R_{\pi \tau }=\{(x,y)\in X\times Y^*: \pi (x)=\tau (y)\}$ , and $\sigma $ and $\pi ^*$ are the restrictions to $X^*$ of the projections of $X\times Y^*$ onto X and $Y^*$ , respectively. Also $\tau : Y^*\rightarrow Y$ is an almost one-to-one extension such that $\tau ^{-1}(y)=\{\pi ^{-1}(y)\}$ for all $y\in Y_c$ .

Since $\pi $ is proximal but not almost one-to-one, $\pi ^*$ is open proximal but not almost one-to-one. By Theorem 4.6, $\pi ^*$ is weakly mixing, and hence by Theorem 4.2, there is a residual subset $Y_0^*$ of $Y^*$ such that for each $\mathbf {y}\in Y_0^*$ , $(\pi ^*)^{-1}(\mathbf {y})$ contains a dense uncountable strongly scrambled subset $K^*$ such that

$$ \begin{align*}K^*\times K^*\setminus \Delta_{X^*}\subseteq \mathrm{Trans}(R_{\pi^*}).\end{align*} $$

Since $\tau $ is almost one-to-one, $\tau (Y_0^*)$ is a residual subset of Y. Let

$$ \begin{align*}Y_0=Y_c\cap \tau(Y_0^*).\end{align*} $$

Then $Y_0$ is a residual subset of Y. We need to verify that for each $y\in Y_0$ , $\pi ^{-1}(y)$ contains an uncountable strongly scrambled subset K satisfying that for any $x_1 \neq x_2\in K$ ,

$$ \begin{align*} \pi^{-1}(y)\times \pi^{-1}(y)\subseteq \overline{\mathcal{O}}((x_1,x_2),T).\end{align*} $$

Let $y\in Y_0$ . Since $y\in Y_c\cap \tau (Y_0^*) \subseteq Y_c$ , by the property of $\tau $ , we have that $\tau ^{-1}(y)=\{\pi ^{-1}(y)\}$ . Thus, $\mathbf {y}=\pi ^{-1}(y)\in Y_0^*$ . So by the definition of $Y_0^*$ , $(\pi ^*)^{-1}(\mathbf {y})$ contains a dense uncountable strongly scrambled subset $K^*$ such that $K^*\times K^*\setminus \Delta _{X^*}\subseteq \mathrm {Trans}(R_{\pi ^*}).$ Let

$$ \begin{align*}K^*=\{(x_{\alpha}, \mathbf{y}): {\alpha}\in \Lambda, x_{\alpha}\in \mathbf{y}\},\end{align*} $$

where $\Lambda $ is an uncountable index set. Now let $K=\sigma (K^*)=\{x_{\alpha }\}_{{\alpha }\in \Lambda }$ . First note that

$$ \begin{align*}\pi(K)=\pi\sigma(K^*)=\tau\pi^*(K^*)=\tau(\mathbf{y})=y,\end{align*} $$

and hence $K\subseteq \pi ^{-1}(y)$ .

Let $x_1,x_2\in K$ with $x_1\neq x_2$ . We show that $\pi ^{-1}(y)\times \pi ^{-1}(y)\subseteq \overline {\mathcal{O} }((x_1,x_2),T)$ . Let $y_1,y_2\in \pi ^{-1}(y)=\mathbf {y}$ . Then

$$ \begin{align*}(y_1, \mathbf{y}), (y_2, \mathbf{y})\in (\pi^*)^{-1}(\mathbf{y}).\end{align*} $$

Since $((x_1,\mathbf {y}), (x_2, \mathbf {y}))\in K^*\times K^*\setminus \Delta _{X^*} \subseteq \mathrm {Trans}(R_{\pi ^*})$ ,

$$ \begin{align*}((y_1,\mathbf{y}), (y_2, \mathbf{y}))\in \overline{\mathcal{O}}(((x_1,\mathbf{y}), (x_2, \mathbf{y})), T).\end{align*} $$

It follows that $(y_1,y_2)\in \overline {\mathcal{O} }((x_1,x_2),T)$ . Thus, $ \pi ^{-1}(y)\times \pi ^{-1}(y)\subseteq \overline {\mathcal{O} }((x_1,x_2),T).$ The proof is complete.

6.1 Glasner and Weiss’ results

Let $(Y,f)$ be a minimal discrete flow and Z be a compact metric space with metric $d_Z$ . Let $H(Z,Z)$ be the space of all homeomorphisms of Z equipped with the metric

$$ \begin{align*}\rho_Z(g,h)=\sup_{z\in Z}d_Z(g(z),h(z))+ \sup_{z\in Z}d_Z(g^{-1}(z),h^{-1}(z)).\end{align*} $$

With this metric, $H(Z,Z)$ is a complete metric space and a topological group. Let $X=Y\times Z$ . Let $H_s(X,X)$ be the subspace of $H(X,X)$ which consists of homeomorphisms which fix all subspace of X of the form $\{y\}\times Z, y\in Y$ . Such a homeomorphism G is determined by a continuous map ${\alpha }: Y\rightarrow H(Z,Z)$ , by $G(y,z)=(y, {\alpha }(y)(z))$ . Put

$$ \begin{align*}\mathcal{S}(f)=\{G^{-1}\circ f\circ G: G\in H_s(X,X)\}.\end{align*} $$

(Here, f is identified with $f\times \mathrm {id}_Z,$ where $\mathrm {id}_Z$ is the identity map on $Z.$ )

If $\mathcal {G}$ is a subgroup of $H(Z,Z)$ , let $\mathcal {G}_s\subseteq H_s(X,X)$ be the subgroup of those elements of $H_s(X,X)$ which come from cocycles (see definition below) of $\mathcal {G}$ . That is,

$$ \begin{align*}\mathcal{G}_s=\{G\in H_s(X,X): G(y,z)=(y,{\alpha}(y)(z)), {\alpha}\in C(Y,\mathcal{G})\}. \end{align*} $$

Let

$$ \begin{align*}\mathcal{S}_{\mathcal{G}}(f)=\{G^{-1}\circ f\circ G: G\in \mathcal{G}_s\}.\end{align*} $$

According to [Reference Glasner and Weiss19, §1], if we choose $Z=\mathbf {P}^n, n\ge 1$ to be the projective n-space, and let $\mathcal {G}$ be a pathwise connected component of $\mathrm {id}_Z$ in $H(Z,Z)$ , then $\mathcal {G}$ satisfies the conditions of Theorems 6.1 and 6.2. Thus, for an arbitrary minimal infinite flow $(Y,f)$ , there are many minimal homeomorphisms of $Y\times Z$ which are proximal but not almost one-to-one extensions of $(Y,f)$ .

6.2 Skew-product

Let $(Y,f)$ be a discrete flow and $(Z,d_Z)$ a compact metric space. Denote the set of all the homeomorphisms of Z to Z with $H(Z,Z)$ . For $\varphi _1, \varphi _2\in H(Z,Z)$ , set

$$ \begin{align*}D_Z(\varphi _1,\varphi _2)=\sup_{z\in Z}d_Z(\varphi _1(z),\varphi _2(z)).\end{align*} $$

Assume $X=Y\times Z$ and let $\rho _X $ denote the max-metric on the product space $Y\times Z$ ,

$$ \begin{align*}\rho_X((y_1,z_1),(y_2,z_2))=\max\{d_Y(y_1,y_2),d_Z(z_1,z_2)\},\end{align*} $$

where $d_Y$ is the metric of Y.

A continuous map $\sigma : Y\rightarrow H(Z,Z)$ is called a cocycle. By a given cocycle $\sigma $ , one can define

$$ \begin{align*}f_\sigma : X\rightarrow X, (y,z)\mapsto (f(y), \sigma (y)(z)) \quad \text{for all } (y,z)\in X.\end{align*} $$

The new flow $(X,f_\sigma )$ is called a skew-product flow.

For $(y,z)\in X$ , set $f^n_\sigma (y,z)= (f^n (y), \sigma _n(y)(z))$ , where

$$ \begin{align*}\sigma_n(y)= \begin{cases} \sigma(f^{n-1}y)\cdots \sigma(fy)\sigma(y), & n\ge 1; \\ \mathrm{id}_Z, & n=0 ;\\ \sigma(f^ny)^{-1}\cdots \sigma(f^{-2}y)^{-1}\sigma(f^{-1}y)^{-1}, & n<0. \end{cases} \end{align*} $$

6.3 A cocycle

Let $Y=Z(2)=\{0,1\}^{\mathbb N}$ with the metric

$$ \begin{align*}d({\alpha},\beta) = \frac1{\min \{i\in \mathbb N: {\alpha}_i\neq \beta_i\}},\quad {\alpha}=({\alpha}_i)_{i=1}^\infty,\quad \beta=(\beta_i)_{i=1}^\infty \in Z(2).\end{align*} $$

The map

$$ \begin{align*}\tau : Z(2)\rightarrow Z(2)\end{align*} $$

is defined as follows: for every ${\alpha } \in Z(2)$ , $\tau ({\alpha })={\alpha } + 10000\ldots $ , where the addition is modulo $2$ from the left to right. Obviously, $\tau $ is continuous. Moreover, it can be shown that $\tau $ is invertible and $(Z(2),\tau )$ is an equicontinuous minimal flow. The flow $(Z(2),\tau )$ is called an adding machine or odometer. Similarly, one can define ${(Z(k)=\{0,1,\ldots , k-1\}^{\mathbb N}, \tau )}$ .

In the following, an increasing sequence $\{n_i\}_{i=1}^\infty \subseteq \mathbb N$ is fixed. For any ${\alpha }\in Z(2)$ , ${\alpha }$ can be written as

(6.1) $$ \begin{align} {\alpha}={\alpha}^1 {\alpha}^2 {\alpha}^3 \ldots, \quad \text{where } {\alpha}^k \ \text{is a block of }n_k\text{ digits of }{\alpha}. \end{align} $$

Let $e: \bigcup _{j=1}^\infty \{0,1\}^j \rightarrow \mathbb Z_+$ be the evaluation function: for $x=x_1x_2\ldots x_j \in \{0,1\}^j$ ,

$$ \begin{align*}e(x)= x_1+2x_2+2^2x_3+\cdots + 2^{j-1}x_j.\end{align*} $$

For $n\in \mathbb Z_+$ , if $x=x_1x_2\ldots x_j \in \{0,1\}^j$ such that $e(x)=n$ , then let

$$ \begin{align*}\underline n = x_1x_2\ldots x_j 0^\infty \in Z(2)=\{0,1\}^{\mathbb N},\end{align*} $$

where $0^\infty =000\ldots .$ For example, $\underline {0}=0^\infty $ , $\underline {2}=010^\infty $ , $\underline {7}=1110^\infty $ .

Let $Z={\mathbb S}^1=\{\exp (i 2\pi x): 0\le x\le 1\}$ and

(6.2) $$ \begin{align} \Phi=\{\varphi_k^j: 0\le j\le 2^{n_k}-2\}_{k=1}^{\infty}\subseteq H({\mathbb S}^1,{\mathbb S}^1). \end{align} $$

We use $\Phi $ to construct a cocycle $\sigma : Z(2)\rightarrow H({\mathbb S}^1,{\mathbb S}^1)$ :

(6.3) $$ \begin{align} \sigma({\alpha})= \begin{cases} \varphi^{e({\alpha}^k)}_{k}, & {\alpha}\neq 1^\infty\text{ and }{\alpha}^k\text{ is the first block in}\ (6.1)\ \text{containing at least one zero}\\ &\quad\text{digit;} \\ \mathrm{id}_{{\mathbb S}^1}, & {\alpha} =1^\infty. \end{cases} \end{align} $$

Lemma 6.3. If $\Phi $ satisfies the following condition:

(C1) $$ \begin{align} \displaystyle \lim_{k\to \infty} \max_{0\le j\le 2^{n_k}-2} D_{{\mathbb S}^1}(\varphi^{j}_k, \mathrm{id}_{{\mathbb S}^1} ) = 0, \end{align} $$

then $\sigma : Z(2)\rightarrow H({\mathbb S}^1,{\mathbb S}^1)$ is continuous and hence it is a cocycle.

Proof. Let $\beta _n, {\alpha } \in Z(2)$ such that $\beta _n \to {\alpha }, n\to \infty $ . We show that $\lim _{n\to \infty } D_{{\mathbb S}^1}(\sigma (\beta _n), \sigma ({\alpha }))=0$ .

If ${\alpha } \ne 1^\infty $ , then one has $\sigma (\beta _n)=\sigma ({\alpha })$ when n is large enough and $D_{{\mathbb S}^1}(\sigma (\beta _n), \sigma ({\alpha }))=0$ . Now assume that ${\alpha } =1^\infty $ . For $\beta _n$ , we have that $ \beta _n=\beta _n^1 \beta _n^2 \beta _n^3 \ldots $ , where $\beta _n^k$ is a block of $n_k$ digits of $\beta _n$ . Thus, $\sigma (\beta _n)=\varphi _k^{e(\beta _n^k)}$ , where $\beta _n^k$ is the first block containing at least one zero digit. Since $\beta _n\to {\alpha }, n\to \infty $ , and $\lim _{k\to \infty } \max _{0\le j\le 2^{n_k}-2} D_{{\mathbb S}^1}(\varphi ^{j}_k, \mathrm {id}_{{\mathbb S}^1} ) = 0$ , we have that

$$ \begin{align*}\lim_{n\to \infty} D_{{\mathbb S}^1}(\sigma(\beta_n), \mathrm{id}_{{\mathbb S}^1})=0.\end{align*} $$

That is, $\sigma $ is continuous.

Thus, when (C1) is satisfied, $\sigma $ is a cocycle. Hence, we have a skew product flow $(X=Z(2)\times {\mathbb S}^1, F_\Phi )$ :

(6.4) $$ \begin{align} F_\Phi\triangleq \tau_\sigma: X\rightarrow X, \, F_\Phi({\alpha}, y)=(\tau({\alpha}), \sigma({\alpha}) y)= \begin{cases} (\tau({\alpha}), \varphi^{e({\alpha}^k)}_{k}(y)), & {\alpha}\neq 1^\infty; \\ (\underline{0}, y), & {\alpha} =1^\infty, \end{cases} \end{align} $$

where ${\alpha }^k$ is the first block in (6.1) containing at least one zero digit when ${\alpha }\neq 1^\infty $ .

6.4 The form of $F_\Phi ^n$

Let ${\alpha }={\alpha }^1{\alpha }^2\ldots \in Z(2)$ as in (6.1) and $z_0\in {\mathbb S}^1$ . Let

$$ \begin{align*}F^n_\Phi ({\alpha} ,z_0)=(\tau ^n{\alpha},z_n).\end{align*} $$

We need to know the formula of $z_n$ for some special n.

For simplicity, we always assume the following conditions hold:

(C2) $$ \begin{align} \varphi^0_k = \mathrm{id}_{{\mathbb S}^1}\quad \text{for all } k\in \mathbb N, \end{align} $$

and

(C3) $$ \begin{align} \Psi_k= \varphi_k^{2^{n_k}-2} \circ \varphi_k^{2^{n_k}-3} \circ \cdots \circ \varphi^1_k\circ \varphi^0_k=\mathrm{id}_{{\mathbb S}^1}\quad \text{for all } k\in\mathbb N. \end{align} $$

First we have a formula for ${\alpha }= \underline 0$ . By an easy induction, we have the following lemma.

Lemma 6.4. Let $(\underline 0 ,y_0) \in Z(2)\times {\mathbb S}^1$ and $(\underline n , y_n)=F_\Phi ^n (\underline 0 , y_0)$ . Let $m_k=2^{n_1+n_2+\cdots +n_k}$ for all $k\in \mathbb N$ . Then

(6.5) $$ \begin{align} y_{m_k}=\varphi^0_{k+1}\circ\psi_k(y_0)=y_0, \end{align} $$

and

(6.6) $$ \begin{align} y_{s\cdot m_k}=\varphi^{s-1}_{k+1}\circ \varphi^{s-2}_{k+1}\circ \cdots \circ \varphi^1_{k+1}\circ \varphi ^0_{k+1}(y_0),\quad 1\le s <2^{n_{k+1}}. \end{align} $$

Now we see the general case. Let ${\alpha }={\alpha }^1{\alpha }^2\ldots \in Z(2)$ as in (6.1) and $z_0\in {\mathbb S}^1$ . We denote $F^n_\Phi ({\alpha } ,z_0)=(\tau ^n{\alpha },z_n).$ Let

(6.7) $$ \begin{align} \psi_k ^+=\varphi^{e({\alpha}^k)-1}_k\circ\varphi^{e({\alpha}^k)-2}_k \circ \cdots \circ \varphi^{1}_k\circ \varphi^{0}_k\quad \text{for all } k\in \mathbb N, \end{align} $$

and

(6.8) $$ \begin{align} \psi _k ^-=\varphi^{2^{n_k}-2}_k\circ\varphi^{2^{n_k}-3}_k\circ \cdots \circ \varphi^{e({\alpha}^k)+1}_k\circ \varphi^{e({\alpha}^k)}_k \quad \text{for all } k\in \mathbb N. \end{align} $$

By the assumption (C3), one has that $(\psi _k ^+)^{-1}=\psi ^-_k.$

By an easy induction, we have the following lemma.

Lemma 6.5. Let ${\alpha }={\alpha }^1{\alpha }^2\ldots \in Z(2)$ as in (6.1) and $z_0\in {\mathbb S}^1$ . Let $F^n_\Phi ({\alpha } ,z_0)=(\tau ^n{\alpha },z_n).$ Let $m_k=2^{n_1+n_2+\cdots +n_k}$ for all $k\in \mathbb N$ . Then

(6.9) $$ \begin{align} F_\Phi ^{m_k -e({\alpha}^1{\alpha}^2\ldots{\alpha}^k)}({\alpha},z_0)=(0^{n_1+\cdots + n_k}\tau({\alpha}^{k+1}{\alpha}^{k+2}\cdots),z_{m_k -e({\alpha}^1{\alpha}^2\ldots{\alpha}^k)}), \end{align} $$

where $z_{m_k-e({\alpha }^1{\alpha }^2\ldots {\alpha }^k)}=\varphi ^{e({\alpha }^{k+1})}_{k+1}\circ \psi ^-_k\circ \psi ^-_{k-1}\circ \cdots \circ \psi ^-_1 (z_0). $ And

(6.10) $$ \begin{align} F_\Phi^{m_k}({\alpha},z_0)=({\alpha}^1{\alpha}^2\ldots{\alpha}^k\tau({\alpha}^{k+1}{\alpha}^{k+2}\ldots),z_{m_k}), \end{align} $$

where $z_{m_k}=\psi ^+_1\circ \psi ^+_2\circ \cdots \circ \psi ^+_k \circ \varphi ^{e({\alpha }^{k+1})}_{k+1}\circ \psi ^-_k\circ \cdots \circ \psi ^-_1 (z_0).$

Remark 6.6. (i) To simplify the calculation, if we require that

$$ \begin{align*} \varphi^1_k =\varphi ^2_k &=\cdots =\varphi^{2^{n_k -1}-1}_k\triangleq\varphi_k,\\\varphi^{2^{n_k -1}}_k=\varphi^{2^{n_k -1}+1}_k&=\cdots =\varphi^{2^{n_k}-2}\triangleq \varphi^-_k=(\varphi_k)^{-1}, \end{align*} $$

then $\psi ^+_k=(\varphi _k)^{c_k}, \psi ^-_k=(\varphi ^-_k)^{c_k},$ where $c_k=2^{n_k -1}-1-|e({\alpha }^k)-2^{n_k-1}|.$ Thus, (6.10) will be

(6.11) $$ \begin{align}z_{m_k}=(\varphi_1)^{c_1}\circ (\varphi_2)^{c_2}\circ\cdots \circ (\varphi_k)^{c_k}\circ \varphi ^{e({\alpha}^{k+1})}_{k+1}\circ (\varphi ^-_k)^{c_k}\circ \cdots \circ (\varphi ^-_1)^{c_1}(z_0). \end{align} $$

(ii) By (6.6), (6.9), for $j<k, 1\leq s<2^{n_{j+1}}$ , we have

(6.12) $$ \begin{align} F^{m_k-e({\alpha}^1{\alpha}^2\ldots {\alpha}^k)+s\cdot m_j}_\Phi ({\alpha} ,z_0)=(\underline s\tau({\alpha}^{k+1}{\alpha} ^{k+2}\ldots), z_{m_k -e({\alpha}^1{\alpha}^2\ldots {\alpha}^k)+s\cdot m_j}), \end{align} $$

where $z_{m_k -e({\alpha }^1{\alpha }^2\ldots {\alpha }^k)+s\cdot m_j}=\varphi ^{s-1}_{j+1}\circ \varphi ^{s-2}_{j+1}\circ \cdots \circ \varphi ^{1}_{j+1} \circ \varphi ^{0}_{j+1}(z_{m_k -e({\alpha }^1{\alpha }^2\ldots {\alpha }^k)}).$

6.5 Uniform rigidity

A discrete flow $(X,F)$ is rigid if there exists an increasing sequence $\{n_i\}_{i\in \mathbb N}$ in $\mathbb N$ such that $F^{n_i}x$ converges to x as i goes to infinity for every $x\in X$ (that is, $F^{n_i}$ converges pointwisely to the identity map). A flow $(X, F)$ is uniformly rigid if there exists an increasing sequence $\{n_i\}_{i\in \mathbb N}$ in $\mathbb N$ such that $\sup \nolimits _{x\in X} d(x,F^{n_i}x) \to 0$ as i goes to infinity (that is, $F^{n_i}$ converges uniformly to the identity map). Refer to [Reference Glasner and Maon18] for more information about topological rigidity.

Lemma 6.7. Let $\mathcal F$ be a finite subset of $H(X,X)$ , where $(X,\rho _X)$ is a compact metric space. Then for any $\epsilon>0$ , there is a ${\delta }>0$ such that for any continuous map $h: X\rightarrow X$ with $D_X(h,\mathrm {id}_X)<{\delta }$ , we have

$$ \begin{align*}D_X(\psi\circ h \circ \psi^{-1} , \mathrm{id}_X)<\epsilon \quad \text{for all } \psi \in \mathcal F.\end{align*} $$

Proof. For any $\epsilon>0$ , there exists $\delta>0$ such that whenever $\rho _X(x_1,x_2)<\delta ,$

$$ \begin{align*}\rho_X(\psi^{-1}(x_1),\quad\psi^{-1}(x_2)) <\epsilon \quad \text{for all } \psi\in \mathcal F.\end{align*} $$

Then for continuous map $h: X\rightarrow X$ with $D(h,\mathrm {id}_X)<\delta $ , we have that

$$ \begin{align*}\rho_X(h(\psi(x)),\quad\psi(x))<\delta \quad\text{for all } x\in X, \text{ for all } \psi\in \mathcal F.\end{align*} $$

It follows that

$$ \begin{align*}\rho_X(\psi ^{-1}\circ h\circ \psi (x),x)<\epsilon \quad \text{for all } x\in X, \text{ for all } \psi\in \mathcal F. \end{align*} $$

That is,

$$ \begin{align*}D_X(\psi\circ h \circ \psi^{-1} , \mathrm{id}_X)<\epsilon \quad \text{for all } \psi\in \mathcal F.\end{align*} $$

The proof is complete.

Proof. By (6.10), for any $({\alpha }, z_0)\in X=Z(2)\times {\mathbb S}^1 ,$ we get

$$ \begin{align*}&F^{m_k}_\Phi ({\alpha},z_0)\\&\quad=({\alpha}^1{\alpha}^2\ldots {\alpha}^k\tau({\alpha}^{k+1}{\alpha}^{k+2}\ldots), \psi({\alpha}^1{\alpha}^2\ldots {\alpha}^k)\varphi ^{e({\alpha}^{k+1})} _{k+1}\psi^{-1}({\alpha}^1{\alpha}^2\ldots {\alpha}^k)(z_0)),\end{align*} $$

where $\psi ({\alpha }^1{\alpha }^2\ldots {\alpha }^k)=\psi ^+_1\circ \psi ^+_2\circ \cdots \circ \psi ^+_k .$ By the assumption (C1) that

$$ \begin{align*}\lim_{k\to \infty} \max_{0\le j\le 2^{n_k}-2} D_{{\mathbb S}^1}(\varphi^{j}_k, \mathrm{id}_{{\mathbb S}^1}) = 0,\end{align*} $$

and Lemma 6.7, we have that

$$ \begin{align*}\lim_{k\to \infty} D_X(F_\Phi^{m_k},\mathrm{id}_X) =0 ,\end{align*} $$

that is, $(Z(2)\times {\mathbb S}^1,F_\Phi )$ is uniformly rigid.

6.6 Minimality

Now we specify the homeomorphisms $\{\varphi _{2k-1}^j: 1\le j\le 2^{n_{2k-1}}-2\}_{k=1}^\infty $ to make the flow $(Z(2)\times {\mathbb S}^1,F_\Phi )$ minimal. Let

(C4) $$ \begin{align} \varphi_{2k-1}^j(z)= \begin{cases} \exp{(i\theta_k)}\cdot z, & 1\le j \le 2^{n_{2k-1}-1}-1; \\ \exp{(-i\theta_k)}\cdot z, & 2^{n_{2k-1}-1}-1<j\le 2^{n_{2k-1}}-2 , \end{cases} \end{align} $$

where $\theta _k = {2\pi }/({2^{n_{2k-1}-1}-1})$ . That is, $\varphi ^{j}_{2k-1}$ is a rotation of ${\mathbb S}^1$ with angle $\theta _k$ in the anti-clockwise direction if $1\le j \le 2^{n_{2k-1}-1}-1$ and in the opposite direction otherwise. We remark that this construction satisfies the assumption (C3), that is, $\psi _{2k-1} =\mathrm {id}.$

Proof. Let ${\alpha }\in Z(2)$ and ${\mathbb S}^1_{{\alpha }}={{\alpha }}\times {\mathbb S}^1.$ First we show that for any $y_0\in {\mathbb S}^1$ ,

$$ \begin{align*}\overline{\mathcal{O}}((\underline 0,y_0),F_\Phi)=Z(2)\times {\mathbb S}^1.\end{align*} $$

By (6.6), for $1\le s<2^{n_{2k-1}}$ , one has that

$$ \begin{align*}F^{s\cdot m_{2k-2}}_\Phi(\underline 0,y_0)=(0^{n_1+n_2+\cdots n_{2k-2}}\underline s0^\infty ,y_{s\cdot m_{2k-2}}),\end{align*} $$

where $y_{s\cdot m_{2k-2}}=\varphi ^{s-1}_{2k-1}\circ \varphi ^{s-2}_{2k-1}\circ \cdots \circ \varphi ^1_{2k-1}\circ \varphi ^0_{2k-1}(y_0).$ As a result, $\{y_{s\cdot m_{2k-2}}: 1\le s \le 2^{n_{2k-1}-1}-1\}$ is $\theta _k = {2\pi }/({2^{n_{2k-1}-1}-1})-$ dense in ${\mathbb S}^1$ . (Recall the definition of $\epsilon $ -dense subsets: if S is a subset of a metric space, then B in S is $\epsilon $ -dense in S for given $\epsilon>0$ if for any $s\in S$ , there is $b\in B$ such that the distance between s and b is $<\epsilon $ .) Moreover, for any $y\in {\mathbb S}^1,$ there exists $1\le s_k\le 2^{n_{2k-1}-1}-1$ such that

$$ \begin{align*}\lim_{k\to \infty}y_{s_k\cdot m_{2k-2}}=y.\end{align*} $$

Thus,

$$ \begin{align*}F^{s_k\cdot m_{2k-2}}_\Phi(\underline 0 ,y_0)\to (\underline 0,y), k\to\infty.\end{align*} $$

Thus, ${\mathbb S}^1_{\underline 0} \subseteq \overline {\mathcal{O} }((\underline 0,y_0),F_\Phi ).$ Since $F^n_\Phi ({\mathbb S}^1_{\underline 0})={\mathbb S}^1_{\underline n},$ we have

$$ \begin{align*}Z(2)\times {\mathbb S}^1=\overline{\bigcup_{n\in \mathbb Z _{+}} {\mathbb S}^1_{\underline n}} \subseteq \overline{\mathcal{O}}((\underline 0,y_0),F_\Phi).\end{align*} $$

Thus,

$$ \begin{align*}\overline{\mathcal{O}}((\underline 0,y_0),F_\Phi)=Z(2)\times {\mathbb S}^1 \ \text{ for all } y_0\in {\mathbb S}^1.\end{align*} $$

Now we show that for any $({\alpha } ,z_0 )\in Z(2)\times {\mathbb S}^1,$ one has

$$ \begin{align*}\overline{\mathcal{O}} (({\alpha}, z_0),F_\Phi)=Z(2)\times {\mathbb S}^1.\end{align*} $$

As $(Z(2),\tau )$ is minimal, there exists some $\{p_k\}$ such that

$$ \begin{align*}\lim _{k\to \infty}\tau ^{p_k} ({\alpha})=\underline 0.\end{align*} $$

Without loss of generality, we may assume that

$$ \begin{align*}\lim_{k\to \infty} F^{p_{k}}_\Phi({\alpha},z_0 ) =(\underline 0, y_0)\end{align*} $$

for some $y_0 \in {\mathbb S}^1.$ Thus, $Z(2)\times {\mathbb S}^1=\overline {\mathcal{O} }( (\underline 0, y_0),F_\Phi ) \subseteq \overline {\mathcal{O} } (({\alpha } , z_0),F_\Phi ).$ Thus,

$$ \begin{align*}\mathcal{O} (({\alpha}, z_0),F_\Phi)=Z(2)\times {\mathbb S}^1.\end{align*} $$

Hence, $(Z(2)\times {\mathbb S}^1,F_\Phi )$ is minimal. Moreover, by Proposition 6.8, it is uniformly rigid. The proof is complete.

6.7 Proximality

In this subsection, we specify the homeomorphisms $\{\varphi _{2k}^j: 1\le j\le 2^{n_{2k}}-2\}_{k=1}^\infty $ to make the extension $\pi : (Z(2)\times {\mathbb S}^1, F_\Phi )\to (Z(2),\tau ), \ ({\alpha } , z_0)\mapsto {\alpha }$ to be proximal. Recall that ${\mathbb S}^1=\{\exp (i 2\pi x): 0\le x<1\}$ . Let $\varphi _{2k}^j: {\mathbb S}^1\rightarrow {\mathbb S}^1$ be defined as follows:

(C5) $$ \begin{align} \varphi_{2k}^j( \exp(i{2\pi x }))= \begin{cases} \exp(i{2\pi x^{t_k} }), & 1< j \le 2^{n_{2k}-1}-1; \\ \exp( {i2\pi x^{1/{t_k}}}), & 2^{n_{2k}-1}-1<j\le 2^{n_{2k}}-2 , \end{cases} \end{align} $$

where $\{t_k\}_{k\in \mathbb N}$ is a decreasing sequence such that $t_k>1$ , $\lim _{k\to \infty } t_k=1$ and $\lim _{k\to \infty } t_k^{2^{n_{2k}}}=+\infty $ . For example, $\{t_k=3^{2^{-{n_{2k}}/{2}}}\}_{ k\in \mathbb N}$ satisfies the condition. Notice that the construction of $\varphi _{2k} ^j$ also satisfies the convention (C3), that is, $\psi _{2k}=\mathrm {id}.$

To verify that $\pi $ is proximal, we begin with the following lemma.

Proof. Let $(x_1,x_2)\in R_\pi $ and $\pi (x_1)=\pi (x_2)=y_1 .$ Since $(Y,T)$ is minimal, we can find $p\in E(X,T) $ such that

$$ \begin{align*}py_1=y_0 .\end{align*} $$

So $px_1,px_2\in \pi ^{-1}(y_0).$ According to the assumption, there exists $q\in E(X,T)$ such that

$$ \begin{align*}qpx_1=qpx_2.\end{align*} $$

It follows that

$$ \begin{align*}(x_1,x_2)\in P(X,T).\end{align*} $$

Thus, $\pi $ is proximal.

Proof. First notice that (C4) and (C5) imply (C1) and (C3). It follows that $(Z(2)\times {\mathbb S}^1, F_\Phi )$ is uniformly rigid and minimal by Proposition 6.9.

By Lemma 6.10, it suffices to show that for arbitrary $z_0^{(1)}$ , $z_0^{(2)} \in {\mathbb S}^1$ , one has

$$ \begin{align*}((\underline 0 ,z_0^{(1)}),(\underline 0 ,z_0^{(2)}))\in P(Z(2)\times {\mathbb S}^1,F_\Phi).\end{align*} $$

Let $z_0^{(1)}=\exp {(i{2\pi x_1 })}$ , $z_0^{(2)} =\exp {(i{2\pi x_2 })}$ , where $0\le x_1,x_2<1$ . Let $d_{{\mathbb S}^1}$ be the metric of ${\mathbb S}^1$ . Let $F^{m}_\Phi ((\underline 0 ,z_0^{(j)}))=(\underline 0 ,z_m^{(j)})$ for $j=1,2$ and $m\in \mathbb N$ . Let $m_k=2^{n_1+n_2+\cdots +n_k}$ for all $k\in \mathbb N$ . Then by Lemma 6.4,

$$ \begin{align*} \lim _{k\to \infty}\kern-1pt d_{{\mathbb S}^1}\kern-0.7pt(z^{(1)}_{2^{n_{2k}-1}\cdot m_{2k-1}}\kern-1pt,z^{(2)}_{2^{n_{2k}-1}\cdot m_{2k-1}}\kern-1.3pt)\kern1.2pt{=}\kern1.2pt\lim _{k\to \infty} \kern-2pt d_{{\mathbb S}^1} (\exp(i{2\pi x_1^{t_k^{(2^{n_{2k}-1}-1)}} }\kern-1pt), \exp(i{2\pi x_2^{t_k^{(2^{n_{2k}-1}-1)}} }\kern-1pt)\kern-0.3pt). \end{align*} $$

Since $\lim _{k\to \infty } t_k^{2^{n_{2k}}}\kern1.2pt{=}+\infty $ and $0\kern1.2pt{\le}\kern1.2pt x_j\kern1.2pt{<}\kern1.2pt1, j\kern1.2pt{=}\kern1.2pt1,2$ , we have that ${\lim _{k\to \infty } x_j^{t_k^{(2^{n_{2k}-1}-1)}}\kern1pt{=}\kern1.5pt0}$ for $j=1,2$ . Thus,

$$ \begin{align*} \begin{split} & \lim _{k\to \infty}d_{{\mathbb S}^1}(z^{(1)}_{2^{n_{2k}-1}\cdot m_{2k-1}},z^{(2)}_{2^{n_{2k}-1}\cdot m_{2k-1}})\\ &\quad =\lim _{k\to \infty} d_{{\mathbb S}^1}\kern-2pt (\exp(i{2\pi x_1^{t_k^{(2^{n_{2k}-1}-1)}} }), \exp(i{2\pi x_2^{t_k^{(2^{n_{2k}-1}-1)}} }))\\ &\quad=d_{{\mathbb S}^1}(1,1)=0. \end{split} \end{align*} $$

In particular,

$$ \begin{align*}\lim_{k\to\infty} \rho_{X}((F_\Phi\times F_\Phi)^{2^{n_{2k}-1}\cdot m_{2k-1}}((\underline 0 ,z_0^{(1)}),(\underline 0 ,z_0^{(2)})))=0.\end{align*} $$

It follows that $((\underline 0 ,z_0^{(1)}),(\underline 0 ,z_0^{(2)}))\in P(Z(2)\times {\mathbb S}^1,F_\Phi )$ .

6.8 Proximal extensions and n-weakly mixing extensions

In §4.1, we mentioned that the fact that $\pi : (X,T)\rightarrow (Y,T)$ is weakly mixing does not imply that $\pi $ is totally weakly mixing. In this subsection, we give such examples. The main result of this subsection is the following theorem.

Proof of (1) of Theorem 6.12

Suppose that $\Phi $ satisfies (C2), (C4), and (C5). We show that the extension

$$ \begin{align*}\pi: (Z(2)\times {\mathbb S}^1, F_\Phi )\to (Z(2),\tau), \ ({\alpha} , z)\mapsto {\alpha}\end{align*} $$

is weakly mixing but not $3$ -weakly mixing. Since $\pi $ is open proximal by Proposition 6.11, it is weakly mixing by Theorem 4.6. Now we show that $\pi $ is not $3$ -weakly mixing, that is, $(R_\pi ^3,F_\Phi ^{(3)})$ is not transitive.

We may regard $R_\pi ^3$ as $Z(2)\times {\mathbb S}^1 \times {\mathbb S}^1\times {\mathbb S}^1.$ Suppose $(R_\pi ^3,F_\Phi ^{(3)})$ is transitive, and let $({\alpha },x_1,x_2,x_3)\in Z(2)\times {\mathbb S}^1 \times {\mathbb S}^1\times {\mathbb S}^1$ be a transitive point. Let $y_1, y_2,y_3$ be distinct points of ${\mathbb S}^1$ and $\beta \in Z(2)$ . Then $(\beta , y_1,y_2,y_3)$ is in the orbit closure of the transitive point $({\alpha },x_1,x_2,x_3)$ . By the construction of $\Phi $ ((C4) and (C5)), $F_\Phi $ preserves orientation, that is, for all ${\alpha }\in Z(2)$ , $F_\Phi $ maps $\{{\alpha }\}\times {\mathbb S}^1$ to $\{\tau {\alpha }\}\times {\mathbb S}^1$ such that it sends the unit circle with the anti-clockwise orientation into the unit circle with the anti-clockwise orientation. It follows that $(\beta , y_1,y_3,y_2)$ can not be in the orbit closure of the transitive point $({\alpha },x_1,x_2,x_3)$ . Thus, $(R_\pi ^3,F_\Phi ^{(3)}) $ is not transitive.

For the proof of (2) of Theorem 6.12, we need some preparations. We use the notation in §6.1.

By the same proof of [Reference Glasner and Weiss19, Theorem 4], we can show the following: if $\mathcal {G}$ is a pathwise connected subgroup of $H(Z,Z)$ such that $(Z^n,\mathcal {G})$ is a transitive flow, then for a residual subset $\mathcal {R}_n \subseteq \overline {\mathcal {S}_{\mathcal {G}}(f)}$ , $(X,F)$ is an n-weakly mixing extension of $(Y,f)$ for every $F\in \mathcal {R}_n$ . Let ${\mathcal R}=\bigcap _{n=2}^\infty {\mathcal R}_n$ . Then we have the following result which slightly generalizes Theorem 6.13.

According to [Reference Glasner and Weiss19], if we choose $Z=\mathbf {P}^n, n\ge 2$ to be the projective n-space, and let $\mathcal {G}$ be a pathwise connected component of $\mathrm {id}_Z$ in $H(Z,Z)$ , then $\mathcal {G}$ satisfies the conditions of Theorems 6.1, 6.2, and 6.14. Thus, for an arbitrary minimal infinite flow $(Y,f)$ , there are many minimal homeomorphisms of $Y\times Z$ which are proximal and totally weakly mixing extensions of $(Y,f)$ . And we have proved (2) of Theorem 6.12.

6.9 Almost proximal

In this subsection, we modify the construction of homeomorphisms $\{\varphi _{2k}^j: 1\le j\le 2^{n_{2k}}-2\}_{k=1}^\infty $ to make the extension

$$ \begin{align*}\pi: (Z(2)\times {\mathbb S}^1, F_\Phi )\to (Z(2),\tau), \quad ({\alpha} , z_0)\mapsto {\alpha}\end{align*} $$

almost proximal.

Let $n\in \mathbb N $ be a fixed number. Let $\omega =\exp {(i({2\pi }/{n}))}$ . We always write intervals on ${\mathbb S}^1$ anti-clockwise, so $[z_1,z_2]$ denotes the anti-clockwise closed interval beginning at $z_1$ and ending at $z_2$ :

$$ \begin{align*}S_q=[\omega^{q-1}, \omega^q]=\bigg[\exp{\bigg(i\frac{2\pi}{n} (q-1) \bigg)}, \exp{\bigg(i\frac{2\pi}{n} q \bigg)}\bigg], \quad q\in\{1,2,\ldots, n\}.\end{align*} $$

Then

$$ \begin{align*}{\mathbb S}^1=S_1\cup S_2\cup\cdots \cup S_n.\end{align*} $$

Thus, ${\mathbb S}^1$ is divided into n closed intervals equally. For each $q\in \{1,2,\ldots ,n\}$ , we may regard $S_q$ as $[0,1]$ (via the map $S_q=[\exp {(i({2\pi }/{n}) (q-1) )}, \exp {(i({2\pi }/{n}) q )}]\rightarrow [0,1], \exp {(i2\pi x)}\mapsto nx-(q-1)$ , where $(({q-1})/{n})\le x\le {q}/{n}$ ) and let the map $\varphi _{2k}^j|_{S_q}: S_q\rightarrow S_q$ be isomorphic to $g_k(x)=x^{t_k}: [0,1]\rightarrow [0,1]$ when $1\le j \le 2^{n_{2k}-1}-1$ ; and $g_k^{-1}(x)=x^{1/t_k}: [0,1]\rightarrow [0,1] $ when $2^{n_{2k}-1}-1<j\le 2^{n_{2k}}-2$ , where $\{t_k\}_{k\in \mathbb N}$ is a decreasing sequence tending to $1$ and $\lim _{k\to \infty } t_k^{2^{n_{2k}}}=+\infty $ .

To be precise, for each $q\in \{1,2,\ldots ,n\}$ , when $z=\exp {(i2\pi x)}\in S_q$ , $({q-1})/{n}\le x\le {q}/{n}$ , $\varphi _{2k}^j: S_q\rightarrow S_q$ is defined as follows:

(C6) $$ \begin{align} \varphi_{2k}^j(z)&= \varphi_{2k}^j(\exp{(i2\pi x)})\nonumber\\&= \begin{cases} \exp\bigg({i2\pi \dfrac{(nx-(q-1))^{t_k}+(q-1)}{n}}\bigg), & 1\le j \le 2^{n_{2k}-1}-1; \\ \exp\bigg( {i2\pi \dfrac{(nx-(q-1))^{1/t_k}+(q-1)}{n}}\bigg), & 2^{n_{2k}-1}-1< j\le 2^{n_{2k}}-2 , \end{cases} \end{align} $$

where $\{t_k\}$ is a decreasing sequence tending to $1$ and $\lim _{k\to \infty } t_k^{2^{n_{2k}}}=+\infty $ . Thus, for each $q\in \{1,2,\ldots , n\}$ , $\varphi _{2k}^j|_{S_q}: S_q\rightarrow S_q$ has exactly two fixed points. Notice that the construction of $\varphi _{2k} ^j$ also satisfies the convention (C3), that is, $\psi _{2k}=\mathrm {id} .$

Note that when $n=1, S_1$ is exactly ${\mathbb S}^1$ and (C6) coincides with (C5).

Proposition 6.16. Suppose that $\Phi $ satisfies (C2), (C4), and (C6). Then the extension

$$ \begin{align*}\pi: (Z(2)\times {\mathbb S}^1, F_\Phi )\to (Z(2),\tau), \quad ({\alpha} , z)\mapsto {\alpha}\end{align*} $$

is an almost proximal extension of uniformly rigid minimal flows, and the cardinality of maximal almost periodic set in each fiber is n.

Proof. First notice that (C4) and (C6) imply (C1) and (C3). It follows that $(Z(2)\times {\mathbb S}^1, F_\Phi )$ is uniformly rigid and minimal by Proposition 6.9.

Let ${\alpha }\in Z(2)$ and $z_0\in {\mathbb S}^1$ . Let

$$ \begin{align*}x_j=\bigg({\alpha}, z_0 \exp{\bigg(i\frac{2\pi}{n}(j-1)\bigg)}\bigg)=({\alpha}, z_0\omega^{j-1})\in \pi^{-1}{\alpha}),\quad 1\le j\le n.\end{align*} $$

First we show that $\{x_1, x_2,\ldots , x_n\}$ is an almost periodic set. To attain that aim, we show $(x_1,x_2,\ldots , x_n)$ is a minimal point of $(X^n, F_\Phi ^{(n)})$ , where $F^{(n)}_\Phi =F_\Phi \times \cdots \times F_\Phi $ (n times).

By the conditions (C4) and (C6), we have

(6.13) $$ \begin{align} \rho_X(F_\Phi(x_{j}), F_\Phi(x_{j+1}))=\rho_X(x_j, x_{j+1})=\frac{2\pi}{n}, \quad 1\le j\le n-1. \end{align} $$

Let $(y_1,y_2,\ldots , y_n)$ be a minimal point of $\overline {\mathcal{O} }((x_1,x_2,\ldots , x_n), F_\Phi ^{(n)})$ with $y_1=x_1$ . (First we choose any minimal point $(y_1', y_2',\ldots , y_n')$ in $\overline {\mathcal{O} }((x_1,x_2,\ldots , x_n), F_\Phi ^{(n)})$ . Since $(X,F_\Phi )$ is minimal, there is some $p\in \mathbf {M}$ such that $py_1'=x_1$ . Let $(y_1,y_2,\ldots , y_n)=p(y_1', y_2',\ldots , y_n')$ . Then $(y_1,y_2,\ldots , y_n)$ is a minimal point of $\overline {\mathcal{O} }((x_1,x_2,\ldots , x_n), F_\Phi ^{(n)})$ with $y_1=x_1$ .) By (6.13),

$$ \begin{align*}\rho_X(y_j,y_{j+1})=\rho_X(x_j, x_{j+1})=\frac{2\pi}{n}, \quad 1\le j\le n-1.\end{align*} $$

Since $F_\Phi $ preserves orientation, we have

$$ \begin{align*}[x_1,x_2]=[y_1, y_2]=[x_1,y_2],\end{align*} $$

and hence $x_2=y_2$ . By the same reason, we have $x_j=y_j$ for $3\le j\le n$ . Thus, $(x_1,x_2,\ldots , x_n)=(y_1,y_2,\ldots , y_n)$ is minimal.

Next we show that for each ${\alpha }\in Z(2), z_1,z_2\in {\mathbb S}^1$ , if $\rho _X(({\alpha },z_1),({\alpha },z_2))<{2\pi }/{n}$ , then $({\alpha },z_1),({\alpha },z_2)$ are proximal. Without loss of generality, we may assume that ${\alpha }=\underline {0}$ and $z_1,z_2\in [1,\omega ).$ By (C6), for $z=\exp {(i2\pi x)}\in S_1=[1,\omega ]$ , ( $0\le x\le {1}/{n}$ )

$$ \begin{align*} \varphi_{2k}^j(z)= \varphi_{2k}^j(\exp{(i2\pi x)})= \begin{cases} \exp\bigg({i2\pi \dfrac{(nx)^{t_k}}{n}}\bigg), & 1\leq j \le 2^{n_{2k}-1}-1; \\ \exp\bigg( {i2\pi \dfrac{(nx)^{1/{t_k}}}{n}}\bigg), & 2^{n_{2k}-1}-1<j\le 2^{n_{2k}}-2. \end{cases} \end{align*} $$

Let $m_k$ and $y_m$ be defined as in Lemma 6.4. According to (6.6), for $1\le s<2^{n_{2k}}$ ,

$$ \begin{align*}F^{s\cdot m_{2k-1}}_\Phi(\underline 0,y_0)=(0^{n_1+n_2+\cdots n_{2k-1}}\underline s0^\infty ,y_{s\cdot m_{2k-1}}),\end{align*} $$

where $y_{s\cdot m_{2k-1}}=\varphi ^{s-1}_{2k}\circ \varphi ^{s-2}_{2k}\circ \cdots \circ \varphi ^1_{2k}\circ \varphi ^0_{2k}(y_0).$ Let $z_1=\exp {(i2\pi a_1)}, z_2=\exp {(i2\pi a_2)}$ and $s=2^{n_{2k}-1}$ , where $0\le a_1,a_2<{1}/{n}$ . Then for $j=1,2$ , by conditions on $\{t_k\}_{k\in \mathbb N}$ and $na_j<1$ , we have that $\lim _{k\to \infty }(na_j)^{t_k^{2^{n_{2k}-1}}}=0$ , and

$$ \begin{align*}F^{sm_{2k-1}}_\Phi (\underline{0},z_j)=\bigg(0^{n_1+n_2+\cdots n_{2k-1}}\underline s0^\infty, \exp\bigg({i2\pi \frac{(na_j)^{t_k^{2^{n_{2k}-1}}}}{n}}\bigg)\bigg)\to (\underline{0}, 1),\quad k\to\infty. \end{align*} $$

Thus,

$$ \begin{align*}((\underline 0,z_1),(\underline 0,z_2))\in P(Z(2)\times {\mathbb S}^1,F_\Phi).\end{align*} $$

To sum up, we have showed that for ${\alpha }\in Z(2)$ , $A\subseteq \pi ^{-1}({\alpha })$ is an almost periodic set with maximal cardinality if and only if $A=\{({\alpha }, z_0 \exp {(i({2\pi }/{n})(j-1))})=({\alpha }, z_0\omega ^{j-1}): 1\le j\le n\}$ for some $z_0\in {\mathbb S} ^1$ . The proof is complete.