May 28, 2026
Motivated by Berg’s notion of quasi-disjointness for ergodic systems, we introduce and investigate the concept of quasi-disjointness for minimal systems. Several equivalent characterizations are provided. We prove that quasi-disjointness is preserved under taking factors, proximal extensions, and group extensions. As a consequence, we establish that every minimal PI system is quasi-disjoint from all minimal systems. In addition, some variant of quasi-disjointness, namely strong quasi-disjointness is also introduced and examined. Particularly, we prove that each AI system is strongly quasi-disjoint from all minimal systems.
A topological dynamical system (or simply a system) is a pair \((X,T)\), where \(X\) is a compact metric space and \(T\) is a homeomorphism on \(X\). Given two systems \((X,T)\) and \((Y,S)\), a joining of \((X,T)\) and \((Y,S)\) is an invariant closed subset of \(X\times Y\) that has full projections onto both \(X\) and \(Y\). If \(X\times Y\) is the only joining of \((X,T)\) and \((Y,S)\), then we say that they are disjoint, denoted by \(X\perp Y\).
The notion of disjointness was introduced simultaneously in ergodic theory and topological dynamics by Furstenberg in his seminal paper [1]. In a sense, disjointness captures a certain form of independence between two systems. In [1], two main open questions regarding disjointness were raised: one asks for a characterization of systems that are disjoint from all minimal systems, and the other for a characterization of those that are disjoint from all distal systems. The second question was resolved by Peterson [2], who showed that the class of systems disjoint from all distal systems consists precisely of the minimal weakly mixing systems. Research on the first question has since attracted considerable effort; see for example, [3]–[7] and the references therein. Recently, Górska, Lemańczyk and de la Rue gave a characterization of measure preserving systems that are disjoint from all ergodic systems [8], a result later reproved in a shorter way by Glasner and Weiss [9]. In the work [10], the authors provide several intrinsic characterizations of topological dynamical systems that are disjoint from all minimal systems.
Perhaps motivated by Furstenberg’s second question, Berg introduced the notion of quasi-disjointness among ergodic systems in [11] and further properties of quasi-disjointness are obtained in [12]. Recently, Moreira, Richter and Robertson generalize Berg’s definition of quasi-disjointness to ergodic systems acted by countable groups [13]. In this paper, we study quasi-disjointness in topological dynamical systems.
For a minimal system \((X,T)\), let \(X_{eq}\) denote its maximal equicontinuous factor. For two minimal systems \((X,T)\) and \((Y,S)\), let \(Eq(X,Y)\) denote the common maximal equicontinuous factor of \((X,T)\) and \((Y,S)\) (see Definition 20). Let \(\pi_{X}: X\rightarrow X_{eq}\) and \(\pi_{Y}: Y\rightarrow Y_{eq}\) be the factor maps. We say that two minimal systems \((X,T)\) and \((Y,S)\) are quasi-disjoint if \(X\times Y\) is the only joining \(J\) of \(X\) and \(Y\) satisfying \(\pi_{X}\times \pi_{Y}(J)=X_{eq}\times Y_{eq}\), denoted by \(X\perp_{Q} Y\). This definition is motivated by notion given in [13] for measure preserving systems.
A related notion of quasi-disjointness is a counterpart of Berg’s notion. Let \(\alpha: X\rightarrow Eq(X,Y), \beta: Y\rightarrow Eq(X,Y)\) the factor maps and \(\gamma: X\times Y\rightarrow Eq(X,Y), (x,y)\mapsto \alpha(x)-\beta(y)\). In [11], Berg suggested that \((X,T)\) and \((Y,S)\) are quasi-disjoint if there is a residual set \(\Omega\subset Eq(X,Y)\) such that \(\gamma^{-1}(z)\) is a minimal subset for each \(z\in \Omega\). In this case, we say that \(X\) is strongly quasi-disjoint from \(Y\), denoted by \(X\perp_{SQ} Y\). We will see that strong quasi-disjointness implies quasi-disjointness. For an equivalent characterization, we have the following result.
Theorem 1. Let \((X,T)\) and \((Y,S)\) be minimal systems. Let \(\alpha: X\rightarrow Eq(X,Y), \beta: Y\rightarrow Eq(X,Y)\) be the factor maps and \(\gamma(x,y)=\alpha(x)-\beta(y)\). Then the following assertions are equivalent:
There is some point \(z\in Eq(X,Y)\) such that \(\gamma^{-1}(z)\) has a unique minimal subset.
There is a dense \(G_{\delta}\) subset \(\Omega\subset Eq(X,Y)\) such that for each \(z\in \Omega\), \(\gamma^{-1}(z)\) has a unique minimal subset.
\(X\perp_{Q} Y\).
Similar characterization of strong quasi-disjointness is given in Theorem 28, i.e. for minimal systems \((X,T)\) and \((Y,S)\), \(X\perp_{SQ} Y\) if and only if there is some point \(z\in Eq(X,Y)\) such that \(\gamma^{-1}(z)\) is a minimal subset.
We will show that (strong) quasi-disjointness is preserved by taking factors.
Theorem 2. Let \((X,T), (Y,S)\) be minimal systems and \(\pi: (X,T)\rightarrow (Z,R)\) be a factor map.
If \(X\perp_{Q} Y\) then \(Z\perp_{Q} Y\);
If \(X\perp_{SQ} Y\) then \(Z\perp_{SQ} Y\).
Further, we will show that quasi-disjointness is preserved by equicontinuous extensions, proximal extensions and taking inverse limits. Consequently, it turns out that minimal PI systems (see Section 2 for the definition) are quasi-disjoint from all minimal systems. Moreover, we prove that minimal AI systems (see Section 2 for the definition) are strongly quasi-disjoint from all minimal systems by using different approach. That is, we have
Theorem 3. Every minimal PI system is quasi-disjoint from all minimal systems and every minimal AI system is strongly quasi-disjoint from all minimal systems.
We remark that it remains an open question if quasi-disjointness and strong quasi-disjointness are the same property, see the last section for a discussion.
In section 2, we give some notions and lemmas used later. In section 3, we establish the relation between quasi-disjointness and strong quasi-disjointness and show Theorem 1. In section 4, we show that quasi-disjointness is preserved by proximal extensions and taking factors and show Theorem 2. In section 5, we show that quasi-disjointness is preserved by equicontinuous extensions. In section 6, we study systems (strong) quasi-disjoint from all minimal systems and show Theorem 3. In section 7, we give some remarks and ask some questions.
We would like to thank Wen Huang and Song Shao for useful discussions.
A topological dynamical system (or system for short) is a pair \((X,T)\), where \(X\) is a compact metric space and \(T: X\rightarrow X\) is a homeomorphism on \(X\). Throughout the paper, we use \(\rho_{X}\) (or \(\rho\) when there is no risk of ambiguity) to denote the metric on \(X\).
Let \((X,T)\) be a system and \(x\in X\). The orbit of \(x\) is \(\{T^{n}x: n\in\mathbb{Z}\}\), which is denoted by \(orb_{T}(x)\) or \(orb(x)\). A system is topologically transitive (or transitive for short) if there is some point whose orbit is dense and such a point is called a transitive point. A system is minimal if the orbit of every point is dense. A point \(x\in X\) is a minimal point if the restriction of \(T\) on \(\overline{orb_{T}(x)}\) is a minimal subsystem.
Let \((X,T)\) and \((Y,S)\) be two systems. If there is a continuous onto map \(\pi: X\rightarrow Y\) such that \(\pi\circ T=S\circ\pi\), then we say that \((X,T)\) is an extension of \((Y,S)\) and \((Y,S)\) is a factor of \((X,T)\). In this case, we also say that \(\pi\) is a factor map or a homomorphism. Further, if \(\pi\) is one to one, then we say that \((X,T)\) and \((Y,S)\) are topologically conjugate or isomorphic.
Let \((X,T)\) be a system. A pair \((x,y)\) of points in \(X\) is said to be proximal if there is a subsequence \((n_i)\) in \(\mathbb{Z}\) such that \(\rho(T^{n_i}x, T^{n_i}y)\rightarrow 0\) as \(i\rightarrow \infty\). We use \({\boldsymbol{P}}(X)\) to denote the set of proximal pairs in \(X\). The system \((X,T)\) is distal if every pair of distinct points \(x,y\in X\) are not proximal. \((X,T)\) is equicontinuous if for every \(\epsilon>0\) there is some \(\delta>0\) such that \(\rho(T^{n}x, T^{n}y)<\epsilon\) for every \(n\in\mathbb{Z}\) whenever \(\rho(x,y)<\delta\).
Let \(\pi: (X,T)\rightarrow (Y,S)\) be an extension between two systems. If for every \(x_1,x_2\in X\) with \(\pi(x_1)=\pi(x_2)\) are proximal then we say that \(\pi\) is a proximal extension. If for every \(x_1\neq x_2\in X\) with \(\pi(x_1)=\pi(x_2)\) are not proximal then we say that \(\pi\) is a distal extension. If for every \(\epsilon>0\) there is some \(\delta>0\) such that for every \(x_1,x_2\in X\) with \(\pi(x_1)=\pi(x_2)\), one has \(\rho(T^{n}x_1, T^{n}x_2)<\epsilon\) for every \(n\in \mathbb{Z}\), then we say that \(\pi\) is an equicontinuous extension. If the set \(\{x\in X: |\pi^{-1}\pi(x)|=1\}\) is residual in \(X\), then we say that \(\pi\) is an almost one to one extension.
Let \(\pi: (X,T)\rightarrow (Y,S)\) be an extension between minimal systems. Suppose that there is a countable ordinal \(\eta\) and a family of systems and homomorphisms \(\{\pi_{\alpha\beta}:(X_{\alpha}, T_{\alpha})\rightarrow (X_{\beta}, T_{\beta}): \beta<\alpha\leq \eta\}\) such that
\(Y=X_{0}, X=X_{\eta}, \pi=\pi_{\eta 0}\),
if \(\gamma<\beta<\alpha\leq \eta\) then \(\pi_{\alpha\gamma}=\pi_{\beta\gamma}\pi_{\alpha\beta}\),
if \(\alpha\leq \eta\) is a limit ordinal, then \(X_{\alpha}=\underset{\longleftarrow}{\lim}_{\beta<\alpha}X_{\beta}\).
We say that \(\pi\) is
an I-extension if \(\pi_{\alpha+1,\alpha}: X_{\alpha+1}\rightarrow X_{\alpha}\) is an equicontinuous extension for each \(\alpha<\eta\);
an AI-extension if \(\pi_{\alpha+1,\alpha}: X_{\alpha+1}\rightarrow X_{\alpha}\) is an equicontinuous or almost one to one extension for each \(\alpha<\eta\);
a strictly PI-extension if \(\pi_{\alpha+1,\alpha}: X_{\alpha+1}\rightarrow X_{\alpha}\) is an equicontinuous or a proximal extension for each \(\alpha<\eta\).
A minimal system \((X,T)\) is a PI system if there is a minimal system \((\tilde{X}, T)\) and a proximal extension \(\theta: \tilde{X}\rightarrow X\) such that \(\tilde{\pi}: \tilde{X}\rightarrow Y=\{pt\}\) is a strictly PI-extension ([14]).
It follows from Furstenberg’s structure theorem of minimal distal systems that a minimal system is distal if and only if it is an \({\boldsymbol{I}}\)-extension of a trivial system ([15]). A minimal system \((X,T)\) is point-distal if there is a point \(x\in X\) such that \((x,y)\) is not proximal for every \(y\in X\) with \(y\neq x\). Veech showed in [16] that a minimal system is point-distal if and only if it is an AI-extension of a trivial system.
Let \((X,T)\) be a system. We say \((x,y)\) is proximal if \(\inf_{n\in{\mathbb{Z}}} \rho(T^nx,T^ny)=0\), and \(x\in X\) is a distal point if \(x\) is only proximal to itself. Veech [16] showed that if a minimal system is AI then set of distal points is residual in \(X\). Moreover, if \(x\) is a distal point then \((x,y)\) is a minimal point of \(X\times Y\) for any \(y\) in a minimal system, see [15]. For more on the structure of minimal systems, one may see [17], [18].
A continuous map \(\pi: X\rightarrow Y\) between two topological spaces is semiopen if for every nonempty open subset \(U\) of \(X\), the interior of \(\pi(U)\) is not empty.
The following lemmas are well-known, which will be used frequently.
Lemma 4. [19]If \(\pi:(X,T)\rightarrow (Y,S)\) is a homomorphism between minimal systems, then \(\pi\) is semiopen.
Lemma 5. [19]If \(\pi:(X,T)\rightarrow (Y,S)\) is a homomorphism between minimal distal systems, then \(\pi\) is open.
Lemma 6. Let \(\pi:X \rightarrow Y\) be a semiopen factor map between two systems \((X,T)\) and \((Y,S)\). If \(\Omega\) is a residual subset of \(Y\), then \(\pi^{-1}\Omega\) is residual in \(X\).
Proof. The semiopeness of \(\pi\) implies that if \(D\subset Y\) is a dense set then \(\pi^{-1}D\) is dense in \(X\). Then the lemma follows. ◻
The following lemma was established in [16] for the factor map of minimal systems. We emphasize that Veech’s proof relied solely on the semi-openness of the map. So the similar arguments yields that the lemma holds true for semiopen maps.
Lemma 7. Let \(\pi: X\rightarrow Y\) be a semiopen factor map between two systems \((X,T)\) and \((Y,S)\). If \(\Sigma\) is a residual subset of \(X\), then there is a residual subset \(\Omega\) of \(Y\) such that \(\pi^{-1}(y)\cap \Sigma\) is residual in \(\pi^{-1}(y)\) for each \(y\in \Omega\).
Let \(\pi: X\rightarrow Y\) be a continuous onto map between compact metric spaces. A point \(x\in X\) is called an open point of \(\pi\) if for any neighborhood \(U\) of \(x\), \(\pi(x)\) is an interior point of \(\pi(U)\). The following result is well-known.
Lemma 8. Let \(\pi: X\rightarrow Y\) be a continuous onto map between compact metric spaces. If \(\pi\) is semiopen then the set of open points of \(\pi\) is residual in \(X\).
Lemma 9. [20]Let \(X,Y,Z\) be compact metric spaces and \(\phi: X\rightarrow Y, \psi: Y\rightarrow Z\) be continuous onto maps. If both \(\phi\) and \(\psi\) are semiopen, then there is a residual subset \(\Omega\) of \(Z\) such that for each \(z\in \Omega\), the restriction \(\phi: \phi^{-1}\psi^{-1}(z)\rightarrow \psi^{-1}(z)\) is semiopen.
Proof. Let \(\Sigma\) be the set of open points of \(\phi\). By Lemma 8, \(\Sigma\) is residual in \(X\). By Lemma 7, there is residual subset \(\Omega\) of \(Z\) such that \(\phi^{-1}\psi^{-1}(z)\cap \Sigma\) is residual in \(\phi^{-1}\psi^{-1}(z)\) for each \(z\in \Omega\). We claim that the restriction \(\phi: \phi^{-1}\psi^{-1}(z)\rightarrow \psi^{-1}(z)\) is semiopen for each \(z\in \Omega\). For this, we fix some \(z\in \Omega\). Let \(U\) be a nonempty open subset of \(X\) with \(U\cap \phi^{-1}\psi^{-1}(z)\neq\emptyset\). Since \(\phi^{-1}\psi^{-1}(z)\cap \Sigma\) is residual in \(\phi^{-1}\psi^{-1}(z)\), there is some \(x\in U\cap \phi^{-1}\psi^{-1}(z)\cap \Sigma\). Recall that \(x\) is an open point of \(\phi\). There is an open neighborhood \(V\) of \(\phi(x)\) in \(Y\) such that \(V\subset \phi(U)\). Thus \[\phi(x)\in V\cap \psi^{-1}(z)\subset \phi(U)\cap \psi^{-1}(z)=\phi(U\cap \phi^{-1}\psi^{-1}(z) ).\] This implies that \(x\) is an interior point of \(\phi(U\cap \phi^{-1}\psi^{-1}(z) )\) under the relative topology of \(\psi^{-1}(z)\). Since \(U\) is chosen arbitrarily, we conclude that the restriction \(\phi: \phi^{-1}\psi^{-1}(z)\rightarrow \psi^{-1}(z)\) is semiopen. ◻
Let \((X,T)\) be a system. A pair \((x,y)\in X\times X\) is regionally proximal if there are sequences \((x_i), (y_i)\) in \(X\) and a sequence \((n_i)\) in \(\mathbb{Z}\) such that \[x_i\rightarrow x, \;\;y_i\rightarrow y \;\text{ and }\;\rho(T^{n_i}x_i, T^{n_i}y_i)\rightarrow 0, \text{ as } i\rightarrow\infty.\] Let \({\boldsymbol{R}P}(X)\) denote the set of regionally proximal pairs. For \(x\in X\), the regionally proximal cell of \(x\) is \(\{y\in X: (x,y)\in{\boldsymbol{R}P}(X)\}\), which is denoted by \({\boldsymbol{R}P}[x]\).
For a minimal system \((X,T)\), it is known that \({\boldsymbol{R}P}(X)\) is a closed invariant equivalence relation ([21]). Further, the quotient \(X_{eq}:=X/{\boldsymbol{R}P}(X)\) is the maximal equicontinuous factor of \(X\), which means that if \(\pi: X\rightarrow Y\) is factor and \((Y,S)\) is equicontinuous then \(Y\) is a factor of \(X_{eq}\) such that the following commuting diagram holds. \[\begin{tikzcd} X \arrow[d, "\pi" '] \arrow[r, "\pi_{X}"] & X_{eq} \arrow[ld, "\phi"] \\ Y & \end{tikzcd}\] Regionally proximal relations have the following lifting property.
Lemma 10. [22] Let \(\pi: (X,T)\rightarrow (Y,S)\) be an extension between minimal systems. Then one has \(\pi\times\pi({\boldsymbol{R}P(X)})={\boldsymbol{R}P}(Y)\).
The following characterization of regionally proximal relation shown by Veech will used later.
Lemma 11. [21]Let \((X,T)\) be a minimal system. Then \((x,y)\in {\boldsymbol{R}P}(X)\) if and only if there is a sequence \((n_i)\) in \(\mathbb{Z}\) and \(z\in X\) such that \[T^{n_i}x\rightarrow z \;\;\text{and }\;\;T^{-n_i}z\rightarrow y.\]
This leads to the following lifting property.
Lemma 12. Let \(\pi:(X,T)\rightarrow(Y,T)\) be an extension between minimal systems. Then for any \((y,y')\in{\boldsymbol{R}P}(Y)\) and \(x\in \pi^{-1}(y)\), there is some \(x'\in \pi^{-1}(y')\) such that \((x,x')\in{\boldsymbol{R}P}(X)\).
Proof. By Lemma 11, there is a sequence \((n_i)\) in \(\mathbb{Z}\) and \(y^{*}\in Y\) such that \[T^{n_i}y\rightarrow y^* \;\;\text{and }\;\;T^{-n_i}y^*\rightarrow y'.\] By passing to some subsequence, we may assume that \(T^{n_i}x\rightarrow x^{*}\in X\). Then \(x^{*}\in\pi^{-1}(y^*)\). Further, we may assume that \(T^{-n_i}x^*\rightarrow x'\in X\). Then \(x'\in\pi^{-1}(y')\). Using Lemma 11 again, we conclude that \((x,x')\in{\boldsymbol{R}P}(X)\). ◻
Let \((X,T)\) and \((Y,S)\) be minimal systems. Let \(\lambda\) be an invariant measure on \(Y\) and \(N\) be a closed invariant subset of \(X\times Y\). For \(x\in X\), let \(N[x]=\{y\in Y:\;(x,y)\in N\}\). Then the following lemma holds (see [19]).
Lemma 13.
For any \(x,x'\in X\), \(\lambda(N[x])=\lambda(N[x'])\).
If \(D_{N}\) is defined on \(X\times X\) by \(D_{N}(x,x')=\lambda(N[x]\Delta N[x'])\), then \(D_{N}\) is continuous and \(T\times T\)-invariant.
If \(K_{N}\) is the equivalence relation on \(X\) defined by \(D_{N}\), that is \((x,x')\in K_{N}\) if \(D_{N}(x,x')=0\), then \(K_{N}\) is closed and \(T\times T\)-invariant and \({\boldsymbol{R}P}(X)\subset K_{N}\).
Let \(x\in X, V\) open in \(Y\), and let \(N=\overline{orb_{T\times S}(\{x\}\times V)}\). Then \({\boldsymbol{R}P}[x]\times V\subset N\).
Lemma 14. Let \((X,T)\) and \((Y,S)\) be minimal systems. Let \(\pi: X\rightarrow X_{eq}\) be the factor map to the maximal equicontinuous factor. If \(W\) is an invariant open set in \(X\times Y\) such that \((\pi\times {\rm id}) W\) is dense in \(X_{eq}\times Y\), then \(W\) is dense in \(X\times Y\).
Proof. Let \(U,V\) be nonempty open subsets of \(X\) and \(Y\), respectively. Since \(\pi\) is semiopen, \((\pi\times {\rm id})(U\times V)\) has nonempty interior. Thus \((\pi\times {\rm id}) W\cap (\pi\times {\rm id})(U\times V)\neq\emptyset\). Then there is some \((x,y)\in W\) such that \((\pi(x),y)\in \pi(U)\times V\). In other words, there is some \(x'\in X\) with \((x,x')\in {\boldsymbol{R}P}(X)\) such that \((x,y)\in W\) and \((x',y)\in U\times V\). Since \(W\) is open, there is an open neighborhood \(V'\) of \(y\) such that \(\{x\}\times V'\subset W\). By Lemma 13-(4), \(\{x'\}\times V'\subset N:=\overline{orb_{T\times S}(\{x\}\times V')}\). In particular, \((x',y)\in N\). Since \(W\) is invariant, one has \((x',y)\in N\subset \overline{W}\). Thus \((U\times V)\cap \overline{W}\). As \(U,V\) are chosen arbitrarily, we conclude that \(W\) is dense in \(X\times Y\). ◻
A special case of [23] on characterizing regionally proximal relation is as following: For a minimal system \((Y,S)\), \((y,y')\in {\boldsymbol{R}P}(Y)\) if and only if for any minimal equicontinuous system \((X,T)\), any \(x\in X\), any neighborhood \(U\) of \(x\) and any neighborhood \(V\) of \(y'\), \[N(x, U)\cap N(y, V):=\{n\in\mathbb{Z}: T^{n}x\in U, S^{n}y\in U\}\neq\emptyset.\] As a corollary, we have
Lemma 15. Let \((X, T)\) and \((Y,S)\) be minimal systems. If \((X,T)\) is equicontinuous, then for any \(x\in X\) and \(y\in Y\), \(\{x\}\times {\boldsymbol{R}P}[y]\subset \overline{orb_{T\times S}(x,y)}\).
The following lemma is a special case of [24].
Lemma 16. Let \((X,T)\) and \((Y,S)\) be minimal systems. Then there is a dense \(G_{\delta}\) subset \(\Omega\) of \(X\times Y\) such that \({\boldsymbol{R}P}[x]\times {\boldsymbol{R}P}[y]\subset \overline{orb_{T\times S}(x,y)}\) for each \((x,y)\in \Omega\).
We have the following corollary that will be used in Section 6.
Corollary 17. Let \((X,T), (Y,S)\) be minimal systems and \(\pi_{X}: X\rightarrow X_{eq}, \pi_{Y}: Y\rightarrow Y_{eq}\) be the factor maps to their maximal equicontinuous factors, respectively. If \(\pi_{X}\) is open, then there is a dense \(G_{\delta}\) subset \(\Omega\) of \(X\times Y\) such that for each \((x,y)\in\Omega\),
\(M_{x,y}:=(\pi_{X}\times\pi_{Y})^{-1}\left(\overline{orb_{T\times S}(\pi_{X}(x),\pi_{Y}(y))} \right)\) is a transitive subsystem of \(X\times Y\) and
\((x,y)\) is a transitive point of this subsystem \(M_{x,y}\).
Proof. By Lemma 16, there is dense \(G_{\delta}\) subset \(\Omega\) of \(X\times Y\) such that \({\boldsymbol{R}P}[x]\times {\boldsymbol{R}P}[y]\subset \overline{orb_{T\times S}(x,y)}\) for each \((x,y)\in \Omega\). We claim that this \(\Omega\) satisfies our requirements. To this end, it suffices to show that \[\overline{orb_{T\times S}(x,y)}=M_{x,y},\;\; \forall (x,y)\in \Omega.\]
Take any \((x,y)\in \Omega\). Clearly, \(\overline{orb_{T\times S}(x,y)}\subset M_{x,y}\). Next we show the other inclusion. Let \(N_{x,y}=\overline{orb_{T\times S}(\pi_{X}(x),\pi_{Y}(y))}\).
Now we fix any \((x',y')\in X\times Y\) with \((\pi_{X}(x'),\pi_{Y}(y'))\in N_{x,y}\).
. For any \(v\in {\boldsymbol{R}P}[y']\), there is some \(u\in {\boldsymbol{R}P}[x']\) with \((u,v)\in \overline{orb_{T\times S}(x,y)}\).
Proof of Claim 1. Let \(L_{x,y}=(\pi_{X}\times {\rm id})(\overline{orb_{T\times S}(x,y)})\). Then \(\overline{orb_{T\times S}(x,y)}\xrightarrow{\pi_{X}\times {\rm id}} L_{x,y}\xrightarrow{{\rm id}\times\pi_Y} N_{x,y}\) are homomorphisms. Thus there is some \(y''\in \pi_{Y}^{-1}(y')={\boldsymbol{R}P}[y']\) such that \((\pi_{X}(x'), y'')\in L_{x,y}\). By Lemma 15, we have that \[\{\pi_{X}(x')\}\times {\boldsymbol{R}P}[y']=\{\pi_{X}(x')\}\times {\boldsymbol{R}P}[y'']\subset \overline{orb_{T\times S}(\pi_{X}(x'), y'')}\subset L_{x,y}.\] This implies that for any \(v\in {\boldsymbol{R}P}[y']\), there is some \(u\in {\boldsymbol{R}P}[x']\) with \((u,v)\in \overline{orb_{T\times S}(x,y)}\). ◻
. For any \(v\in {\boldsymbol{R}P}[y']\), \({\boldsymbol{R}P}[x']\times\{v\}\subset \overline{orb_{T\times S}(x,y)}\).
Proof of Claim 2. Fix \(v\in {\boldsymbol{R}P}[y']\). By Claim 1, there is some \(u\in {\boldsymbol{R}P}[x']\) with \((u,v)\in \overline{orb_{T\times S}(x,y)}\). Then there is a subsequence \((n_i)\) in \(\mathbb{Z}\) such that \(T^{n_i}\times S^{n_i}(x,y)\rightarrow (u,v)\). Since \((x,y)\in \Omega\), \({\boldsymbol{R}P}[x]\times {\boldsymbol{R}P}[y]\subset \overline{orb_{T\times S}(x,y)}\). In particular, \({\boldsymbol{R}P}[x]\times \{y\}\subset \overline{orb_{T\times S}(x,y)}\). Note that \(T^{n_i}\pi_{X}(x)\rightarrow \pi_{X}(x')\). Since \(\pi_{X}\) is open, the map \(X_{eq}\rightarrow 2^{X}, z\mapsto \pi_{X}^{-1}(z)\) is continuous. Thus \(T^{n_i}{\boldsymbol{R}P}[x]\rightarrow {\boldsymbol{R}P}[x']\) and hence \[{\boldsymbol{R}P}[x']\times \{v\}=\lim_{i\rightarrow\infty}(T\times S)^{n_i} {\boldsymbol{R}P}[x]\times \{y\}\subset \overline{orb_{T\times S}(x,y)} .\] ◻
Now it follows from Claim 2 that \({\boldsymbol{R}P}[x']\times {\boldsymbol{R}P}[y']\subset \overline{orb_{T\times S}(x,y)}\). Note that \[M_{x,y}=\bigcup\{ {\boldsymbol{R}P}[x']\times {\boldsymbol{R}P}[y']: (x',y')\in X\times Y \text{ with } (\pi_{X}(x'),\pi_{Y}(y'))\in N_{x,y}\}.\] Since \((x',y')\) are chosen arbitrarily, we conclude that \(M_{x,y}\subset \overline{orb_{T\times S}(x,y)}\) and hence they coincide. ◻
Let \((X,T)\) and \((Y,S)\) be two systems. A joining of \((X,T)\) and \((Y,S)\) is a \(T\times S\)-invariant closed subset in \(X\times Y\) that projects onto each coordinate. \((X,T)\) and \((Y,S)\) are disjoint if whenever there is a system \((Z, R)\) with \(\phi: Z\rightarrow X\) and \(\psi: Z\rightarrow Y\), then there is a homomorphism \(\theta: Z\rightarrow X\times Y\) such that \(\phi=p_{X}\theta\) and \(\psi=p_{Y}\theta\), where \(p_{X}: X\times Y\rightarrow X\) and \(p_{Y}: X\times Y\rightarrow Y\) are projections. We then write \(X\perp Y\). An equivalent characterization of disjointness is that \(X\perp Y\) if and only if \(X\times Y\) is the only one joining ([1]).
Two systems \((X,T)\) and \((Y,S)\) are weakly disjoint if \((X\times Y, T\times S)\) is transitive, which is denoted by \(X\curlywedge Y\).
Definition 18. Two minimal systems \((X,T)\) and \((Y,S)\) are quasi-disjoint, denoted by \(X\perp_{Q} Y\), if \(X\times Y\) is the only joining of \(X\) and \(Y\) that projects onto \(X_{eq}\times Y_{eq}\).
Proposition 19. Let \((X,T)\) and \((Y,S)\) be minimal systems. Then \(X\perp Y\) if and only if \(X\perp_{Q} Y\) and \(X\curlywedge Y\).
Proof. It is clear that \(X\perp Y\) implies that \(X\perp_{Q} Y\) and \(X\curlywedge Y\). Now suppose that \(X\perp_{Q} Y\) and \(X\curlywedge Y\). Then \(X_{eq}\perp Y_{eq}\). Thus any joining \(J\) of \(X\) and \(Y\) projects onto \(X_{eq}\times Y_{eq}\). Since \(X\perp_{Q} Y\), we have \(J=X\times Y\) and hence \(X\perp Y\). ◻
Let \(X\) be a compact metric space. Let \(2^X\) be the collection of nonempty closed subsets of \(X\). One may define a metric on \(2^X\) as follows: \[H(A,B) = \inf \{\varepsilon>0: A\subseteq B_\varepsilon(B), B\subseteq B_\varepsilon(A)\}\] where \(B_\varepsilon(A)=\{x\in X: \rho(x, A)<\varepsilon\}\). The metric \(H\) is called the Hausdorff metric of \(2^X\), and \(2^X\) is called the hyperspace of \(X\).
Let \(\{A_i\}_{i=1}^\infty\) be an arbitrary sequence of subsets of \(X\). Define \[\liminf A_i=\{x\in X: \text{for any neighbourhood U of x, U\cap A_i\neq \emptyset for all but finitely many i}\};\] \[\limsup A_i=\{x\in X: \text{for any neighbourhood U of x, U\cap A_i\neq \emptyset for infinitely many i}\}.\] We say that \(\{A_i\}_{i=1}^\infty\) converges to \(A\), denoted by \(\lim_{i\to \infty} A_i=A\), if \[\liminf A_i=\limsup A_i=A.\] Now let \(\{A_i\}_{i=1}^\infty\subseteq 2^X\) and \(A\in 2^X\). Then \(\lim_{i\to\infty} A_i=A\) if and only if \(\{A_i\}_{i=1}^\infty\) converges to \(A\) in \(2^X\) with respect to the Hausdorff metric.
Let \(X,Y\) be two compact metric spaces. Let \(F: Y\rightarrow 2^X\) be a map and \(y\in Y\). We say that \(F\) is upper semi-continuous (u.s.c.) at \(y\) if whenever \(\lim y_i=y\), one has that \(\limsup F(y_i)\subseteq F(y)\). We say \(F\) is lower semi-continuous (l.s.c.) at \(y\) if whenever \(\lim y_i=y\), one has that \(\liminf F(y_i)\supset F(y)\). If \(F\) is u.s.c. (l.s.c.) at every point of \(Y\), then we say that \(F\) is u.s.c. (l.s.c.).
It is easy to verify that \(F: Y\rightarrow 2^X\) is u.s.c. at \(y\in Y\) if and only if for each \(\varepsilon>0\) there exists a neighbourhood \(U\) of \(y\) such that \(F(U)\subseteq B_\varepsilon(F(y))\); and \(F: Y\rightarrow 2^X\) is l.s.c. at \(y\in Y\) if and only if for each \(\varepsilon>0\) there exists a neighbourhood \(U\) of \(y\) such that \(F(y)\subseteq B_\varepsilon(F(y'))\) for all \(y'\in U\).
Definition 20. Let \((X,T)\) and \((Y,T)\) be minimal systems. A system \((Z,R)\) is a maximal common equicontinuous factor of \(X\) and \(Y\) if there are homomorphisms \(\alpha: X\rightarrow Z\) and \(\beta: Y\rightarrow Z\) such that if there is an equicontinuous system \((W,S)\) and homomorphisms \(\phi: X\rightarrow W\) and \(\psi: Y\rightarrow W\), then there are homomorphisms \(\eta_{i}: Z\rightarrow W, i=1,2\) such that \(\phi=\eta_1\alpha\) and \(\psi=\eta_2\beta\). \[\begin{tikzcd} X \arrow[rd, "\alpha"] \arrow[rdd,"\phi"'] & & Y \arrow[ld, "\beta" '] \arrow[ldd,"\psi"] \\ & Z\arrow[d,"\eta_1" ', "\eta_2"] & \\ & W & \end{tikzcd}\]
To show the existence of the maximal common equicontinuous factor, we recall another description of maximal equicontinuous factor. Let \((X,T)\) be a minimal system. A continuous function \(f\) on \(X\) with \(|f|=1\) is an eigenfunction of \(T\) if there is a nonzero \(\lambda\in\mathbb{C}\) such that \(f(Tx)=\lambda f(x)\) for every \(x\in X\). In this case, \(\lambda\) is called an eigenvalue of \(T\). Now let \(\Lambda\) be the collection of eigenvalues of \(T\). For each \(\lambda\in \Lambda\), let \(f_{\lambda}\) be the corresponding eigenfunction. Let \(\widehat{\Lambda}\) be the Pontrjagin dual of \(\Lambda\). Further, let \(\theta\) be the inclusion map from \(\Lambda\) to the unit circle \(\mathbb{S}^{1}\). Then \(\theta\) is a character of \(\Lambda\) and we identify it with an element of \(\widehat{\Lambda}\). Note that for each \(x\in X\), \(\Lambda\rightarrow \mathbb{S}^{1}, \lambda\mapsto f_{\lambda}(x)\) is a character of \(\Lambda\). One can verify that \[\pi: X\rightarrow \widehat{\Lambda},\;x\mapsto f_{\lambda}(x)\] is a factor map between \((X, T)\) and \((\widehat{\Lambda}, R_{\theta})\), where \(R_{\theta}: \widehat{\Lambda}\rightarrow\widehat{\Lambda}, z\mapsto z+\theta\). Moreover, \((\widehat{\Lambda}, R_{\theta})\) is isomorphic to the maximal equicontinuous factor of \((X,T)\) (See [25] for more details).
Lemma 21. Let \((X,T)\) and \((Y,S)\) be minimal systems. Then there exists a unique maximal common equicontinuous factor up to isomorphisms.
Proof. It is clear from the definition that any two maximal common equicontinuous factors are isomorphic. Thus it suffices to show the existence.
Let \(\Lambda_{X}, \Lambda_{Y}\) be the sets of eigenvalues of \((X,T)\) and \((Y,S)\), respectively. The maximal equicontinuous factors \((\widehat{\Lambda_{X}}, R_{\theta_1}), (\widehat{\Lambda_{Y}}, R_{\theta_2})\) of \((X,T)\) and \((Y,S)\), where \(\alpha: \Lambda_{X}\rightarrow \mathbb{S}^{1}\) and \(\beta: \Lambda_{Y}\rightarrow \mathbb{S}^{1}\) are inclusions. Let \(\Lambda=\Lambda_{X}\cap\Lambda_{Y}\) and let \(\theta: \Lambda\rightarrow \mathbb{S}^{1}\) be the inclusion. Then \((\widehat{\Lambda}, R_{\theta})\) is a common factor of \((\widehat{\Lambda_{X}}, R_{\theta_1}), (\widehat{\Lambda_{Y}}, R_{\theta_2})\) and hence also a common factor of \((X,T)\) and \((Y,S)\). For \(\lambda\in \Lambda\), we let \(f_{\lambda}\) and \(g_{\lambda}\) be the eigenfunctions corresponding to \(\lambda\) of \(T\) and \(S\), respectively. Then \[\alpha: X\rightarrow \widehat{\Lambda}, x\mapsto f_{\lambda}(x) \;\;\text{and}\;\;\beta: Y\rightarrow \widehat{\Lambda}, y\mapsto g_{\lambda}(y)\] are homomorphisms.
We claim that \((\widehat{\Lambda}, R_{\theta})\) is a maximal common equicontinuous factor of \((X,T)\) and \((Y,S)\). Suppose that \((W,R)\) is an equicontinuous system and there are homomorphisms \(\phi: X\rightarrow W\) and \(\psi: Y\rightarrow W\). Let \(\Gamma\) be the set of eigenvalues of \((W,R)\). Since \((W,R)\) is minimal and equicontinuous, \(W\cong \widehat{\Gamma}\). Further, \(\Gamma\) is a subgroup of \(\Lambda\) since \(W\) is a common factor of \(X\) and \(Y\). By the Pontrjagin dual, there are group homomorphisms \(\eta: \widehat{\Lambda}\rightarrow \widehat{\Gamma}\). Take \(x_0\in X\) and define \[\eta_1: \widehat{\Lambda}\rightarrow \widehat{\Gamma}, \chi\mapsto \eta(\chi)-\eta(f_{\lambda}(x_0))+\phi(x_0).\] Then \(\eta_1\) is a homomorphism between \((\widehat{\Lambda}, R_{\theta})\) and \((W=\widehat{\Gamma}, R)\). It follows from the minimality that \(\eta_1\alpha=\phi\). In the similar way, we can define \(\eta_2: \widehat{\Lambda}\rightarrow \widehat{\Gamma}\) which satisfies \(\eta_2\beta=\psi\). This completes the proof. ◻
By Lemma 21, we may use \(Eq(X,Y)\) to denote the maximal common equicontinuous factor of minimal systems \((X,T)\) and \((Y,S)\). Further, let \(\alpha: X\rightarrow Eq(X,Y)\) and \(\beta: Y\rightarrow Eq(X,Y)\) be the homomorphisms. The map \(\gamma: X\times Y\rightarrow Eq(X,Y)\) is defined by \[\gamma(x,y)=\alpha(x)-\beta(y).\]
Lemma 22. The map \(\gamma: X\times Y\rightarrow Eq(X,Y)\) is semiopen.
Proof. Let \(\pi_{X}: X\rightarrow X_{eq}\) and \(\pi_{Y}: Y\rightarrow Y_{eq}\) be factor maps. By Lemma 4, \(\pi_X\) and \(\pi_Y\) are semiopen and hence \(\pi_{X}\times \pi_{Y}: X\times Y\rightarrow X_{eq}\times Y_{eq}\) is semiopen. Recall that \(\phi_{X}: X_{eq}\rightarrow Eq(X,Y)\) and \(\phi_{Y}: Y_{eq}\rightarrow Eq(X,Y)\) are open (Lemma 5). Since \(Eq(X,Y)\) is a compact group, the map \(\sigma: Eq(X,Y)\times Eq(X,Y)\rightarrow Eq(X,Y), (g,h)\mapsto g-h\) is open. Now it follows that the map \(\gamma=\sigma\circ (\phi_{X}\times \phi_{Y})\circ (\pi_{X}\times \pi_{Y})\) is semiopen. ◻
We give a more precise characterization of \(Eq(X,Y)\) and the map \(\gamma\). Suppose that \((X,T)\) and \((Y,S)\) are minimal equicontinuous. Then \(X\) and \(Y\) are compact abelian groups. We use \(e_X\) and \(e_{Y}\) to denote the units of \(X\) and \(Y\), respectively. Let \[H:=\overline{\{(T^{n}e_X, S^{n}e_Y): n\in\mathbb{Z}\}},\] which is closed subgroup of \(X\times Y\). Define \(R: (X\times Y)/H\rightarrow (X\times Y)/H\) by \[R((x,y)+H)=(Tx,y)+H=(x,S^{-1}y)+H.\] We claim that \(((X\times Y)/H, R)\) is the maximal common factor of \((X,T)\) and \((Y,S)\). For this, define \[\alpha: X\rightarrow (X\times Y)/H,\;x\mapsto (x, e_Y)+H\] and \[\beta: Y\rightarrow (X\times Y)/H,\;y\mapsto (e_X, -y)+H.\] Then \(\alpha\) and \(\beta\) are factor maps from \((X,T)\) and \((Y,S)\) to \(((X\times Y)/H,R)\), respectively. Note that \((x,y)+H\) is a minimal set in \(X\times Y\) for each \((x,y)\in X\times Y\). Thus \(((X\times Y)/H, R)\) is the maximal common factor of \((X,T)\) and \((Y,S)\). Now \(\gamma: X\times Y\rightarrow (X\times Y)/H, (x,y)\mapsto (x,y)+H\) is the factor map between \((X\times Y, T\times S)\rightarrow ((X\times Y)/H, {\rm id})\). Moreover, \[\gamma(x,y)=\alpha(x)-\beta(y)\] and \(\gamma^{-1}((x,y)+H)=(x,y)+H\).
Now suppose that \((X,T)\) and \((Y,S)\) are minimal systems. Let \(\pi_{X}: X\rightarrow X_{eq}\) and \(\pi_{Y}: Y\rightarrow Y_{eq}\) be the factor maps. Then we use \(\phi_{X}: X_{eq}\rightarrow Eq(X,Y)\) and \(\phi_{Y}: Y\rightarrow Eq(X,Y)\) to denote the homomorphisms. Further, let \(\alpha=\phi_{X}\pi_{X}, \beta=\phi_{Y}\pi_{Y}\) and \(\gamma: X\times Y\rightarrow Eq(X,Y), (x,y)\mapsto \alpha(x)-\beta(y)\).
\[\begin{tikzcd} X\arrow[r,"\pi_{X}"]\arrow[rr, bend left=30, "\alpha"] & X_{eq}\arrow[r, "\phi_{X}"] & Eq(X,Y)& Y_{eq}\arrow[l, "\phi_{Y}"'] & Y\arrow[l, "\pi_{Y}"']\arrow[ll, bend right=30, "\beta"'] \end{tikzcd}\]
According to the illustration above, we have the following remark.
Remark 23. Let \((X,T), (Y,S)\) be minimal systems and \(\gamma: X\times Y\rightarrow Eq(X,Y)\) be defined as above. Then for each \(z\in Eq(X,Y)\), \[\gamma^{-1}(z)=(\pi_{X}\times \pi_{Y})^{-1}(\overline{orb_{T\times S}(\pi_{X}(x), \pi_{Y}(y))}),\;\forall (x,y)\in \gamma^{-1}(z),\] where \(\pi_{X}: X\rightarrow X_{eq}\) and \(\pi_{Y}: Y\rightarrow Y_{eq}\) are the factor maps.
Lemma 24. Let \(\pi: (X,T)\rightarrow (Z,{\rm id})\) be a factor map, where \(Z\) consists of fixed points. Then \[\Omega:=\{z\in Z: \pi^{-1}(z) \text{ has a unique minimal subset}\}\] is a \(G_{\delta}\) set of \(Z\).
Proof. For \(\epsilon>0\), let \[\Omega_{\epsilon}:=\{z\in Z: \exists \text{ minimal subsets }M, M'\subset \pi^{-1}(z) \text{ such that }\rho(M, M')\geq \epsilon\}.\] We claim that \(\Omega_{\epsilon}\) is closed in \(Z\) for each \(\epsilon>0\). For this, we fix \(\epsilon>0\) and take a sequence \((z_n)\) in \(\Omega_{\epsilon}\) that converges to \(z\in Z\). We need to show that \(z\in \Omega_{\epsilon}\). By the definition of \(\Omega_{\epsilon}\), there are minimal subsets \(M_n, M_n'\subset \pi^{-1}(z_n)\) with \(\rho(M_n, M_n')\geq \epsilon\) for each \(n\in\mathbb{N}\). By passing to some subsequences, we may assume that \(M_n\rightarrow M\) and \(M_n'\rightarrow M'\) in \(2^{X}\) as \(n\) tends to \(\infty\). Clearly, \(M\) and \(M'\) are nonempty invariant closed subsets of \(\pi^{-1}(z)\). It remains to show that \(\rho(M,M')\geq\epsilon\). To this end, we take \(x\in M\) and \(x'\in M'\). Then there are sequences \(x_n\in M_n\) and \(x_n'\in M_n'\) such that \(x_n\rightarrow x\) and \(x_n'\rightarrow x'\). Then \[\rho(x,x')=\lim_{n\rightarrow \infty}\rho(x_n,x_n')\geq \liminf_{n\rightarrow\infty}\rho(M_n,M_n')\geq \epsilon.\] Since \(x\) and \(x'\) are chosen arbitrarily, we conclude that \(\rho(M,M')\geq \epsilon\). Since \(M, M\) are invariant closed subsets, there are minimal subsets \(N\subset M\) and \(N'\subset M'\). Thus \(\rho(N,N')\geq \rho(M,M')\geq \epsilon\). This shows that \(\Omega_{\epsilon}\) is closed in \(Z\).
Now it is clear that \[\Omega=\bigcup_{k=1}^{\infty} (Z\setminus \Omega_{1/k}).\] Thus \(\Omega\) is a \(G_{\delta}\) subset of \(Z\). ◻
Lemma 25. Let \(\pi: (X,T)\rightarrow (Z,{\rm id})\) be a factor map, where \(Z\) consists of fixed points. Then \[\Omega:=\{z\in Z: \pi^{-1}(z) \text{ is a minimal set}\}\] is a \(G_{\delta}\) set of \(Z\).
Proof. For each \(z\in Z\), set \[\mathcal{M}_{z}:=\{ E: E \text{ is a nonempty invariant closed subset of } \pi^{-1}(z)\}.\] Let \(H\) be the Hausdorff metric on \(2^{X}\). Then \(\pi^{-1}(z)\) is minimal if and only if \(\mathcal{M}_{z}=\{\pi^{-1}(z)\}\) if and only if \(\mathop{\mathrm{diam}}_{H}(\mathcal{M}_{z})=0\). Now for each \(k\in\mathbb{N}\), let \[\Omega_{k}:=\{z\in Z: \mathop{\mathrm{diam}}_{H}(\mathcal{M}_{z})\geq 1/k\}.\] We claim that \(\Omega_k\) is closed in \(Z\) for each \(k\in\mathbb{N}\). For this, we fix \(k\in\mathbb{N}\). Take a sequence \((z_n)\) in \(\Omega_k\) that converges to some point \(z\in Z\). Since \(z_n\in\Omega_k\), there are nonempty invariant closed subsets \(M_n,M_n'\subset \pi^{-1}(z_n)\) such that \(H(M_n,M_n')\geq 1/k\). By passing to some subsequences, we may assume that \(M_n\rightarrow M, M_n'\rightarrow M'\) in \(2^X\). Then it clear that \(M, M'\subset \pi^{-1}(z)\), since \(z_n\rightarrow z\). Moreover, \(H(M,M')\geq 1/k\). Thus \(z\in \Omega_{k}\). This shows that \(\Omega_{k}\) is closed in \(Z\).
Clearly, \[\Omega=\bigcup_{k=1}^{\infty}(Z\setminus \Omega_k).\] Thus \(\Omega\) is a \(G_{\delta}\) subset of \(Z\). ◻
Lemma 26. If a system \((X,T)\) has at least two minimal subsets, then there is some \(\epsilon>0\) such that for each \(x\in X\), there is a minimal set contained in \(X\setminus B(x,\epsilon)\).
Proof. By the assumption, there are two different minimal subsets \(M_1\) and \(M_2\) in \(X\). Let \(\delta=\inf\{\rho(x_1,x_2): x_1\in M_1,x_2\in M_2\}>0\) and take \(\epsilon=\frac{\delta}{3}\). Then for any \(x\in X\), \[\max(\rho(x, M_1), \rho(x, M_2)) >\epsilon.\] Thus there is a minimal set contained in \(X\setminus B(x,\epsilon)\). ◻
Lemma 27. Let \(\pi: (X,T)\rightarrow (Z,{\rm id})\) be a semiopen factor map, where \(Z\) consists of fixed points. If for each \(z\in Z\), \(\pi^{-1}(z)\) has at least two minimal subsets, then there is an invariant closed proper subset \(E\subsetneq X\) such that \(\pi(E)=Z\).
Proof. For each \(\epsilon>0\), let \[\Omega_{\epsilon}:=\{z\in Z: \forall x\in \pi^{-1}(z), \exists \text{ a subsystem } M_{x}\subset \pi^{-1}(z)\setminus B(x,\epsilon)\}.\] Clearly, \(\Omega_{\epsilon_2}\subset\Omega_{\epsilon_1}\) for any \(0<\epsilon_1\leq \epsilon_2\). According to Lemma 26, one has \[\label{eq1} Z=\bigcup_{\epsilon>0}\Omega_{\epsilon}=\bigcup_{k=1}^{\infty}\Omega_{1/k}.\tag{1}\]
First, we claim that \(\overline{\Omega_{\epsilon}}\subset \Omega_{\epsilon/2}\), for each \(\epsilon>0\). For this, we fix \(\epsilon>0\) and take a sequence \((z_n)\) in \(\Omega_{\epsilon}\) that converges to \(z\in Z\). Further, we may assume that \((\pi^{-1}(z_n))\) converges in \(2^{X}\), saying \(M=\lim_{n\rightarrow \infty}\pi^{-1}(z_n)\). Clearly, \(M\subset \pi^{-1}(z)\). We will show that \(z\in \Omega_{\epsilon/2}\).
Take \(x\in \pi^{-1}(z)\) and we divide it into two cases.
. \(x\in M\). Then there is \(x_n\in \pi^{-1}(z_n)\) such that \(x_n\rightarrow x\). Fix a \(\delta>0\). Then there is some \(N\in\mathbb{N}\) such that \(x_n\subset B(x, \delta)\) for any \(n\geq N\). Then \[B(x, \epsilon-\delta)\subset B(x_n, \epsilon),\;\;\forall n\geq N.\] Since \(z_n\in \Omega_{\epsilon}\), there is some nonempty invariant closed subset \(M_n\subset \pi^{-1}(z_n)\setminus B(x_n, \epsilon)\) for each \(n\in \mathbb{N}\). In particular, \[M_n\subset X\setminus B(x, \epsilon-\delta),\;\forall n\geq N.\] Thus \(M':=\limsup_{n\rightarrow \infty} M_n\subset X\setminus B(x, \epsilon-\delta)\). It is clear that \(M'\) is an invariant closed subset that is contained in \(\pi^{-1}(z)\). Since \(\delta>0\) is arbitrary, we conclude that there is a nonempty invariant closed subset \(M_x\) that is contained in \(\pi^{-1}(z)\setminus B(x,\epsilon)\).
. \(x\in \pi^{-1}(z)\setminus M\). If \(B(x,\epsilon/2)\cap M=\emptyset\), then \(M\) is an invariant closed subset satisfying \(M\subset \pi^{-1}(z)\setminus B(x,\epsilon/2)\). If \(B(x,\epsilon/2)\cap M\neq\emptyset\), then we take \(x'\in B(x,\epsilon/2)\cap M\) and hence \[B(x,\epsilon/2)\subset B(x',\epsilon).\] Then it follows from Case 1 that there is some nonempty invariant closed subset \(M''\subset \pi^{-1}(z)\setminus B(x', \epsilon)\). In particular, \(M''\subset \pi^{-1}(z)\setminus B(x, \epsilon/2)\).
According to both cases above, we conclude that \(z\in \Omega_{\epsilon/2}\) and hence \(\overline{\Omega_{\epsilon}}\subset \Omega_{\epsilon/2}\).
Now it follows from (1 ) and Baire’s Category theorem that there is some \(k\in\mathbb{N}\) such that \(\overline{\Omega_{1/k}}\) has a nonempty interior. Further, by the claim above, the interior of \(\Omega_{1/2k}\) is not empty. Thus there is a nonempty open set \(V\) contained in \(\Omega_{1/2k}\).
Finally, we can construct the desired invariant closed set \(E\) of \(X\) that projects onto \(Z\). Take some \(z_0\in V\) and \(x_0\in \pi^{-1}(z)\). Then there is some \(\delta\in(0,1/4k)\) such that \(\pi(B(x_0, \delta))\subset V\). Since \(\pi\) is semiopen, the interior of \(\pi(B(x_0, \delta))\) in \(Z\) is not empty. Let \(U\) denote the interior of \(\pi(B(x_0, \delta))\) in \(Z\) and let \(W=\pi^{-1}(U)\cap B(x_0,\delta)\). For each \(z\in U\), we take some \(x_{z}\in W\cap \pi^{-1}(z)\). Then there is some nonempty invariant closed subset \(M_{z}\) satisfying \[M_z\subset \pi^{-1}(z)\setminus B(x_z, 1/2k)\subset X\setminus W,\] since \(B(x_0,\delta)\subset B(x_z, 1/2k)\). Set \[E=\overline{(\bigcup_{z\in U} M_z) \cup (\bigcup_{z\in Z\setminus U}\pi^{-1}(z))}.\] Clearly, \(E\) is an invariant closed subset of \(X\) and \(\pi(E)=Z\). Further, \(E\subsetneq X\) since \(B(x_0,\delta)\cap E=\emptyset\). ◻
Proof. Let \(\Omega:=\{z\in Eq(X,Y): \gamma^{-1}(z) \text{ has a unique minimal subset}\}\).
(1) \(\Rightarrow\) (2) By Lemma 24, it suffices to show that \(\Omega\) is dense in \(Z\). By the assumption, there is some point \(z_0\in\Omega\).
Let \(R\) be the minimal translation on \(Eq(X,Y)\) as the common factor of \(X_{eq}\) and \(Y_{eq}\). Then \(\{R^{k}(z_0): k\in\mathbb{Z}\}\) is dense in \(Eq(X,Y)\). We claim that \(\{R^{k}(z_0): k\in\mathbb{Z}\}\subset \Omega\). For this, we show that \(T^{k}\times {\rm id}\) is a conjugation between \(\gamma^{-1}(z_0)\) and \(\gamma^{-1}(R^{k}z_0)\). For each \((x,y)\in \gamma^{-1}(z_0)\), \[\gamma(T^kx, y)=\alpha(T^k x)-\beta(y)=R^{k}(\alpha(x))-\beta(y)=R^{k}(\alpha(x)-\beta(y))=R^{k}(z_0)\] and hence \(T^{k}\times {\rm id}(\gamma^{-1}(z_0))\subset \gamma^{-1}(R^{k}z_0)\). Similarly, \(T^{-k}\times {\rm id}(\gamma^{-1}(R^{k}z_0))\subset \gamma^{-1}(z_0)\). Thus \(T^{k}\times {\rm id}\) is a homeomorphism between \(\gamma^{-1}(z_0)\) and \(\gamma^{-1}(R^{k}z_0)\). Since \(T^{k}\times {\rm id}\) commutes with \(T\times S\), we conclude that \(\gamma^{-1}(z_0)\) and \(\gamma^{-1}(R^{k}z_0)\) are conjugate by \(T^{k}\times {\rm id}\). Thus \(\gamma^{-1}(R^kz_0)\) also has a unique minimal subset. This shows that \(\{R^{k}(z_0): k\in\mathbb{Z}\}\subset \Omega\) hence \(\Omega\) is dense in \(Eq(X,Y)\).
(2) \(\Rightarrow\) (3) Let \(J\in\mathcal{J}(X,Y)\) be a joining of \(X\) and \(Y\) that projects onto \(X_{eq}\times Y_{eq}\). Thus \(\gamma(J)=Eq(X,Y)\). Then for each \(z\in Eq(X,Y)\), \(\gamma^{-1}(z)\cap J\) is a \(T\times S\)-invariant nonempty closed subset. Hence there is a minimal subset contained in \(\gamma^{-1}(z)\cap J\). For each \(z\in \Omega\), let \(M_{z}\) be the unique minimal subset contained in \(\gamma^{-1}(z)\). Thus \[\label{eq323462} \bigcup_{z\in\Omega}M_{z}\subset J.\tag{2}\]
Take \(z_0\in \Omega\) and \((x_0,y_0)\in M_{z_0}\). We claim that \(\Sigma:=\{(T^{m}x_0, S^{n}x_0): (m,n)\in\mathbb{Z}^{2}\}\) is contained in \(J\). Fix \((m,n)\in\mathbb{Z}^{2}\). On the one hand, it is clear that \((T^{m}x_0, S^{n}x_0)\) is a \(T\times S\)-minimal point. On the other hand, \[\begin{align} \gamma(T^mx_0, S^ny_0)&=\alpha(T^m x_0)-\beta(S^ny_0)=R^{m}(\alpha(x_0))-R^{n}\beta(y_0)\\ &=R^{m-n}(\alpha(x_0)-\beta(y_0))=R^{m-n}(z_0). \end{align}\] According to the proof of (1) \(\Rightarrow\) (2), \(R^{m-n}(z_0)\in\Omega\). Thus \((T^{m}x_0, S^{n}y_0)\in M_{R^{m-n}z_0}\). Now it follows from (2 ) that \(\Sigma\subset J\).
Since both \((X,T)\) and \((Y,S)\) are minimal, \(\Sigma\) is dense in \(X\times Y\). Thus \(J=X\times Y\). Hence \(X\perp_{Q} Y\).
(3) \(\Rightarrow\) (1) Let \(\pi_X: X\rightarrow X_{eq}\) and \(\pi_{Y}: Y\rightarrow Y_{eq}\) be factor maps. Suppose that \(\gamma^{-1}(z)\) has at least two minimal subsets for each \(z\in Eq(X,Y)\). Applying Lemma 27 to \(\gamma: X\times Y\rightarrow Eq(X,Y)\), there is an invariant closed subset \(E\subsetneq X\times Y\) such that \(\gamma(E)=Eq(X,Y)\). Clearly, \(E\) is a joining of \(X\times Y\) and \(\pi_X\times \pi_Y(E)=X_{eq}\times Y_{eq}\). But this contradicts the quasi-disjointness of \(X\) and \(Y\). This contradiction implies that there is some point \(z\in Eq(X,Y)\) such that \(\gamma^{-1}(z)\) has a unique minimal subset. ◻
In a similar way, we have the following result.
Theorem 28. Let \((X,T)\) and \((Y,S)\) be minimal systems. Let \(\alpha: X\rightarrow Eq(X,Y), \beta: Y\rightarrow Eq(X,Y)\) be the factor maps and \(\gamma(x,y)=\alpha(x)-\beta(y)\). Then the following assertions are equivalent:
There is some point \(z\in Eq(X,Y)\) such that \(\gamma^{-1}(z)\) is a minimal subset.
There is a dense \(G_{\delta}\) subset \(\Omega\subset Eq(X,Y)\) such that for each \(z\in Z\), \(\gamma^{-1}(z)\) is a minimal subset.
\(X\perp_{SQ} Y\).
Proof. It suffices to show that (1) implies (2).
Let \(\Omega:=\{z\in Eq(X,Y): \gamma^{-1}(z) \text{ is a minimal subset}\}\) and \(R\) be the minimal translation on \(Eq(X,Y)\) as the common factor of \(X_{eq}\) and \(Y_{eq}\). By Lemma 25, it suffices to show that \(\Omega\) is dense in \(Z\). By assumption, there is some \(z_0\in \Omega\). According to the proof of (1) \(\Rightarrow\) (2) of Theorem 1, \(\gamma^{-1}(z_0)\) is conjugate to \(\gamma^{-1}(R^{k}z_0)\), for each \(k\in\mathbb{Z}\).. Thus \(\gamma^{-1}(R^{k}z_0)\) is also minimal for each \(k\in\mathbb{Z}\). Thus \(\{ R^{k}(z_0): k\in\mathbb{Z}\}\subset \Omega\). Clearly, \(\{ R^{k}(z_0): k\in\mathbb{Z}\}\) is dense in \(Eq(X,Y)\) and hence \(\Omega\) is dense in \(Eq(X,Y)\). Therefore, \(\Omega\) is a dense \(G_{\delta}\) subset of \(Eq(X,Y)\). ◻
In this section, we show Theorem 2, i.e. both quasi-disjointness and strong quasi-disjointness are preserved by taking factors. In addition, quasi-disjointness is preserved by proximal extensions.
Lemma 29. [26]Suppose that \(X'\) is a proximal extension of minimal system \(X\) and \(X'\) has a distal factor \(Y\). Then \(Y\) is a factor of \(X\).
A subset \(A\subset {\mathbb{Z}}\) is syndetic if it has bounded gaps, is thick if there is a sequence \((n_i)\subset {\mathbb{Z}}\) such that it contains \(\cup_{i=1}^\infty \{n_i+1, \ldots,n_i+i\}.\) It is clear that a syndetic suset and a thick subset have non-empty intersections. It is classical that the following lemma holds for disjointness (see [27]).
Lemma 30. Let \((X,T)\) and \((Y,S)\) be minimal systems. Suppose that \(\phi: (X',T)\rightarrow (X,T)\) is a minimal proximal extension. If \(X\perp_{Q} Y\) then \(X'\perp_Q Y\).
Proof. Suppose that \(X\perp_{Q} Y\) and let \(J'\) be a joining of \(X'\) and \(Y\) that projects onto \(X'_{eq}\times Y_{eq}\). Then \(J:=\phi\times{\rm id}(J')\) is a joining of \(X\) and \(Y\) that projects onto \(X_{eq}\times Y_{eq}\). By Lemma 29, \(X'_{eq}=X_{eq}\). By the quasi-disjointness of \(X\) and \(Y\), one has \(J=X\times Y\).
. If \((x',y)\in X'\times Y\) is a \(T\times S\)-minimal point, then \((x',y)\in J'\).
Proof of Claim 1. Fix a minimal point \((x',y)\in X'\times Y\). Since \(\phi\times{\rm id}(J')=X\times Y\). There is some \(x''\in X'\) such that \(\phi(x')=\phi(x'')\) and \((x'',y)\in J'\). For any \(\epsilon>0\), it follows from the minimality of \((x',y)\) that \(\{n\in\mathbb{Z}:\;\rho(T^nx',x')<\epsilon/2, \rho(S^ny, y)<\epsilon\}\) is syndetic and it follows from the proximality of \((x',x'')\) that \(\{n\in\mathbb{Z}:\;\rho(T^nx',T^nx'')<\epsilon/2\}\) is thick. Thus there is some \(n\in\mathbb{Z}\) such that \[\rho(T^{n}x'', x')\leq \rho(T^{n}x'', T^{n}x')+\rho(T^{n}x',x')<\epsilon\;\;\text{and}\;\;\rho(T^{n}y,y)<\epsilon.\] This implies that \((x',y)\in \overline{orb_{T\times S}(x'',y)}\). Since \(J'\) is \(T\times S\)-invariant, \(\overline{orb_{T\times S}(x'',y)}\subset J'\). Thus \((x',y)\in J\). ◻
Note that the set of \(T\times S\)-minimal points in dense in \(X'\times Y\). Thus it follows from Claim 1 that \(J'=X'\times Y\). Hence \(X'\perp_{Q} Y\). ◻
Remark 31. A special case in Lemma 30 is \(X'\rightarrow X\) is an almost one to one extension.
Theorem 32. Let \((X,T),(Y,S)\) be minimal systems and \(\phi:(X,T)\rightarrow (Z,R)\) be a factor. If \(X\perp_{Q} Y\) then \(Z\perp_{Q} Y\).
Proof. Let \(J\) be joining of \(Z\) and \(Y\) that projects onto \(Z_{eq}\times Y_{eq}\). Define \[\widetilde{J}=\{(x,y)\in X\times Y: \;(\phi(x),y)\in J\}.\] Clearly, \(\widetilde{J}\) is a joining of \(X\) and \(Y\). We claim that \(\widetilde{J}\) projects onto \(X_{eq}\times Y_{eq}\). This is equivalent to that \[({\boldsymbol{R}P}[x]\times {\boldsymbol{R}P}[y])\cap \widetilde{J}\neq\emptyset, \;\;\forall (x,y)\in X\times Y.\] Since \(J\) projects onto \(Z_{eq}\times Y_{eq}\), we have \[({\boldsymbol{R}P}[z]\times {\boldsymbol{R}P}[y])\cap \widetilde{J}\neq\emptyset, \;\;\forall (z,y)\in Z\times Y.\] Fix \((x,y)\in X\times Y\) and let \(z=\phi(x)\). Then there is \(z'\in {\boldsymbol{R}P}[z]\) and \(y'\in {\boldsymbol{R}P}[y]\) such that \((z',y')\in J\). By Lemma 12, there is \(x'\in \phi^{-1}(z')\) such that \((x, x')\in{\boldsymbol{R}P}(X)\). Thus \((x', y')\in \widetilde{J}\) and \((x', y')\in {\boldsymbol{R}P}[x]\times {\boldsymbol{R}P}[y]\). This implies that \(\widetilde{J}\) projects onto \(X_{eq}\times Y_{eq}\). Since \(X\perp_{Q} Y\), we have \(\widetilde{J}=X\times Y\). Thus \(J=Z\times Y\) and hence \(Z\perp_{Q} Y\). ◻
Lemma 33. Let \((X,T)\) and \((Y,S)\) be minimal systems. If \(\pi: X\rightarrow X'\) is a factor, then \(Eq(X',Y)\) is a factor of \(Eq(X,Y)\) and we have the following commuting diagrams. \[\label{eq4461} \begin{tikzcd} X \arrow[r,"\pi_{X}"] \arrow[d,"\pi" ']\arrow[rr, bend left=30, "\alpha"] & X_{eq} \arrow[d, "\pi' "'] \arrow[r, "\phi_{X}"]& Eq(X,Y)\arrow[d,"\psi"'] \\ X' \arrow[r,"\pi_{X'}"] \arrow[rr, bend right=30,"\alpha' "]& X'_{eq} \arrow[r,"\phi_{X'}"] & Eq(X',Y) \end{tikzcd} \qquad\quad \begin{tikzcd} X\times Y\ar[d,"\pi\times {\rm id}"left] \ar[r,"\gamma" above] & Eq(X,Y) \ar[d,"\psi"]\\ X'\times Y \ar[r, "\gamma~' "above]& Eq(X',Y) \end{tikzcd}\qquad{(1)}\]
Proof. Let \(X_{eq}=G, X'_{eq}=G'\) and \(Y_{eq}=H\), which are compact abelian metric groups. Let \[\Gamma=\overline{\{(T^{n}e_{G}, S^{n}e_{H}): n\in\mathbb{Z}\}} \;\;\text{and}\;\; \Gamma'=\overline{\{(T^{n}e_{G'}, S^{n}e_{H}): n\in\mathbb{Z}\}}.\] Then \(Eq(X,Y)=(G\times H)/\Gamma\) and \(Eq(X', Y)=(G'\times H)/\Gamma'\).
Recall that the transformations on \(Eq(X,Y)\) and \(Eq(X',Y)\) are defined by \[R: (G\times H)/\Gamma\rightarrow (G\times H)/\Gamma,\; R((x,y)+\Gamma)=(Tx,y)+\Gamma=(x,S^{-1}y)+\Gamma\] and \[R: (G'\times H)/\Gamma'\rightarrow (G'\times H)/\Gamma', \; R((x',y)+\Gamma')=(Tx',y)+\Gamma'=(x',S^{-1}y)+\Gamma'.\]
Let \(\pi': X_{eq}\rightarrow X'_{eq}\) be the factor map. Then we have \(\pi'\times {\rm id}(\Gamma)=\Gamma'\). Thus we can define \(\psi: (G\times H)/\Gamma\rightarrow (G'\times H)/\Gamma'\) by \(\psi((x,y)+\Gamma)=(\pi'(x),y)+\Gamma'\). Clearly, \[\begin{align} \psi R((x,y)+\Gamma))&=\psi((x,S^{-1}y)+\Gamma)=(\pi'(x), S^{-1}y)+\Gamma'\\ &=R ((\pi'(x), y)+\Gamma')=R\psi((x,y)+\Gamma)), \end{align}\] for any \((x,y)+\Gamma\in (G\times H)/\Gamma\). Thus \(\psi\) is a factor map.
Next we verify the commuting diagrams. It suffices to show the following ones. \[\begin{tikzcd} X_{eq} \arrow[d, "\pi' "'] \arrow[r, "\phi_{X}"]& Eq(X,Y)\arrow[d,"\psi"'] \\ X'_{eq} \arrow[r,"\phi_{X'}"] & Eq(X',Y) \end{tikzcd} \qquad\quad \begin{tikzcd} X\times Y\ar[d,"\pi\times {\rm id}"left] \ar[r,"\gamma" above] & Eq(X,Y) \ar[d,"\psi"]\\ X'\times Y \ar[r, "\gamma~' "above]& Eq(X',Y) \end{tikzcd}\]
Recall that \(\phi_{X}: X_{eq}\rightarrow Eq(X,Y)\) is define by \(\phi_{X}(x)=(x, e_{H})+\Gamma\) and \(\phi_{X'}: X'_{eq}\rightarrow Eq(X',Y)\) is define by \(\phi_{X'}(x')=(x', e_{H})+\Gamma'\). Thus \[\psi\phi_{X}(x)=\psi((x, e_{H})+\Gamma)=(\pi'(x),e_{H})+\Gamma'=\phi_{X'}(\pi'(x))=\phi_{X'}\pi'(x)\] for any \(x\in X_{eq}\). This shows that \(\psi\phi_{X}=\phi_{X'}\pi'\).
For the second diagram, recall that \(\phi_{Y}:Y_{eq}\rightarrow Eq(X,Y)\) is define by \(\phi_{Y}(y)=(e_{G}, -y)+\Gamma\) and \(\phi'_{Y}: Y_{eq}\rightarrow Eq(X',Y)\) is define by \(\phi'_{Y}(y)=(e_{G'}, -y)+\Gamma'\). For any \((x,y)\in X\times Y\), one has \[\begin{align} \psi\gamma(x,y)&=\psi(\phi_{X}\pi_{X}(x)-\phi_{Y}\pi_{Y}(y))=\psi((\pi_{X}x, -\pi_{Y}y)+\Gamma)\\ &=(\pi'\pi_{X}x, -\pi_{Y}y)+\Gamma'=(\pi_{X'}\pi x, -\pi_{Y}y)+\Gamma'\\ &=\phi_{X'}(\pi x)- \phi'_{Y}(y)=\gamma'(\pi x, y)=\gamma' (\pi\times{\rm id})(x,y). \end{align}\] Thus the second commuting diagram holds. ◻
Theorem 34. Let \((X,T)\) and \((Y,S)\) be minimal systems. Suppose that \(\pi: X\rightarrow X'\) is a factor. If \(X\perp_{SQ} Y\) then \(X'\perp_{SQ} Y\).
Proof. Notions are the same with the ones in Lemma 33.
Since \(X\perp_{SQ} Y\), there is some \(z\in Eq(X,Y)\) such that \(\gamma^{-1}(z)\) is minimal. Let \(z'=\psi(z)\in Eq(X',Y)\). By Lemma 33, we have \[\label{eq32factor} \gamma'^{-1}(z')=(\pi\times {\rm id})\gamma^{-1}(z).\tag{3}\] Thus \(\gamma'^{-1}(z')\) is minimal and hence \(X'\perp_{SQ} Y\). ◻
We remark that it follows from (3 ) that if \(\gamma^{-1}(z)\) has a unique minimal set then so does \(\gamma'^{-1}(z')\). By Theorem 1, if \(X\perp_{Q} Y\) then \(X'\perp_{Q}Y\). This yields another proof of Theorem 32.
Lemma 35. Let \((X,T)\) and \((Y,S)\) be minimal systems. Suppose that \(\pi: (X,T)\rightarrow (X',T)\) is an almost one to one extension. If \(X'\perp_{SQ} Y\) then \(X\perp_{SQ} Y\).
Proof. It is clear that \(X_{eq}=X'_{eq}\) and hence \(Eq(X', Y)=Eq(X,Y)\). We use the notations as in (?? ). Since \(X'\perp_{SQ} Y\), there is a dense \(G_{\delta}\) subset \(\Omega_1\) of \(Eq(X,Y)\) such that \(\gamma'^{-1}(z)\) is \(T\times S\)-minimal for each \(z\in \Omega\). Note that both \(\pi\times {\rm id}: X\times Y \rightarrow X'\times Y\) and \(\gamma': X'\times Y \rightarrow Eq(X,Y)\) are semiopen. By Lemma 9, there is a dense \(G_{\delta}\) subset \(\Omega_2\) of \(Eq(X,Y)\) such that the restriction \(\pi\times {\rm id}: \gamma^{-1}(z)\rightarrow \gamma'^{-1}(z)\) is semiopen for each \(z\in \Omega_2\). Now for each \(z\in \Omega_1\cap \Omega_2\), \(\gamma'^{-1}(z)\) is minimal and \(\pi\times {\rm id}: \gamma^{-1}(z)\rightarrow \gamma'^{-1}(z)\) is semiopen. Thus \(\gamma^{-1}(z)\) is also minimal for each \(z\in \Omega_1\cap \Omega_2\). This shows that \(X\perp_{SQ} Y\). ◻
In this section, we show that quasi-disjointness preserved by equicontinuous extensions which is needed for the proof of Theorem 3.
Let \(\pi: (X,T)\rightarrow (Y,S)\) be an extension. Suppose there is a compact group \(G\) acting on \(X\) continuously that commutes with \(T\) such that \(Y=X/G\), i.e., \(\pi^{-1}\pi(x)=Gx=\{gx: g\in G\}\). Then we say that \(\pi\) is a (compact) group extension by compact group \(G\). To indicate the commutativity of \(G\) and \(T\), we write the \(G\)-action on the right in the sequel.
It is clear that group extensions are equicontinuous extensions. The following lemma reveal their relation.
Lemma 36. [19]Let \(\pi: (X,T)\rightarrow (Y,S)\) be an extension between minimal systems. Then \(\pi\) is an equicontinuous extension if and only if there is a minimal system \((Z,R)\) and homomorphisms \(\tilde{\pi}: Z\rightarrow Y\) and \(\phi: Z\rightarrow X\) with \(\pi\phi=\tilde{\pi}\) and \(\tilde{\pi}\) is a compact group extension. \[\begin{tikzcd} Z\arrow[rr, "\phi",] \arrow[rd, "\tilde{\pi}"'] & & X\arrow[ld, "\pi"]\\ & Y& \end{tikzcd}\]
By Theorem 32, to show that the quasi-disjointness preserved by equicontinuous extensions, it suffices to show it holds for group extensions.
Lemma 37. Let \((X,T)\) be a minimal system. Suppose \(\pi: (X,T)\rightarrow (Y,S)\) be a group extension by a compact group \(G\). Then there is a closed normal subgroup \(H\) of \(G\) and an intermediate factor \((X,T)\overset{\phi}{\rightarrow} (Z,R)\overset{\psi}{\rightarrow} (Y,S)\) such that the following commuting diagram holds and
both \(\phi\) is an \(H\)-extension, \(\psi\) is a \(H\backslash G\)-extension and \(\pi=\psi\circ\phi\);
\(\psi'\) is a \(H\backslash G\)-extension;
\(X_{eq}=Z_{eq}\).
\[\begin{tikzcd}[sep=2.5cm] X\arrow[r, "\phi", "H-extension"']\arrow[d, "\pi_X" ']\arrow[rr, bend left=20, "\pi"] & Z \arrow[r, "\psi", "H\backslash G-extension"'] \arrow[d, "\pi_{Z}" '] & Y \arrow[d, "\pi_{Y}"'] \\ X_{eq}\arrow[r, equal, "\phi'"] & Z_{eq} \arrow[r, "\psi' ","H\backslash G-extension"'] & Y_{eq} \end{tikzcd}\]
Proof. Claim 1. \({\boldsymbol{R}P}(X)\) is \(G\)-invariant.
Proof of Claim 1. Take \((x,y)\in{\boldsymbol{R}P}(X)\) and \(g\in G\). We need to show \((xg, yg)\in{\boldsymbol{R}P}(X)\). Since \((x,y)\in{\boldsymbol{R}P}(X)\), there are sequences \((x_i),(y_i)\) in \(X\) and \((n_i)\) in \(\mathbb{Z}\) such that \[x_i\rightarrow x, y_i\rightarrow y \text{ and } \rho(T^{n_i}x_i, T^{n_i}y_i)\rightarrow 0.\] Then we have \[x_ig\rightarrow xg, y_ig\rightarrow yg \text{ and } \rho(T^{n_i}x_ig, T^{n_i}y_ig)\rightarrow 0.\] Thus \((xg, yg)\in{\boldsymbol{R}P}(X)\). ◻
. \(H:=\{g\in G: (x,xg)\in {\boldsymbol{R}P}(X), \forall x\in X\}\) is a normal closed subgroup of \(G\) and \(H=\{g\in G: \exists x_0\in X, (x_0,x_0g)\in {\boldsymbol{R}P}(X)\}\)
Proof of Claim 2. Clearly, \(e_{G}\in H\) and if \(h\in H\) then \(h^{-1}\in H\). Now take \(h_1,h_2\in H\) and fix \(x\in X\). Then \((xh_1, xh_1h_2)\in {\boldsymbol{R}P}(X)\) and \((x,xh_1)\in {\boldsymbol{R}P}(X)\). Since \({\boldsymbol{R}P}(X)\) is an equivalence relation, we have \((x, xh_1h_2)\in {\boldsymbol{R}P}(X)\). Since \(x\) is chosen arbitrarily, we conclude that \(H\) is a subgroup of \(G\). Since \({\boldsymbol{R}P}(X)\) is closed in \(X\times X\), it is clear that \(H\) is a closed subgroup of \(H\).
Next we show that \(H\) is normal in \(G\). For this, take \(g\in G\) and \(h\in H\). We need to show \(ghg^{-1}\in H\). Fix \(x\in X\). Then \((xg, xgh)\in {\boldsymbol{R}P}(X)\). By Claim 1, we have \((x,xghg^{-1})=(xgg^{-1}, xghg^{-1})\in{\boldsymbol{R}P}(X)\). Thus \(ghg^{-1}\in H\) and hence \(H\) is normal in \(G\).
Finally, we show that \[\{g\in G: (x,xg)\in {\boldsymbol{R}P}(X), \forall x\in X\}=\{g\in G: \exists x_0\in X, (x_0,x_0g)\in {\boldsymbol{R}P}(X)\}.\] It suffices to show that if \((x_0,x_0g)\in {\boldsymbol{R}P}(X)\) for some \(x_0\in X\) then \((x,xg)\in {\boldsymbol{R}P}(X)\) for any \(x\in X\). Suppose that \((x_0,x_0g)\in {\boldsymbol{R}P}(X)\) and \(x\in X\). Since \((X,T)\) is minimal, there is a sequence \((k_i)\) in \(\mathbb{Z}\) such that \(T^{k_i}x_0\rightarrow x\). Since \({\boldsymbol{R}P}(X)\) is \(T\)-invariant, we have \((T^{k_i}x_0, T^{k_i}x_0g)\in{\boldsymbol{R}P}(X)\) for each \(k_i\). Since \({\boldsymbol{R}P}(X)\) is closed, we have that \((x, xg)=\lim_{i\rightarrow\infty}(T^{k_i}x_0, T^{k_i}x_0g)\in {\boldsymbol{R}P}(X)\). ◻
Now let \(Z=X/H\) and let \(\phi: X\rightarrow Z\) be the quotient map. Since the \(H\)-action on \(X\) commutes with \(T\), \(\phi\) is also a factor map. For \(x\in X\), \(\pi^{-1}\pi(x)=xG:=\{xg: g\in G\}\) and \(\phi^{-1}\phi(x)=xH\). Define \(\psi: Z\rightarrow Y\) by \(\psi(xH)=\pi(xH(Hg))\) for \(x\in X\) and \(Hg\in H\backslash G\). Then \(\psi\) is a factor map between \(Z\) and \(Y\) and \(\pi=\psi\circ\phi\).
By the property of maximal equicontinuous factors, \(Z_{eq}\) is a factor of \(X_{eq}\) and \(Y_{eq}\) is a factor of \(Z_{eq}\). In additional, the commuting diagram holds. It remains to verify that \(X_{eq}=Z_{eq}\) and \(Z_{eq}\) is an \(H\backslash G\)-extension of \(Y_{eq}\).
To show that \(X_{eq}=Z_{eq}\), it suffices to show that for any \((x_1,x_2)\in X\times X\), \((x_1,x_2)\in {\boldsymbol{R}P}(X)\) if and only if \((\phi(x_1),\phi(x_2))\in {\boldsymbol{R}P}(Z)\). If \((x_1,x_2)\in {\boldsymbol{R}P}(X)\) then it follows from Lemma 10 that \((\phi(x_1),\phi(x_2))\in {\boldsymbol{R}P}(Z)\). If \((\phi(x_1),\phi(x_2))\in {\boldsymbol{R}P}(Z)\) then it follows from Lemma 10 that there is \(x_1'\in \phi^{-1}\phi (x_1)\) and \(x_2'\in \phi^{-1}\phi (x_2)\) such that \((x_1',x_2')\in {\boldsymbol{R}P}(X)\). Since \(X\) is an \(H\)-extension of \(Z\), there are \(h_1,h_2\in H\) such that \(x_1'=x_1h_1\) and \(x_2'=x_2h_2\). By Claim 1, we have \((x_1,x_2h_2h_1^{-1})=(x_1h_1,x_2h_2)h_1^{-1}\in {\boldsymbol{R}P}(X)\). By the definition of \(H\), we have \((x_2, x_2h_2h_1^{-1})\in {\boldsymbol{R}P}(X)\). Then the equivalence of \({\boldsymbol{R}P}\) implies that \((x_1,x_2)\in {\boldsymbol{R}P}(X)\). This shows that \(X_{eq}=Z_{eq}\).
Finally, we show that \(Z_{eq}\) is an \(H\backslash G\)-extension of \(Y_{eq}\). We first show that \((z, zHg)\notin {\boldsymbol{R}P}(Z)\) for any \(z\in Z\) and \(Hg\in H\backslash G\) with \(Hg\neq e_{H\backslash G}\). If \((z, zHg)\notin {\boldsymbol{R}P}(Z)\), then there is some \(x\in X\) and \(h\in H\) such that \((x,xhg)\in {\boldsymbol{R}P}(X)\). By Claim 2, we have \(hg\in H\) and hence \(Hg=e_{H\backslash G}\). This contradicts our choice and hence \((z, zHg)\notin {\boldsymbol{R}P}(Z)\). Next we show that \(\psi'^{-1}(w)=w(H\backslash G)=\{wHg: g\in G\}\) for each \(w\in Z_{eq}\). Clearly, \(w(H\backslash G)\subset \psi'^{-1}(w)\). Suppose that \(w'\in \psi'^{-1}(w)\). Take \(z\in \pi_{Z}^{-1}(w)\) and \(z'\in \pi_{Z}^{-1}(w')\). Then \(\pi_{Y}\psi(z)=\pi_{Y}\psi(z')\). Thus \((\psi(z),\psi(z'))\in{\boldsymbol{R}P}(Y)\). By Lemma , there is some \(z_1\in \psi^{-1}\psi(z)\) and \(z_2\in \psi^{-1}\psi(z')\) with \((z_1,z_2)\in {\boldsymbol{R}P}(Z)\). Since \(Z\) is a \(H\backslash G\)-extension of \(Y\), there are some \(g_1,g_2\in G\) such that \(z_1=zHg_1\) and \(z_2=z'Hg_2\). Then \((zHg_1, z'Hg_2)\in {\boldsymbol{R}P}(Z)\) implies that \((zHg_1g_2^{-1},z')=(zHg_1, z'Hg_2)Hg_2^{-1}\in {\boldsymbol{R}P}(Z)\). Thus \[w'=\pi_{Z}(z')=\psi(zHg_1g_2^{-1})=\psi(z)Hg_1g_2^{-1}=wHg_1g_2^{-1}.\] Then we have \(\psi'^{-1}(w)\subset w(H\backslash G)\) and hence they are equal. Therefore, \(Z_{eq}\) is a \(H\backslash G\)-extension of \(Y_{eq}\). ◻
Lemma 38. Let \((X,T)\) and \((Z,S)\) be minimal systems. Suppose that \(\pi: X\rightarrow Y\) is a group extension by a compact group \(G\). If \(Y\perp_{Q} Z\) and \(X_{eq}=Y_{eq}\), then \(X\perp_{Q} Z\).
Proof. Let \(J\) be a joining of \(X\) and \(Z\) that projects onto \(X_{eq}\times Z_{eq}\). Then \(\widetilde{J}:=(\pi\times{\rm id})(J)\) is a joining of \(Y\) and \(Z\). Since \(Y_{eq}\) is a factor of \(X_{eq}\), \(\widetilde{J}\) projects onto \(Y_{eq}\times Z_{eq}\). Then it follows from \(Y\perp_{Q}Z\) that \(\widetilde{J}=Y\times Z\).
We write the action of \(G\) on \(X\) from the right and assume that \(G\) acts on \(X\) freely. For each \(g\in G\), let \[J_{g}:=J(g\times{\rm id}).\] Since \(G\)-action commutes with \(T\), we conclude that \(J_g\) is also a joining of \(X\) and \(Z\) that projects onto \(X_{eq}\times Z_{eq}\). Thus \(\widetilde{J_{g}}:=(\pi\times {\rm id})(J_g)=Y\times Z\) by the quasi-disjointness of \(Y\) and \(Z\).
Now let \(V\) be a closed subset of \(G\) with a nonempty interior. Then there are \(g_1,g_2,\ldots,g_n\in G\) such that \(G=Vg_1\cup Vg_2\cup\cdots\cup Vg_n\). Let \[J_{V}:=\bigcup_{g\in V}J_g,\] which is closed in \(X\times Z\). Then \(J_{V}(g_1\times{\rm id})\cup\cdots\cup J_V(g_n\times{\rm id})=X\times Z\). Thus there is some \(g_i\) such that \(J_{V}(g_i\times{\rm id})\) has a nonempty interior and hence \(J_{V}\) has a nonempty interior.
. \((\pi\times{\rm id})(W)\) is dense in \(Y\times Z\), where \(W\) is the interior of \(J_{V}\).
Proof of Claim 1. Note that the interior of \(J_V(g\times {\rm id})\) is \(W(g\times{\rm id})\) for each \(g\in G\). Since \(J_{V}(g_1\times{\rm id})\cup\cdots\cup J_V(g_n\times{\rm id})=X\times Z\), we conclude that \(W':=\bigcup_{i=1}^{n}W(g_i\times{\rm id})\) is dense in \(X\times Z\). Clearly, \((\pi\times{\rm id})(W)=(\pi\times{\rm id})(W')\). Since \(\pi\times{\rm id}\) is surjective, we conclude that \((\pi\times{\rm id})(W)\) is dense in \(Y\times Z\). ◻
Now it follows from Claim 1 that \(((\pi_{Y}\circ \pi)\times {\rm id})(W)\) is dense in \(Y_{eq}\times Z\), where \(\pi_{Y}: Y\rightarrow Y_{eq}\) is the factor map. Since \(X_{eq}=Y_{eq}\), we have that \(\pi_{X}=\pi_{Y}\circ \pi\),where \(\pi_{Y}: Y\rightarrow Y_{eq}\) is the factor map. Thus \((\pi_{X}\times {\rm id})(W)\) is dense in \(X_{eq}\times Z\). Clearly, \(W\) is \(T\times S\)-invariant, since \(J_{V}\) is \(T\times S\)-invariant. By Lemma 14, \(W\) is dense in \(X\times Y\). Since \(J_{V}\) is closed, we conclude that \(J_{V}=X\times Z\).
There is a sequence \((V_k)\) of closed neighborhoods of \(e_{G}\) such that \(\{e_{G}\}=\bigcap_{k=1}^{\infty}V_{k}\). Then we have \(J=\bigcap_{k=1}^{\infty}J_{V_k}=X\times Z\). This shows that \(X\perp_{Q}Z\). ◻
Lemma 39. Let \((X,T)\) and \((Z,S)\) be minimal systems. Suppose that \(\pi: X\rightarrow Y\) is a group extension by a compact group \(G\). If \(Y\perp_{Q} Z\) and \(\pi': X_{eq}\rightarrow Y_{eq}\) is also a group extension by \(G\) such that the following commuting diagram holds, then \(X\perp_{Q} Z\). \[\begin{tikzcd} X \arrow[r, "\pi"] \arrow[d, "\pi_{X}"'] & Y \arrow[d, "\pi_Y"] \\ X_{eq} \arrow[r, "\pi'"] & Y_{eq} \end{tikzcd}\]
Proof. Let \(\gamma: X\times Z \rightarrow Eq(X,Z)\) and \(\gamma': Y\times Z\rightarrow Eq(Y,Z)\) as before. Since \(Y\perp_{Q} Z\), it follows from Theorem 1 that there is some \(v\in Eq(Y,Z)\) such that \(\gamma'^{-1}(v)\) has a unique minimal set \(N\). Let \(\psi: Eq(X,Z)\rightarrow Eq(Y,Z)\) be the factor map and take \(u\in \psi^{-1}(v)\).
We claim that \(\gamma^{-1}(u)\) has a unique minimal set. Then \(X\perp_{Q} Z\) follows from Theorem 1. To the contrary, suppose there are distinct minimal sets \(M_1\) and \(M_2\) contained in \(\gamma^{-1}(u)\). Since \(\pi_{X}\times\pi_{Z}(\gamma^{-1}(u))\) is a minimal set in \(X_{eq}\times Z_{eq}\), we have \(\pi_{X}\times\pi_{Z}(M_1)=\pi_{X}\times\pi_{Z}(M_2)\). Take \((x_1,z_1)\in M_1\). Then there is some \((x_2,z_2)\in M_2\) with \(\pi_{X}(x_1)=\pi_{X}(x_2)\) and \(\pi_{Z}(z_1)=\pi_{Z}(z_2)\). Note that both \(\pi\times{\rm id}(M_1)\) and \(\pi\times{\rm id}(M_2)\) are minimal sets in \(\gamma'^{-1}(v)\). Thus \(\pi\times{\rm id}(M_1)=\pi\times{\rm id}(M_2)=N\). This implies that \(z_1=z_2\) and there is some \(g\in G\) such that \(x_2=x_1g\). Since \(\pi': X{eq}\rightarrow Y_{eq}\) is also a group extension by group \(G\), \((x_1,x_1g)\notin{\boldsymbol{R}P}(X)\) unless \(g=e_{G}\). But \(M_1\) and \(M_2\) are distinct, we have \(x_1\neq x_2\) and hence \((x_1,x_2)\neq{\boldsymbol{R}P}(X)\). This contradicts that \(\pi_{X}(x_1)=\pi_{X}(x_2)\). Therefore, \(\gamma^{-1}(u)\) has a unique minimal set. ◻
Now combining Lemma 37, 38 and 39, we conclude that quasi-disjointness is preserved by group extensions.
Proposition 40. Let \((X,T)\) and \((Z,S)\) be minimal systems. Suppose that \(\pi: X\rightarrow Y\) is a group extension by a compact group \(G\). If \(Y\perp_{Q} Z\) then \(X\perp_{Q} Z\).
Finally, combining Lemma 36 and Proposition 40 we conclude that quasi-disjointness is preserved by equicontinuous extensions.
Theorem 41. Let \((X,T)\) and \((Z,S)\) be minimal systems. Suppose that \(\pi: X\rightarrow Y\) is an equicontinuous extension. If \(Y\perp_{Q} Z\) then \(X\perp_{Q} Z\).
Proof. By Lemma 36, there is a minimal system \((\tilde{Z}, T)\) and homomorphisms \(\tilde{\pi}: \tilde{Z}\rightarrow Y\) and \(\phi: \tilde{Z}\rightarrow X\) with \(\pi\phi=\tilde{\pi}\) and \(\tilde{\pi}\) is a compact group extension. By Proposition 40, one has that \(\tilde{Z}\perp_{Q} Z\). Further, by Theorem 32, we have \(X\perp_{Q}Z\). ◻
In this section, based on the preparation in the previous sections, we are ready to show Theorem 3, i.e. minimal PI systems are quasi-disjoint from all minimal systems and minimal AI systems are strongly quasi-disjoint from all minimal systems.
Since quasi-disjointness is preserved under equicontinuous extensions (Theorem 41), proximal extensions (Lemma 30), and taking factors (Theorem 32), the structure theory for PI systems implies that it suffices to prove quasi-disjointness is also preserved under taking inverse limits.
Lemma 42. Let \((X,T)=\underset{\longleftarrow}{\lim}(X_n, T)\) be an inverse limit of minimal systems and \((Y,S)\) be a minimal system. If \(X_{n}\perp_{Q} Y\) for each \(n\in\mathbb{N}\), then \(X\perp_{Q} Y\).
Proof. Let \(J\) be a joining of \(X\) and \(Y\) that projects onto \(X_{eq}\times Y_{eq}\). For each \(n\in\mathbb{N}\), let \(\phi_n: X\rightarrow X_n\) be the canonical factor map. It is clear that \(J_n:=(\phi_n\times {\rm id})(J)\) is also a joining of \(X_n\times Y\), for each \(n\in\mathbb{N}\). Since \(X_n\) is a factor of \(X\) for each \(n\in\mathbb{N}\), \((X_n)_{eq}\) is also a factor of \(X_{eq}\). Thus each \(J_n\) projects onto \((X_n)_{eq}\times Y\). By the quasi-disjointness of \(X_n\) with \(Y\), one has \(J_{n}=X_n\times Y\). Clearly, \(J=\underset{\longleftarrow}{\lim} J_n\). Thus \(J=X\times Y\) and hence \(X\perp_{Q} Y\). ◻
Theorem 43. Every minimal PI system is quasi-disjoint from any minimal system.
Proof. Let \((X,T)\) be a minimal PI system and \((Y,S)\) be a minimal system. Then there is a minimal strict PI system \((X', T')\) which is a proximal extension of \((X,T)\). Since a trivial system is quasi-disjoint from any minimal system and \(X'\) is constructed from a trivial system by taking equicontinuous extensions, proximal extensions and inverse limits, it follows from Theorem 41, 30, 42 that \(X'\perp_{Q} Y\). By Theorem 32 , \(X\perp_{Q} Y\). ◻
To end the subsection we state a remark. Since every weakly mixing system is weakly disjoint from any minimal system, it follows from Proposition 19 that a minimal system is quasi-disjoint from every minimal weakly mixing system if and only if it is disjoint from every minimal weakly mixing system. Glasner constructed in [28] a non-PI system that is disjoint from all minimal weakly mixing systems. In particular, it is quasi-disjoint from all minimal weakly mixing systems. In [29], the authors characterize the structure of transitive systems disjoint from minimal weakly mixing systems. But we do not know how characterize minimal systems that are quasi-disjoint from all minimal weakly mixing systems.
Theorem 44. Every minimal distal system is strongly quasi-disjoint from any minimal system.
Proof. Let \((X,T)\) be a minimal distal system and \((Y,S)\) be a minimal system. Let \(\gamma: X\times Y\rightarrow Eq(X,Y)\) be as defined in subsection 3.1. Since \(X\) is distal, the factor map \(\pi_{X}: X\rightarrow X_{eq}\) is open (Lemma 5). By Corollary 17, there is dense \(G_{\delta}\) subset \(\Omega\subset X\times Y\) such that for each \((x,y)\in\Omega\),
\(M_{x,y}:=(\pi_{X}\times\pi_{Y})^{-1}\left(\overline{orb_{T\times S}(\pi_{X}(x),\pi_{Y}(y))} \right)\) is a transitive subsystem of \(X\times Y\) and
\((x,y)\) is a transitive point of this subsystem \(M_{x,y}\).
By remark 23, we know that \(\gamma^{-1}(z)=M_{x,y}\) for each \((x,y)\in X\times Y\), where \(z=\gamma(x,y)\). Since \(X\) is distal, it follows from [30] that \((x,y)\) is a minimal point in \(X\times Y\). Thus \(\gamma^{-1}(z)\) is minimal for each \((x,y)\in \Omega\) with \(z=\gamma(x,y)\). This shows that \(X\perp_{SD} Y\). ◻
We will strength the above conclusion using a different approach.
Theorem 45. Let \((X,T)\) and \((Y,S)\) be minimal systems.
If \(X\perp_{SQ} Y\) then the set of minimal points of \(X\times Y\) is residual in \(X\times Y\).
If \((X,T)\) is PI and the set of minimal points of \(X\times Y\) is residual in \(X\times Y\), then \(X\perp_{SQ} Y\).
Proof. (1) Assume that \(X\perp_{SQ} Y\). By Theorem 28 there is a residual set \(\Omega'\subset Eq(X,Y)\) such that for any \(z\in \Omega'\), \(\gamma^{-1}(z)\) is a minimal subset of \(X\times Y\). Set \(\Omega=\gamma^{-1}(\Omega')\). Then \(\Omega\) is residual in \(X\times Y\) by Lemma 6 as \(\gamma\) is semiopen (Lemma 22). Note that each \((x,y)\in \Omega\) is minimal.
(2) Let \(\Omega_1\) be the set of minimal points of \(X\times Y\). Since \((X,T)\) is PI, we get that \(X\perp_Q Y\) by Theorem 1. Thus there is a residual set \(\Omega'\subset Eq(X,Y)\) such that for any \(z\in \Omega'\), \(\gamma^{-1}(z)\) contains a unique minimal subset of \(X\times Y\). Moreover, there is a residual set \(\Omega''\subset Eq(X,Y)\) such that for any \(z\in \Omega''\), \(\gamma^{-1}(z)\cap \Omega_1\) is a dense set of \(\Omega_1\) by Lemma 7. Set \[\Omega=\gamma^{-1}(\Omega'\cap \Omega'').\] Then \(\Omega\) is residual in \(X\times Y\) by Lemma 6 as \(\gamma\) is semiopen .
Fix \((x,y)\in \Omega\). Then \(\gamma(x,y)\in \Omega'\cap \Omega''\). It follows that \(W=\gamma^{-1}\gamma(x,y)\) contains a unique minimal subset of \(X\times Y\) and the set of minimal points of \(X\times Y\) in dense in \(W\). Thus, \(W\) is minimal. Put \(z=\gamma(x,y)\). Then Theorem 28 implies that \(X\perp_{SQ} Y\). This ends the proof. ◻
Corollary 46. Let \((X,T)\) be a minimal PI system. Then the set of minimal points of \(X\times Y\) is a residual subset of \(X\times Y\) for any minimal system \((Y,S)\) if and only if \(X\) is strongly quasi-disjoint from all minimal systems.
Consequently, each minimal AI system is strongly quasi-disjoint from all minimal systems.
Proof. The first statement follows from Theorem 45.
To show the second statement, we note that if \((X,T)\) is AI then the set of distal points of \(X\) (denoted by \(X'\)) is residual, see Subsection 2.1. This implies that for any minimal system \((Y,S)\), the set of minimal points of \(X\times Y\) contains \(X'\times Y\) (see Subsection 2.1), and hence is residual in \(X\times Y\). Thus, Theorem 45-(2) implies that \(X\perp_{SQ} Y\). ◻
In this paper, we use maximal equicontinuous factors to define the quasi-disjointness. One may also use maximal distal factor to define another kinds of quasi-disjointness. But we will that show these two notions coincide.
For a minimal system \((X,T)\), we use \(X_{dis}\) to denote the maximal distal factor of \(X\).
Definition 47. Two minimal systems \((X,T)\) and \((Y,S)\) are distally quasi-disjoint, denoted by \(X\perp_{DQ} Y\), if \(X\times Y\) is the only joining of \(X\) and \(Y\) that projects onto \(X_{dis}\times Y_{dis}\).
Theorem 48. Let \((X,T),(Y,S)\) be minimal systems. Then \(X\perp_{Q} Y\) if and only if \(X\perp_{QD} Y\), that is the product \(X\times Y\) is the only joining of \(X\) and \(Y\) that projects onto \(X_{dis}\times Y_{dis}\).
Proof. (\(X\perp_{Q} Y\Rightarrow X\perp_{DQ} Y\)) Let \(J\) be a joining of \(X\) and \(Y\) that projects onto \(X_{dis}\times Y_{dis}\). Then it is clear that \(J\) projects onto \(X_{eq}\times Y_{eq}\). Since \(X\perp_{Q} Y\), one has \(J=X\times Y\). Thus \(X\perp_{DQ} Y\).
(\(X\perp_{DQ} Y\Rightarrow X\perp_{Q} Y\)) Let \(J\) be a joining of \(X\) and \(Y\) that projects onto \(X_{eq}\times Y_{eq}\). Let \(J_{dis}\) be the projection of \(J\) to \(X_{dis}\times Y_{dis}\). Clearly, \(J_{dis}\) is a joining of \(X_{dis}\) and \(Y_{dis}\). On the other hand, \((X_{dis})_{eq}=X_{eq}\) and \((Y_{dis})_{eq}=Y_{eq}\). By Theorem 3, \(X_{dis}\perp_{Q}Y_{dis}\). Since \(J_{dis}\) projects onto \(X_{eq}\times Y_{eq}\), one has \(J_{dis}=X_{dis}\times Y_{dis}\). Further, one has \(J=X\times Y\) since \(X\perp_{DQ} Y\). ◻
To end the paper we ask some open questions. We have shown that quasi-disjointness is preserved by proximal extensions (Lemma 30) and strong quasi-disjointness is preserved by almost one to one extensions (Lemma 35). We think that strong quasi-disjointness is not preserved by proximal extensions. We ask the following question.
Question 1. Let \((X,T)\) be a minimal system which is a non-trivial proximal extension of \(X_{eq}\). Is there a such system such that the set of minimal points of \(X\times X\) is not a residual subset of \(X\times X\)?
We strongly believe that such a system exists. If it is the case then quasi-disjointness and strong quasi-disjointness are different, since \(X\perp_Q X\) by Theorem 1, and at the same time \(X\not \perp_{SQ} X\) by Theorem 45.
A related question is the following, where the notion of weakly mixing RIC extension one may refer to [19] or [17].
Question 2. Let \((X,T)\) be minimal and \(\pi:X\rightarrow X_{eq}\) be a non-trivial proximal or a weakly mixing RIC extension. Is it true that there is a residual set \(\Omega\subset X\) such that for each \(z\in \Omega\), \(\gamma^{-1}(z)=(\pi\times \pi)^{-1}(\overline{orb_{T\times T}(\pi(x), \pi(y))})\) is not minimal?
We remark that the question has an affirmative answer when \(y\) is in the orbit of \(x\).
In [4], [7], the authors asked whether the collection of systems disjoint from all minimal systems has the product property. For quasi-disjointness, we also do not know whether the product property holds.
Question 3. Is it true that \(X_1\perp_{Q} Y, X_2\perp_Q Y\) and \(X_1\times X_2\) minimal implies \(X_1\times X_2\perp_{Q} Y\)?
In this paper we only consider the quasi-disjointness between minimal systems. We hope this can be generalized to transitive systems or general systems.
Question 4. How to generalize the quasi-disjointness to transitive systems?