Null systems in the non-minimal case

TL;DR

This paper explores properties of null dynamical systems, demonstrating conditions under which they are equicontinuous or mean equicontinuous, and constructs examples with specific proximal relations.

Contribution

It establishes new links between nullness, distalness, and equicontinuity, and constructs examples illustrating these properties in dynamical systems.

Findings

Null and distal systems are equicontinuous.

Null systems with closed proximal relations are mean equicontinuous.

Constructed examples with specific proximal relations.

Abstract

In this paper, it is shown that if a dynamical system is null and distal, then it is equicontinuous. It turns out that a null system with closed proximal relation is mean equicontinuous. As a direct application, it follows that a null dynamical system with dense minimal points is also mean equicontinuous. Meanwhile, a distal system with trivial $\text{Ind}_{fip}$-pairs, and a non-trivial regionally proximal relation of order $\infty$ is constructed.

Equations

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}\sum _{i=0}^{n-1}d(T^{i}x,T^{i}y)<\epsilon .$$

$$P_m^{(l)}=\{\overline{p}=(p_1,p_2,\dots ,p_l):p_i\in {\mathbb{Z}}_{+},0\le p_i\le m,i=1,\dots ,l\},$$

$$\{{\bar{s}}_n^{(k)}+P_n^{(l)}:k=1,2,\dots \}\subset T^{(l)}.$$

$$h_{top}^{\mathrm{A}}(T,\mathcal{U})=\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}\log \mathrm{N}(\underset{i=1}{\overset{n}{\bigvee }}{T}^{-t_i}\mathcal{U})$$

$$h_{top}^{\mathrm{A}}(T)=\underset{\mathcal{U}}{\sup }{h}_{top}^{\mathrm{A}}(T,\mathcal{U}),$$

$$\bigcap _{i=1}^k{T}^{-t_i}{U}_{s(i)}\ne \varnothing ,$$

$$\{{\bar{s}}_n^{(k)}+P_n^{(l)}:k=1,2,\dots \}\subset F_1.$$

$$\{{\bar{s}}_d^{(k)}+P_d^{(l)}:k=1,2,\dots \}\subset F_2.$$

$$F_1\cap F_2\supseteq$$

$$\supseteq \{{\bar{s}}_d^{(k)}+{\overline{m}}_k+P_n^{(l)}:\text{for some}{\overline{m}}_k\in P_m^{(l)},k=1,2,\dots \}.$$

$$A_s^{(l)}=\{\bar{n}=(n_1,n_2,\dots ,n_l)\in {\mathbb{Z}}_{+}^l:(T^{n_1}y,\dots ,T^{n_l}y)\in V^{(s)}\}=\prod _{i=1}^{l}N(y,V_{s(i)})$$

$$B_s^{(l)}=$$

$$\supset \{\bar{n}=(n_1,n_2,\dots ,n_l)\in {\mathbb{Z}}_{+}^l:(T^{n_1}y,T^{n_2}y,\dots ,T^{n_l}y)\in U^{(s)}\}$$

$$\supset \{\bar{n}+P_N^{(l)}:\bar{n}\in A_s^{(l)}\}.$$

$$\bigcap _{s\in \{1,\dots ,n{\}}^l}{B}_s^{(l)}\ne \varnothing .$$

$$\bigcap _{i=1}^l{T}^{-n_i}{U}_{s(i)}\ne \varnothing$$

$$\underset{k\to \infty }{\lim }(x_k,y_k)=(x,y)\text{and}\underset{k\to \infty }{\lim }(T^{n_k}{x}_k,T^{n_k}{y}_k)=(z,z).$$

$$\underset{k\to \infty }{\lim }d((T\times T{)}^{n_k}(x_k,y_k),(T\times T{)}^{n_k}(T^{-n_k}z,T^{-n_k}z))=\underset{k\to \infty }{\lim }d((T^{n_k}{x}_k,T^{n_k}{y}_k),(z,z))=0.$$

$$R=\{(\mathbf{u},\mathbf{v})\in \tilde{X}\times \tilde{X}:\mathbf{u},\mathbf{v}\in \overline{Orb(\mathbf{x},\tilde{T})}\}\cup \{(\mathbf{u},\mathbf{v})\in \tilde{X}\times \tilde{X}:\mathbf{u},\mathbf{v}\in \overline{Orb(\mathbf{w},\tilde{T})}\}\cup {\Delta }_{\tilde{X}}.$$

$$\underset{k\to \infty }{\lim }d(y_k,x_1)=0\text{and}\underset{k\to \infty }{\lim }d(T^{n_k}{y}_k,x_2)=0,$$

$$\underset{k\to \infty }{\lim }(x_k,y_k)=(x,y)\text{and}\underset{k\to \infty }{\lim }(T\times T{)}^{n_k}(x_k,y_k)=(z,z).$$

$$\frac{1}{n_k}\sum _{i=0}^{n_k-1}d(T^i{x}_k,T^i{y}_k)\ge {\epsilon }_0.$$

$${\int }_{X\times X}d(x,y)\text{d}{\mu }_k⟶{\int }_{X\times X}d(x,y)\text{d}\mu$$

$${\int }_{X\times X}d(x,y)\text{d}{\mu }_k=\frac{1}{n_k}\sum _{i=0}^{n_k-1}d(T^i{x}_k,T^i{y}_k)\ge {\epsilon }_0$$

$$\mu (\text{supp}(\mu )\setminus {\Delta }_X)>0.$$

$${\mu }_k(B_l)=\frac{1}{n_k}\#\{0\le i\le n_k-1:(T^i{x}_k,T^i{y}_k)\in B_l\},$$

$$\underset{l\to \infty }{\lim }(x_{k_l},y_{k_l})=(z,z)\text{and}\underset{l\to \infty }{\lim }(T^{i_l}{x}_{k_l},T^{i_l}{y}_{k_l})=(u,v).$$

$$d(\pi (T^{i_l}{x}_{k_l}),\pi (T^{i_l}{y}_{k_l}))=d(S^{i_l}\pi (x_{k_l}),S^{i_l}\pi (y_{k_l}))<\frac{\epsilon }{3},$$

$$d(\pi (u),\pi (v))\le d(\pi (u),\pi (T^{i_l}{x}_{k_l}))+d(\pi (T^{i_l}{x}_{k_l}),\pi (T^{i_l}{y}_{k_l}))+d((T^{i_l}{y}_{k_l}),\pi (v))<\epsilon$$

$$\underset{k\to \infty }{\lim }{x}_k=z=\underset{k\to \infty }{\lim }{y}_k\text{and}\underset{k\to \infty }{\lim }(T^{n_k}{x}_k,T^{n_k}{y}_k)=(x,y).$$

$$R=\{(\mathbf{a},\mathbf{b})\in \tilde{X}\times \tilde{X}:\mathbf{a},\mathbf{b}\in \overline{Orb(\mathbf{x},\tilde{T})}\text{or}\overline{Orb(\mathbf{z},\tilde{T})}\}\cup {\Delta }_{\tilde{X}}.$$

$$\mathrm{supp}(X,T)=\overline{\bigcup _{\mu \in M(X,T)}\mathrm{supp}(\mu )}=\overline{\bigcup _{\nu \in M^e(X,T)}\mathrm{supp}(\nu )}.$$

Full text

Meanwhile, a distal system with trivial $\text{Ind}_{fip}$-pairs, and a non-trivial regionally proximal relation of order $\infty$ is constructed.

1. Introduction

Let $X$ be a compact metric space with a metric $d$, and let $T$ be a continuous map from $X$ to itself. The pair $(X,T)$ will be called a (topological) dynamical system.

Entropy and related notions are the measurements of the complexity of a dynamical system. Recall that a dynamical system $(X,T)$ is null if the topological sequence entropy is zero for any sequence. For a measurable dynamical system, the nullness is defined similarly. Kushnirenko [20] has shown that a measure-preserving transformation $T$ has a discrete spectrum if and only if it is null. Unfortunately, in the topological setting, there is no such theorem, since Goodman [13] obtained a minimal non-equicontinuous system which is null. Under the minimality assumption, Huang, Li, Shao and Ye in [15] showed that a minimal null system looks like a minimal equicontinuous system. Precisely, if $(X,T)$ is a minimal null system, $\pi:X\rightarrow X_{eq}$ is the factor map to the maximal equicontinuous factor, then $(X,T)$ is uniquely ergodic, $\pi$ is almost 1-1 and is a measure-theoretic isomorphism. One purpose of the paper is to show what a null system would be without the assumption of minimality.

Localizing the notion of sequence entropy, the authors in [15] defined the notion of sequence entropy pairs and showed that for a minimal distal system the set of sequence entropy pairs coincides with the regionally proximal relation, and thus a minimal distal null system is equicontinuous. In this paper, we will show the result remains valid without the minimality assumption, i.e. a distal null system is equicontinuous (Theorem 3.2). In fact we can show more, i.e. a distal, non-equicontinuous system has infinite sequence entropy (Theorem 3.4) which extends the previous result proved under the minimality assumption [15]. We remark that the methods that we use to show the above two results are different from the ones in [15].

In [22] the notion of mean equicontinuity was introduced, which is equivalent to the notion of stable in the mean in the sense of Lyapunov given by Fomin in [6]. A dynamical system $(X,T)$ is called mean equicontinuous if for every $\varepsilon>0$, there exists a $\delta>0$ such that whenever $x,y\in X$ with $d(x,y)<\delta$,

[TABLE]

It was known that a minimal mean equicontinuous system is uniquely ergodic. The authors in [22] proved that if $(X,T)$ is a minimal mean equicontinuous system and $\pi:X\rightarrow X_{eq}$ is the factor map to the maximal equicontinuous factor, then $\pi$ is a measure-theoretic isomorphism (it is not necessarily almost 1-1, see [5]). We refer to [8, 9, 10, 11, 16] for further study on mean equicontinuity and related subjects. Just recently, the authors of the current paper in [23] showed that many results (related to mean equicontinuity) proved before under the minimality assumption hold for general systems. In this paper, we will show that a null dynamical system with a closed proximal relation is mean equicontinuous (Theorem 4.3). A direct consequence is that a null dynamical system with dense minimal points is mean equicontinuous. Note that it is easy to construct a null dynamical system which is not mean equicontinuous. It is natural to ask if a null transitive system is mean equicontinuous. Unfortunately, we can not answer the question in the current paper.

To get an analogue result proved before in ergodic theory, in  [14] the notion of regionally proximal relation of order d (denoted by $\mathbf{RP}^{[d]}$) was introduced by Host, Kra and Maass. We refer to [12, 24] for the further study on the topic. In  [4], Dong, Donoso, Maass, Shao and Ye studied the properties of $\text{Ind}_{fip}$-pair. They showed that if $(X,T)$ is a minimal distal system and $(x,y)\in\mathbf{RP}^{[\infty]}=\cap_{d\geq 1}\mathbf{RP}^{[d]}$, then $(x,y)\in\text{Ind}_{fip}(X,T)$. Since our Theorem 3.2 can be restated as: if $(X,T)$ is distal, $\mathbf{RP}^{[1]}$ is non-trivial, then $\text{Ind}_{f}(X,T)$ is not trivial. It is natural to conjecture that the result proved in [4] is valid without the minimality assumption. It is surprising for us that it is not the case. That is, we construct a distal system such that $\mathbf{RP}^{[\infty]}$ is not trivial, but ${\text{Ind}}_{fip}(X,T)$ is trivial (Example 5.1).

The paper is organized as follows. In Section 2, we gather definitions and prove some initial theorems on the thickly syndetic sets of $\mathbb{Z}_{+}^{l}$. We show that a distal null system is equicontinuous in Section 3. Moreover, it turns out that a distal but not equicontinuous system has infinite sequence entropy. In Section 4, we prove that a null system with closed proximal relation is mean equicontinuous and give some applications. In the final section, we construct a distal system with trivial $\text{Ind}_{fip}$-pairs, and a non-trivial regionally proximal relation of order $\infty$.

Acknowledgments. The authors would like to thank Professor X. Ye and Dr. L. Xu for helping discussions and remarks. We also thank Jian Li, Jie Li and T. Yu for useful suggestions. The authors were supported by NNSF of China (11431012).

2. Preliminaries

In this section we recall some notions and aspects of the theories of topological dynamical systems.

2.1. Cubes in $\mathbb{Z}_{+}^{l}$

Denote by ${\mathbb{Z}_{+}}$ ($\mathbb{N}$, $\mathbb{Z}$, respectively) the set of all non-negative integers (positive integers, integers, respectively).

For $l\in\mathbb{N}$, a cube in $\mathbb{Z}_{+}^{l}$ is defined by

[TABLE]

for some $m\in\mathbb{N}.$

A subset $S^{(l)}=\{\overline{s}^{(k)}=(n^{(k)}_{1},n^{(k)}_{2},\ldots,n^{(k)}_{l}):k=1,2,\ldots\}$ of $\mathbb{Z}_{+}^{l}$ is syndetic if there is a cube $P_{m}^{(l)}$, such that for each element $(a_{1},a_{2},\ldots,a_{l})\in\mathbb{Z}_{+}^{l}$, there is $(p_{1},p_{2},\ldots,p_{l})\in P_{m}^{(l)}$ with $(a_{1}+p_{1},a_{2}+p_{2},\ldots,a_{l}+p_{l})\in S^{(l)}$. We say that $m$ is a gap of $S^{(l)}$. Note that when $l=1$ we recover the classical definition of syndetic set in $\mathbb{Z}_{+}$.

A subset $T^{(l)}=\{\bar{n}^{(k)}=(n^{(k)}_{1},n^{(k)}_{2},\ldots,n^{(k)}_{l}):k=1,2,\ldots\}$ of $\mathbb{Z}_{+}^{l}$ is thickly syndetic if for every cube $P_{n}^{(l)}$, there is a syndetic set $S_{n}^{(l)}=\{\bar{s}_{n}^{(k)}:k=1,2,\ldots\}$ such that

[TABLE]

Let $\mathcal{F}_{ts}^{l}$ be the set of all thickly syndetic sets in $\mathbb{Z}_{+}^{l}$.

2.2. Compact metric spaces

Let $(X,d)$ be a compact metric space. For $x\in X$ and $\varepsilon>0$, denote $B(x,\varepsilon)=\{y\in X\colon d(x,y)<\varepsilon\}$. Denote by the product space $X\times X=\{(x,y)\colon x,y\in X\}$ and the diagonal $\Delta_{X}=\{(x,x)\colon x\in X\}$. Let $U\subset X$, denote the diameter of the set $U$ by $\operatorname{diam}(U)=\sup\{d(x,y):x,y\in U\}$.

2.3. Topological dynamics

Let $(X,T)$ be a dynamical system. The orbit of a point $x\in X$, $\{x,Tx,T^{2}x,\ldots,\}$, is denoted by $Orb(x,T)$. The set of limit points of the orbit $Orb(x,T)$ is called the $\omega$-limit set of $x$, and is denoted by $\omega(x,T)$.

If $A$ is a non-empty closed subset of $X$ and $TA\subset A$, then $(A,T|_{A})$ is called a subsystem of $(X,T)$, where $T|_{A}$ is the restriction of $T$ on $A$. If there is no ambiguity, we will use the notation $T$ instead of $T|_{A}$.

We say that a point $x\in X$ is recurrent if $x\in\omega(x,T)$. The system $(X,T)$ is called (topologically) transitive if $\omega(x,T)=X$ for some $x\in X$, and such a point $x$ is called a transitive point.

The system $(X,T)$ is said to be minimal if every point of $X$ is a transitive point. A subset $Y$ of $X$ is called minimal if $(Y,T)$ forms a minimal subsystem of $(X,T)$. A point $x\in X$ is called minimal if it is contained in a minimal set $Y$ or, equivalently, if the subsystem $(\overline{Orb(x,T)},T)$ is minimal.

For $x\in X$ and $A\subset X$, let $N(x,A)=\{n\in{\mathbb{Z}_{+}}\colon T^{n}x\in A\}$. If $U$ is a neighborhood of $x$, then the set $N(x,U)$ is called the set of return times of the point $x$ to the neighborhood $U$. The following result is well-known, see [7] for example.

Lemma 2.1**.**

Let $(X,T)$ be a dynamical system and $x\in X$. Then $x$ is minimal if and only if $N(x,U)$ is syndetic for every neighborhood $U$ of $x$.

A pair of points $(x,y)\in X\times X$ is said to be proximal if for any $\varepsilon>0$, there exists a positive integer $n$ such that $d(T^{n}x,T^{n}y)<\varepsilon$. Let $P(X,T)$ denote the collection of all proximal pairs in $(X,T)$.

The system $(X,T)$ is called distal if the proximal relation is trivial i.e., $P(X,T)=\Delta_{X}$.

The following proposition summarizes some basic properties of distal system:

Proposition 2.2**.**

[3, 1, Chapters 5 and 7]**

1. (1)

The Cartesian product of a finite family of distal systems is a distal system. 2. (2)

If $(X,T)$ is a distal system and $Y$ is a closed and invariant subset of $X$, then $(Y,T)$ is a distal system.

Recall that a pair of points $(x,y)$ is called regionally proximal if for every $\varepsilon>0$, there exist two points $x^{\prime},y^{\prime}\in X$ with $d(x,x^{\prime})<\varepsilon$ and $d(y,y^{\prime})<\varepsilon$, and a positive integer $n$ such that $d(T^{n}x^{\prime},T^{n}y^{\prime})<\varepsilon$. Let $Q(X,T)$ be the set of all regionally proximal pairs in $(X,T)$.

Clearly, we have $P(X,T)\subset Q(X,T)$.

When $(X,T)$ and $(Y,S)$ are two dynamical systems and $\pi\colon X\to Y$ is a continuous onto map which intertwines the actions (i.e., $\pi\circ T=S\circ\pi$), one says that $(Y,S)$ is a factor of $(X,T)$ or $(X,T)$ is an extension of $(Y,S)$, and $\pi$ is a factor map. If $\pi$ is a homeomorphism, then we say that $\pi$ is a conjugacy and that the dynamical systems $(X,T)$ and $(Y,S)$ are conjugate.

If $\pi\colon(X,T)\to(Y,S)$ is a factor map, then $R_{\pi}=\{(x,x^{\prime})\in X\times X\colon\pi(x)=\pi(x^{\prime})\}$ is closed $T\times T$-invariant equivalence relation, that is $R_{\pi}$ is a closed subset of $X\times X$ and if $(x,x^{\prime})\in R_{\pi}$, then $(Tx,Tx^{\prime})\in R_{\pi}$. Conversely, if $R$ is a closed $T\times T$-invariant equivalence relation on $X$, then the quotient space $X/R$ is a compact metric space and $T$ naturally induces an action on $X/R$ by $T_{R}([x])=[Tx]$. Then $(X/R,T_{R})$ forms a dynamical system and the quotient map $\pi_{R}\colon X\to X/R$ is a factor map. Hence there is a one-to-one correspondence between factors and closed invariant equivalence relations, we will use them interchangeably.

2.4. Invariant measures

Let $(X,T)$ be a dynamical system and $M(X,T)$ be the set of $T$-invariant regular Borel probability measures on $X$. An invariant measure is ergodic if and only if it is an extreme point of $M(X,T)$. Denote by $M^{e}(X,T)$ the set of all ergodic measure.

For $\mu\in M(X,T)$. We define the support of $\mu$ by $\operatorname{supp}(\mu)=\{x\in X:\mu(U)>0\;\text{for any}$ neighborhood $U$ of $x\}.$ The support of a dynamical system $(X,T)$, denoted by $\operatorname{supp}(X,T)$, is the smallest closed subset $C$ of $X$ such that $\mu(C)=1$ for all $\mu\in M(X,T)$.

Let $(X,T)$ be a dynamical system. We call $(X,T)$ an $E$-system if it is transitive and there exists $\mu\in M(X,T)$ such that $\operatorname{supp}(\mu)=X$. We say that $(X,T)$ is uniquely ergodic if $M(X,T)$ consists of a single measure.

2.5. Topological sequence entropy

Let $\mathrm{A}=\{0\leq t_{1}<t_{2}<\cdots\}\subseteq{\mathbb{Z}_{+}}$ be an increasing sequence of natural numbers and $\mathcal{U}$ be a finite cover of $X$. The topological sequence entropy of $\mathcal{U}$ with respect to $(X,T)$ along $\mathrm{A}$ is defined by

[TABLE]

where $\mathrm{N}(\bigvee_{i=1}^{n}T^{-t_{i}}\mathcal{U})$ is the minimal cardinality among all cardinalities of sub-covers of $\bigvee_{i=1}^{n}T^{-t_{i}}\mathcal{U}$. The topological sequence entropy of $(X,T)$ along $\mathrm{A}$ is

[TABLE]

where supremum is taken over all finite open covers of $X$ (that is, made up of open sets). If $\mathrm{A}=\mathbb{N}$ we recover standard topological entropy. In this case we omit the superscript $\mathbb{N}$.

2.6. Sequence entropy $n$-tuple

Given a dynamical system $(X,T)$ and an integer $n\geq 2$, the $n$-th product system is the dynamical system $(X^{(n)},T^{(n)})$, where $X^{(n)}$ is the cartesian product of $X$ with itself $n$ times and $T^{(n)}$ represents the simultaneous action $T$ in each coordinate of $X^{(n)}$. The diagonal of $X^{(n)}$ is denoted by $\Delta_{n}(X)=\{(x,\ldots,x)\in X^{(n)}:x\in X\}.$

Let $(x_{i})_{i=1}^{n}\in X^{(n)}$. A finite cover $\mathcal{U}=\{U_{1},\ldots,U_{k}\}$ of $X$ is said to be *admissible cover * with respect to $(x_{i})_{i=1}^{n}$ if for each $1\leq j\leq k$ there exists $1\leq i_{j}\leq n$ such that $x_{i_{j}}$ is not contained in the closure of $U_{j}$.

Definition 2.3**.**

Let $(X,T)$ be a dynamical system. An $n$-tuple $(x_{i})_{i=1}^{n}\in X^{(n)},n\geq 2$, is called a sequence entropy $n$-tuple if for some $1\leq i,j\leq n,x_{i}\neq x_{j},$ for any admissible open cover $\mathcal{U}$ with respect to $(x_{i})_{i=1}^{n}$ there exists an increasing sequence of natural numbers $\mathrm{A}$ such that $h_{top}^{\mathrm{A}}(T,\mathcal{U})>0.$

We denote by $SE_{n}(X,T)$ the set of sequence entropy $n$-tuples. Denote by $SE_{n}^{e}(X,T)$ the set of essential sequence entropy $n$-tuples, which means $(x_{1},\ldots,x_{n})\in SE_{n}(X,T)$ with $x_{i}\neq x_{j},1\leq i<j\leq n$. Sequence entropy 2-tuples are called sequence entropy pairs. Denote by $SE(X,T)$ the set of all sequence entropy pairs.

Remark 2.4**.**

By the definition, it follows that if $(x_{1},\ldots,x_{n})\in SE_{n}(X,T)$, then we have $(x_{i_{1}},\ldots,x_{i_{k}})\in SE_{k}(X,T)\cup\Delta_{k}(X)$ for any $1\leq i_{1}<\cdots<i_{k}\leq n,k\geq 2$.

We have the following proposition:

Proposition 2.5**.**

[17, Proposition 3.2]** Let $(X,T)$ be a dynamical system.

1. (1)

If $\mathcal{U}=\{U_{1},\ldots,U_{k}\}$ is an open cover of $X$ with $h_{top}^{\mathrm{A}}(T,\mathcal{U})>0$ for some increasing sequence of natural numbers $\mathrm{A}$, then for all $1\leq i\leq n$ there exists $x_{i}\in U_{i}^{c}$ such that $(x_{i})_{i=1}^{n}$ is a sequence entropy $n$-tuple. 2. (2)

$SE_{n}(X,T)\cup\Delta_{n}(X)$* is a closed $T^{(n)}$-invariant subset of $X^{(n)}$.* 3. (3)

Let $\pi:(Y,S)\longrightarrow(X,T)$ be a factor map of dynamical system.

1. (a)

If $(x_{i})_{i=1}^{n}\in SE_{n}(X,T)$, then for all $1\leq i\leq n$ there exists $y_{i}\in Y$ such that $\pi(y_{i})=x_{i}$ and $(y_{i})_{i=1}^{n}\in SE_{n}(Y,S).$ Moreover, if $SE_{n}^{e}(X,T)\neq\emptyset$, then we have $SE_{n}^{e}(Y,S)\neq\emptyset$. 2. (b)

If $(y_{i})_{i=1}^{n}\in SE_{n}(Y,S)$ and $(\pi(y_{i}))_{i=1}^{n}\notin\Delta_{n}(X),$ then $(\pi(y_{i}))_{i=1}^{n}\in SE_{n}(X,T)$.

Proposition 2.6**.**

[19, Theorem 5.9]** Let $(X,T)$ be a dynamical system and $(x_{i})_{i=1}^{n}\in X^{(n)}$ with $x_{i}\neq x_{j},i\neq j.$ If for any neighborhoods $U_{i}$ of $x_{i}$ ($i=1,\ldots,n$) respectively, and any $k\in\mathbb{N}$ there is a sequence $\mathrm{A}=\{0\leq t_{1}<t_{2}<\cdots<t_{k}\}$ of ${\mathbb{Z}_{+}}$ such that for any $s=(s(1),\ldots,s(k))\in\{1,\ldots,n\}^{k}$,

[TABLE]

then $(x_{i})_{i=1}^{n}\in SE_{n}(X,T)$.

Remark 2.7**.**

By this proposition, we know that a system $(X,T)$ is null if and only if $SE(X,T)=\emptyset$.

2.7. Independence and $\mathbf{RP}^{[d]}(X)$

Let $(X,T)$ be a dynamical system and $\mathcal{A}=(A_{1},\ldots,A_{k})$ be a tuple of subsets of $X$. A subset $F\subset\mathbb{Z}_{+}$ is an independence set for $\mathcal{A}$ if for any nonempty finite subset $J\subset F$ and any $s=(s(j):j\in J)\in\{1,2,\ldots,k\}^{|J|}$ we have $\cap_{j\in J}T^{-j}A_{s(j)}$ $\neq\emptyset$. For a finite subset $\{p_{1},\ldots,p_{m}\}$ of $\mathbb{N}$, the finite IP-set generated by $\{p_{1},\cdots,p_{m}\}$ is the set $\{\epsilon_{1}p_{1}+\cdots+\epsilon_{m}p_{m}:\epsilon_{i}\in\{0,1\},1\leq i\leq m\}\backslash\{0\}$. A pair of points $(x,y)\in X\times X$ is an $\text{Ind}_{fip}$-pair if and only if each $\mathcal{A}=(A_{1},A_{2})$, with $A_{1}$ and ${A_{2}}$ neighborhoods of $x$ and $y$ respectively, contains finite IP-independence sets of length $m$ for any $m\in\mathbb{N}$.

A pair $(x,y)\in X\times X$ is said to be regionally proximal of order d if for any $\delta>0$, there exist $x^{\prime},y^{\prime}\in X$ and a vector ${\bf{n}}=(n_{1},\ldots,n_{d})\in\mathbb{Z}_{+}^{d}$ such that $d(x,x^{\prime})<\delta$, $d(y,y^{\prime})<\delta$, and $d(T^{{\bf{n}}\cdot\epsilon}x^{\prime},T^{{\bf{n}}\cdot\epsilon}y^{\prime})<\delta$ for any $\epsilon=(\epsilon_{1},\ldots,\epsilon_{d})\in\{0,1\}^{d}\backslash\{\mathbf{0}\}$, where ${\bf{n}}\cdot\epsilon=\sum_{i=1}^{d}\epsilon_{i}n_{i}$. The set of regionally proximal pairs of order $d$ is denoted by $\mathbf{RP}^{[d]}(X)$. In  [14] the authors showed that if the system is minimal and distal then $\mathbf{RP}^{[d]}(X)$ is an equivalence relation and $(X/\mathbf{RP}^{[d]}(X),T)$ is the maximal $d$-step nilfactor of the system.

2.8. $\mathcal{F}_{ts}^{l}$ is a filter

We recall some notions related to a family. Denote by $\mathcal{P}^{l}=\mathcal{P}^{l}(\mathbb{Z}_{+}^{l})$ the collection of all subsets of $\mathbb{Z}_{+}^{l}$. A subset $\mathcal{F}^{l}$ of $\mathcal{P}^{l}$ is called a family, if it is hereditary upward, i.e., $F_{1}\subset F_{2}$ and $F_{1}\in\mathcal{F}^{l}$ imply $F_{2}\in\mathcal{F}^{l}$. A family $\mathcal{F}^{l}$ is called proper if it is a non-empty proper subset of $\mathcal{P}^{l}$, i.e., neither empty nor all of $\mathcal{P}^{l}$. A proper family $\mathcal{F}^{l}$ is called a filter if $F_{1},F_{2}\in\mathcal{F}^{l}$ implies $F_{1}\cap F_{2}\in\mathcal{F}^{l}$. Recall that $\mathcal{F}_{ts}^{l}$ the set of all thickly syndetic sets in $\mathbb{Z}_{+}^{l}$. Then we have the following theorem.

Proposition 2.8**.**

For $l\in\mathbb{N}$, if $F_{1},F_{2}\in\mathcal{F}_{ts}^{l}$, then we have $F_{1}\cap F_{2}\in\mathcal{F}_{ts}^{l}$.

Proof.

For $l\in\mathbb{N}$, assume that $F_{1},F_{2}\in\mathcal{F}_{ts}^{l}$.

Fix $n\in\mathbb{N}$, as $F_{1}\in\mathcal{F}_{ts}^{l}$, there is a syndetic set $S_{n}^{(l)}=\{\bar{s}_{n}^{(k)}:k=1,2,\ldots\}$ with

[TABLE]

Let $m\in\mathbb{N}$ be a gap of $S_{n}^{(l)}$.

Put $d=m+n$. Again, as $F_{2}\in\mathcal{F}_{ts}^{l}$, there is a syndetic set $S_{d}^{(l)}=\{\bar{s}_{d}^{(k)}:k=1,2,\ldots\}$ with

[TABLE]

For every $k\in\mathbb{N}$, there is a $\overline{m}_{k}\in P_{m}^{(l)}$ with $\bar{s}_{d}^{(k)}+\overline{m}_{k}\in S_{n}^{(l)},$ hence $\bar{s}_{d}^{(k)}+\overline{m}_{k}+P_{n}^{(l)}\subset F_{1}.$

Thus

[TABLE]

Clearly, $\{\bar{s}_{d}^{(k)}+\overline{m}_{k}\}_{k=1}^{\infty}$ is syndetic which implies $F_{1}\cap F_{2}\in\mathcal{F}_{ts}^{l}$. ∎

By induction, we can easily obtain the following corollary.

Corollary 2.9**.**

For $l,n\in\mathbb{N}$, if $F_{1},F_{2},\ldots,F_{n}\in\mathcal{F}_{ts}^{l}$, then $\cap_{i=1}^{n}F_{i}\in\mathcal{F}_{ts}^{l}$.

3. Systems with sequence entropy $n$-tuples

In this section we will give the conditions under which a dynamical system has a sequence entropy $n$-tuple.

Lemma 3.1**.**

Let $X$ be a dynamical system with fixed points $x_{i},i=1,2,\ldots,n,\ n\geq 2$. If for any neighbourhood $W_{i}$ of $x_{i},i=1,2,\cdots,n$, there exists a minimal point $y$ such that for every $i=1,2,\cdots,n$ there is $m_{i}\in{\mathbb{Z}_{+}}$ with $T^{m_{i}}y\in W_{i}$, then $(x_{1},x_{2},\ldots,x_{n})\in SE_{n}^{e}(X,T)$.

Proof.

Let $U_{i}$ be a neighbourhood of $x_{i},i=1,2,\cdots,n$. There is a $\delta>0$ with $B(x_{i},\delta)\subset U_{i},i=1,2,\ldots,n$. For a given $N\in\mathbb{N}$, by the uniform continuity, there is a $\delta^{\prime}>0$ such that $d(T^{j}u,T^{j}v)<\delta$ for every $1\leq j\leq N$ and $u,v\in X$ with $d(u,v)<\delta^{\prime}$.

We can choose a minimal point $y\in X$ and $m_{i}\in{\mathbb{Z}_{+}}$ such that $d(x_{i},T^{m_{i}}y)<\frac{\delta^{\prime}}{2}$ for all $i=1,2,\ldots,n$. Put $V_{i}=B(x_{i},\delta^{\prime})$, then $N(y,V_{i})$ is not empty, $i=1,2,\ldots,n$.

For $l\in\mathbb{N}$ and $s\in\{1,2,\ldots,n\}^{l}$, let $V^{(s)}=\prod_{i=1}^{l}V_{s(i)}$ and $U^{(s)}=\prod_{i=1}^{l}U_{s(i)}$. Then

[TABLE]

is syndetic since $y$ is a minimal point.

For every $\bar{n}=(n_{1},n_{2},\ldots,n_{l})\in A_{s}^{(l)}$, $\bar{p}=(p_{1},p_{2},\ldots,p_{l})\in P_{N}^{(l)}$ and $i=1,2,\ldots,l$, we have $T^{n_{i}}y\in V_{s(i)}$. It follows that $d(T^{n_{i}}y,x_{s(i)})<\delta^{\prime}$ which implies $d(T^{n_{i}+p_{i}}y,x_{s(i)})<\delta$ for every $0\leq p_{i}\leq N$. Thus,

[TABLE]

That is, $B_{s}^{(l)}$ is thickly syndetic.

By Corollary 2.9, it follows that

[TABLE]

Pick $(n_{1},n_{2},\ldots,n_{l})\in\cap_{s\in\{1,\ldots,n\}^{l}}B_{s}^{(l)}$. Without loss of generality, assume $n_{1}<n_{2}<\cdots<n_{l}$. Then,

[TABLE]

for any $s\in\{1,2,\ldots,n\}^{l}$, which implies $(x_{1},x_{2},\ldots,x_{n})\in SE_{n}(X,T)$ by Proposition  2.6. ∎

Theorem 3.2**.**

Let $(X,T)$ be a distal and null system, then it is equicontinuous.

Proof.

Assume that $(X,T)$ is not equicontinuous, then there is $(x,y)\in Q(X,T)\setminus\Delta_{X}$, with $x_{k},y_{k},z\in X$ and $n_{k}\in{\mathbb{Z}_{+}},k=1,2,\cdots$ such that

[TABLE]

As $(X,T)$ is distal, $T$ is invertible and $(X\times X,T\times T)$ is pointwise minimal by Proposition 2.2. Without loss of generality, we may assume that $T^{-n_{k}}z\rightarrow w$, then $((x,y),(w,w))\in Q(X\times X,T\times T)$ since

[TABLE]

Put $\mathbf{x}=(x,y),\mathbf{w}=(w,w),\mathbf{x_{k}}=(x_{k},y_{k}),\mathbf{z}=(z,z),\widetilde{X}=X\times X$ and $\widetilde{T}=T\times T$.

Let

[TABLE]

It is easy to see that $R$ is a $\widetilde{T}\times\widetilde{T}$-invariant closed equivalence relation. Let $Y=\widetilde{X}/R$ (i.e. we collapse $\overline{Orb(\mathbf{x},\widetilde{T})}$ to a point, and collapse $\overline{Orb(\mathbf{w},\widetilde{T})}$ to a point), then $[\mathbf{x}],[\mathbf{w}]$ are fixed points in $Y$.

It is easy to see $\lim_{k\rightarrow\infty}d([\mathbf{x_{k}}],[\mathbf{x}])=0$ and $\lim_{k\rightarrow\infty}d(\widetilde{T}^{n_{k}}[\mathbf{x_{k}}],[\mathbf{z}])=0$ since $[\mathbf{w}]=[\mathbf{z}]$. By Lemma 3.1, it follows that $([\mathbf{x}],[\mathbf{z]})\in SE(Y,\widetilde{T})$ which implies $SE(\widetilde{X},\widetilde{T})\neq\emptyset$. We can choose $((c_{1},c_{2}),(d_{1},d_{2}))\in SE(\widetilde{X},\widetilde{T})$ with $c_{1}\neq d_{1}$ by Proposition 2.5.

Now consider the projection $\pi_{1}$ from $X\times X$ to the first coordinate, again by Proposition 2.5, we have $(c_{1},d_{1})\in SE(X,T).$ It is a contradiction which shows the theorem. ∎

By the proof above, we can show more. That is, if $(X,T)$ is distal but not equicontinuous, then for every $n\geq 2,SE_{n}^{e}(X,T)\neq\emptyset.$

Before showing the statement, we need the following lemma.

Lemma 3.3**.**

Let $(X,T)$ be a distal dynamical system with fixed points $x_{1},x_{2}$. If there exist points $y_{k}\in X$ and $n_{k}\in{\mathbb{Z}_{+}},k=1,2,\cdots$ such that

[TABLE]

then for every $n\geq 2$, $SE_{n}^{e}(X,T)\neq\emptyset$.

Proof.

As $(X,T)$ is distal, $y_{k},k=1,2,\cdots$ are minimal. By Lemma 3.1, it is clear for $n=2$. Let $U_{1}$ and $U_{2}$ be neighbourhoods of $x_{1}$ and $x_{2}$ respectively with $\overline{U_{1}}\cap\overline{U_{2}}=\emptyset$.

Claim 1**.**

Let $U_{1}^{\prime}=U_{1}\cap T^{-1}U_{1}$ and $U_{2}^{\prime}=U_{2}\cap T^{-1}U_{2}.$ There is a $K\in{\mathbb{Z}_{+}}$ such that for every $k\geq K$, we can choose $l_{k}\in{\mathbb{Z}_{+}}$ with $T^{l_{k}}y_{k}\notin U_{1}^{\prime}\cup U_{2}^{\prime}$.

Proof of the Claim.

Clearly, $U_{1}^{\prime}$ and $U_{2}^{\prime}$ are still neighbourhoods of $x_{1}$ and $x_{2}$ respectively since $x_{1}$ and $x_{2}$ are fixed points. As $\lim_{k\rightarrow\infty}d(y_{k},x_{1})=0$, there is a $K\in{\mathbb{Z}_{+}}$ such that $y_{k}\in U_{1}^{\prime}$ for all $k\geq K$.

Fix $k\geq K,$ put $l_{k}=\min\{n\in{\mathbb{Z}_{+}}:T^{n+1}y_{k}\in U_{2}^{\prime}\},$ then we have $T^{l_{k}}y_{k}\notin U_{1}^{\prime}$. Indeed, if $T^{l_{k}}y_{k}\in U_{1}^{\prime}$, we have $T^{l_{k}}y_{k},T^{l_{k}+1}y_{k}\in U_{1}$. By the definition of $l_{k}$, it follows that $T^{l_{k}+1}y_{k}\in U_{2}^{\prime}\subseteq U_{2}$, it is a contradiction. Thus $T^{l_{k}}y_{k}\notin U_{1}^{\prime}\cup U_{2}^{\prime}$. ∎

By considering the limit points of $\{T^{l_{k}}y_{k}\}_{k=K}^{\infty}$, we may assume that $T^{l_{k}}y_{k}\rightarrow x_{3}$ as $k\rightarrow\infty$. Then, it is clear that $x_{3}\neq x_{1},x_{2}.$

Let $R_{1}=\{(u,v)\in X\times X:u,v\in\overline{Orb(x_{3},T)}\}\cup\Delta_{X}$ and it is a $T\times T$-invariant closed equivalence relation. Put $X_{1}=X,X_{2}=X_{1}/R_{1}$, then we have $x_{1},x_{2},x_{3}$ (instead of $[x_{1}],[x_{2}],[x_{3}]$) are fixed points of $X_{2}$ and $\lim_{k\rightarrow\infty}d(T^{l_{k}}y_{k},x_{3})=0$. As $y_{k}$ is minimal in $X_{1}$, so is in $X_{2}$.

Again by Lemma 3.1, we have $(x_{1},x_{2},x_{3})\in SE_{3}(X_{2},T)$ which implies $SE_{3}(X,T)=SE_{3}(X_{1},T)\neq\emptyset,$ by Proposition 2.5. Moreover, we have $SE_{3}^{e}(X,T)\neq\emptyset$ since $x_{3}\neq x_{1},x_{2}.$.

Now assume that $x_{1},x_{2},\ldots,x_{m}$ are fixed points of $X_{m-1}$ and $y_{k}\in X_{m-1},n^{(1)}_{k},n^{(2)}_{k},\ldots,n^{(m)}_{k}\in{\mathbb{Z}_{+}},n^{(1)}_{k}=0,k\in\mathbb{N}$ with $\lim_{k\rightarrow\infty}d(x_{i},T^{n^{(i)}_{k}}y_{k})=0,i=1,2,\ldots,m$.

Choose neighborhoods $U_{1},U_{2},\ldots,U_{m}$ of $x_{1},x_{2},\ldots,x_{m}$ respectively with $\overline{U_{i}}\cap\overline{U_{j}}=\emptyset$ for $1\leq i\neq j\leq m$.

Claim 2**.**

Let $U_{i}^{\prime}=U_{i}\cap T^{-1}U_{i},i=1,2,\ldots,m.$ There is a $K\in{\mathbb{Z}_{+}}$ such that for every $k\geq K$, we can choose $m_{k}\in{\mathbb{Z}_{+}}$ with $T^{m_{k}}y_{k}\notin\cup_{i=1}^{m}U_{i}^{\prime}$.

Proof.

It is similar to the proof of Claim 1. ∎

By considering the limit points of $\{T^{m_{k}}y_{k}\}_{k=K}^{\infty}$, we may assume that $T^{m_{k}}y_{k}\rightarrow x_{m+1}$ as $k\rightarrow\infty$. It is clear that $x_{m+1}\neq x_{i},i=1,2,\ldots,m.$

Let $R_{m-1}=\{(u,v)\in X_{m-1}\times X_{m-1}:u,v\in\overline{Orb(x_{m+1},T)}\}\cup\Delta_{X_{m-1}}$. Then it is a $T\times T$-invariant closed equivalence relation. Put $X_{m}=X_{m-1}/R_{m-1}$, then we have $x_{1},x_{2},\ldots,x_{m+1}$ (instead of $[x_{1}],[x_{2}],\ldots,[x_{m+1}]$) are fixed points of $X_{m}$ and $\lim_{k\rightarrow\infty}d(T^{m_{k}}y_{k},x_{m+1})=0$. Moreover, we have $y_{k}$ is minimal in $X_{m}.$

By Lemma 3.1, we have $(x_{1},x_{2},\ldots,x_{m+1})\in SE_{m+1}(X_{m},T)$ and so $SE_{m+1}^{e}(X,T)\neq\emptyset$ by repeatedly applying Proposition 2.5. Thus the proof is finished by induction. ∎

Now we are ready to show our statement.

Theorem 3.4**.**

Let $(X,T)$ be a distal but not equicontinuous system, then for every $n\geq 2,\ SE_{n}^{e}(X,T)\neq\emptyset.$ Consequently, the sequence entropy of $(X,T)$ is $\infty$ by the result in [18].

Proof.

Assume that $(x,y)\in Q(X,T)\backslash\Delta_{X}$ with $x_{k},y_{k},z\in X$ and $n_{k}\in{\mathbb{Z}_{+}},k=1,2,\ldots$ such that

[TABLE]

As $(X,T)$ is distal, it follows that for every $k\in\mathbb{N},(x_{k},y_{k})$ is minimal in $X\times X$ and $\overline{Orb((x,y),T\times T)}\cap\Delta_{X}=\emptyset$. By collapsing $\overline{Orb((x,y),T\times T)}$ and $\Delta_{X}$ to points respectively, we obtain a factor $(Y,S)$ of $(X\times X,T\times T)$ which satisfies the condition (1). This implies $SE_{n}^{e}(Y,S)\neq\emptyset$ for every $n\geq 2$. Moreover, $SE_{n}^{e}(X\times X,T\times T)\neq\emptyset$ for every $n\geq 2$.

Fix $n\geq 2,$ choose $((u_{i},v_{i}))_{i=1}^{n^{2}}\in SE_{n^{2}}^{e}(X\times X,T\times T)$.

For every $z\in X$, let $I_{z}=\{u_{i}:1\leq i\leq n^{2},u_{i}=z\}$. Clearly, there exists $1\leq m\leq n^{2},z_{j}\in X,j=1,\ldots,m$ such that $I_{z_{j}}\neq\emptyset$.

If $m\geq n$, let $u_{i_{j}}\in I_{z_{j}}$ for some $1\leq i_{j}\leq n^{2}$, then $((u_{i_{j}},v_{i_{j}}))_{j=1}^{n}\in SE_{n}^{e}(X\times X,T\times T)$ by Remark 2.4. Now consider the projection $\pi_{1}$ from $X\times X$ to the first coordinate, it follows that $(u_{i_{j}})_{j=1}^{n}\in SE_{n}^{e}(X,T)$ by Proposition 2.5.

If $m<n$, there exists $1\leq i_{0}\leq m$ with $l=|I_{z_{i_{0}}}|\geq\frac{n^{2}}{m}>n.$

Suppose that $u_{i_{j}}=z_{i_{0}},j=1,\ldots,l$, then $((u_{i_{j}},v_{i_{j}}))_{j=1}^{n}\in SE_{n}^{e}(X\times X,T\times T)$. Now consider the projection $\pi_{2}$ from $X\times X$ to the second coordinate, it follows that $(v_{i_{j}})_{j=1}^{n}\in SE_{n}^{e}(X,T)$ by the Proposition 2.5. Thus we complete the proof. ∎

4. Null systems and mean equicontinuity

In this section we will discuss in which case a null dynamical system is mean equicontinuous. We start with the following characterizations of mean equicontinuous systems.

Before showing our theorem, we need the following lemmas.

Lemma 4.1**.**

[2, Corollary 1]** If $(X,T)$ is a dynamical system and $P(X,T)$ is closed in $X\times X$, then it is a $T\times T$-invariant closed equivalence relation.

Lemma 4.2**.**

[15, Theorem 3.1]** If $(X,T)$ is a transitive not minimal $E$-system, then we have $SE(X,T)\neq\emptyset$.

Now we are ready to show

Theorem 4.3**.**

Let $(X,T)$ be a null system. Then the following conditions are equivalent:

1. (1)

$P(X,T)$* is closed;* 2. (2)

$P(X,T)=Q(X,T);$** 3. (3)

$(X,T)$* is mean equicontinuious.*

Proof.

By [22, Theorem 3.5] we know that (3) implies (2). It is clear that (2) implies (1). It remains to show (1) $\Rightarrow$ (3).

Since $P(X,T)$ is closed in $X\times X$, $P(X,T)$ is a $T\times T$-invariant closed equivalence relation by Lemma  4.1. Let $Y=X/P(X,T)$ and $\pi:X\rightarrow Y$ be the factor map, then $Y$ is the maximal distal factor. As $(X,T)$ is a null system, so is $(Y,T)$ by Proposition 2.5. Moreover, $(Y,T)$ is equicontinuious by Theorem 3.2.

Assume the contrary $(X,T)$ is not mean equicontinuous, then there are points $x_{k},y_{k},z\in X$, positive integers $n_{k}\in\mathbb{N},k=1,2,\cdots$ and $\varepsilon_{0}>0$ such that $\lim_{k\rightarrow\infty}x_{k}=z=\lim_{k\rightarrow\infty}y_{k}$, and for every $k\in\mathbb{N}$, we have:

[TABLE]

Let $\mu_{k}=\frac{1}{n_{k}}\sum_{i=0}^{n_{k}-1}\delta_{(T^{i}x_{k},T^{i}y_{k})}$. We may assume $\mu_{k}\rightarrow\mu$ as $k\rightarrow\infty$ (otherwise consider the subsequence), where $\mu\in M(X\times X,T\times T)$.

We claim that $\mu(\text{supp}(\mu)\setminus\Delta_{X})>0$. Actually, $d(\cdot,\cdot)$ is a continuous function on $X\times X$, then we have when $k\rightarrow\infty$,

[TABLE]

and

[TABLE]

which implies

[TABLE]

By ergodic decomposition, we have $\nu(\text{supp}(\mu)\setminus\Delta_{X})>0$ for some ergodic measure $\nu$ on $X\times X$.

Case 1: $\operatorname{supp}(\nu)$ is not minimal.

It follows that $\operatorname{supp}(\nu)$ is an $E$-system which is not minimal, then $SE(\operatorname{supp}(\nu),T\times T)\neq\emptyset$ by Lemma  4.2, which implies $SE(X,T)\neq\emptyset.$

Case 2: $\operatorname{supp}(\nu)$ is minimal.

Assume that $\operatorname{supp}(\nu)=\overline{Orb((u,v),T\times T)}$, where $(u,v)\notin\Delta_{X}$.

For $l\in\mathbb{N}$, let $B_{l}=\{(x,y)\in X\times X:d((x,y),(u,v))<\frac{1}{l}\}$, then $\mu(B_{l})>0$. Since $0<\mu(B_{l})\leq\liminf_{k\rightarrow\infty}\mu_{k}(B_{l})$ and

[TABLE]

thus there is $k_{l},i_{l}\in\mathbb{N}$ with $0\leq i_{l}\leq n_{k_{l}}-1$ such that $(T^{i_{l}}x_{k_{l}},T^{i_{l}}y_{k_{l}})\in B_{l}.$ Hence we have

[TABLE]

Let $(Z,S)$ be the maximal equicontinuous factor of $(X,T)$ and $\pi\colon(X,T)\to(Z,S)$ be the factor map. Then $R_{\pi}=\{(x,y)\in X\times X:\pi(x)=\pi(y)\}\subset P(X,T)$. Fix $\varepsilon>0$. As $\pi$ is continuous, there is $\delta_{1}>0$ such that $d(\pi(a),\pi(b))<\frac{\varepsilon}{3}$ whenever $a,b\in Z$ with $d(a,b)<\delta_{1}$. As $(Z,S)$ is equicontinuous, there is $\delta_{2}>0$ such that $d(S^{n}a,S^{n}b)<\min\{\frac{\varepsilon}{3},\delta_{1}\}$ for every $n\in\mathbb{N}$ whenever $a,b\in Z$ with $d(a,b)<\delta_{2}$. Put $\delta=\min\{\delta_{1},\delta_{2}\}$. Now choose $l\in\mathbb{N}$ such that $d(x_{k_{l}},y_{k_{l}})<\delta_{2}$ and $d(u,T^{i_{l}}x_{k_{l}})<\delta_{1},d(v,T^{i_{l}}y_{k_{l}})<\delta_{1}.$ Then

[TABLE]

[TABLE]

which implies $\pi(u)=\pi(v)$. Moreover $(u,v)\in P(X,T)$, $\overline{Orb((u,v),T\times T)}\cap\Delta_{X}\neq\emptyset$. It is a contradiction. Thus $(X,T)$ is mean equicontinuous. ∎

Theorem 4.4**.**

For a null system $(X,T)$, if it has dense minimal points, then it is mean equicontinuous.

Proof.

It is sufficient to show that $Q(X,T)=P(X,T)$ by Theorem 4.3.

If there is a pair $(x,y)\in Q(X,T)\setminus P(X,T)$, by the definition of $Q(X,T)$, there are $x_{k},y_{k},z\in X$ and $n_{k}\in\mathbb{N},k=1,2,\ldots$ such that

[TABLE]

As the minimal points for $T$ are dense in $X$, so are for $T\times T$ in $X\times X$ (see [1]).Without loss of generality, we can assume that $(x_{k},y_{k})$ is minimal in $X\times X$ for every $k\in\mathbb{N}.$

Put $\mathbf{x}=(x,y),\mathbf{x_{k}}=(x_{k},y_{k}),\mathbf{z}=(z,z)$ and $\widetilde{X}=X\times X,\widetilde{T}=T\times T.$

Since $(x,y)\notin P(X,T)$, we have $\overline{Orb(\mathbf{x},\widetilde{T})}\cap\overline{Orb(\mathbf{z},\widetilde{T})}=\emptyset$. Let

[TABLE]

It is easy to see that $R$ is a closed $\widetilde{T}\times\widetilde{T}$-invariant equivalence relation on $\widetilde{X}$. Let $Y=\widetilde{X}/R$, it follows that $[\mathbf{x}],[\mathbf{z}]$ are fixed points in $Y$ and $\lim_{k\rightarrow\infty}[\mathbf{x_{k}}]=[\mathbf{z}],\lim_{k\rightarrow\infty}\widetilde{T}^{n_{k}}[\mathbf{x_{k}}]=[\mathbf{x}].$ As $\mathbf{x_{k}}$ is minimal in $\widetilde{X}$, so is $[\mathbf{x_{k}}]$ in $Y$. By Lemma 3.1, we have $([\mathbf{x}],[\mathbf{z}])\in SE(\widetilde{X},\widetilde{T}),$ which implies $SE(X,T)\neq\emptyset$ by Proposition 2.5.

It is a contradiction which shows $Q(X,T)=P(X,T)$. ∎

Corollary 4.5**.**

Let $(X,T)$ be a null system. If $\operatorname{supp}(X,T)=X$, then it is mean equicontinuous.

Proof.

By [21, Lemma 3.3], we have

[TABLE]

Clearly, $(\operatorname{supp}(\nu),T)$ is minimal since a transitive and non-minimal $E$-system is not null [15, Theorem 3.1]. Thus the minimal points for $T$ in $X$ are dense which implies $(X,T)$ is mean equicontinuous by Theorem 4.4. ∎

5. A counterexample

In the last part of this paper, we give the example which is mentioned in the introduction. That is,

Example 5.1**.**

There is a distal system $(X,T)$ (consisting of periodic orbits) such that $\mathbf{RP}^{[\infty]}(X,T)\neq\Delta_{X}$ and $\text{Ind}_{fip}(X,T)=\emptyset.$

Let $A\in[0,1]\times[-1,1]$, denote the horizontal ordinate and vertical ordinate by $x_{A}$ and $y_{A}$ respectively. We define a relation $\sim$ on $[0,1]\times[-1,1]$ by $(x,1)\sim(x,-1)$ for $x\in[0,1]$. Clearly, it is an equivalence relation.

Assume $A(x_{1},y_{1}),B(x_{2},y_{2})\in[0,1]\times[-1,1]/\sim$, let

[TABLE]

it is easy to check that $\rho$ is a metric on $[0,1]\times[-1,1]/\sim$.

We begin to construct $(X,T)$ in $[0,1]\times[-1,1]/\sim$.

Step 1: Basic periodic systems.

First, we construct basic periodic systems $(I_{i},T_{i})$. Put

[TABLE]

The map $T_{i}$ from $I_{i}$ to itself is defined by

[TABLE]

such that $(I_{i},T_{i})$ is a periodic system.

See the following figures for the construction.

y$$x$$C_{1}1B_{1}$$A_{1}$$1$$D_{1}$$D_{1}$$(I_{1},T_{1})

y$$x$$C_{2}$$1$$B_{2}$$1$$A_{2}$$D_{2}$$D_{2}$$(I_{2},T_{2})

y$$x$$1$$B_{3}$$D_{3}$$D_{3}$$A_{3}$$C_{3}$$1$$(I_{3},T_{3})

For every $i$, put $A_{i}=(0,1),B_{i}=(0,0),C_{i}=(1,0),D_{i}=(1,-1)$, and from the construction, it is easy to see that $T^{2^{i+1}}_{i}B_{i}=C_{i},T_{i}^{2i}A_{i}=B_{i}$.

Step 2: Construction of $(X,T)$.

To get the properties, we need glue all the basic periodic systems constructed above. For compactness, it is necessary to compress them first. Precisely, for every $i$, squeeze $I_{i}$ horizontally such that the distance from $B_{i}$ to $C_{i}$ is $\frac{1}{2i(i+1)}$, and horizontally translate $B_{i}$ to $B^{\prime}_{i}(\frac{2i+1}{2i(i+1)},0)$ and $C_{i}$ to $C_{i}^{\prime}(\frac{1}{i},0)$ which is denoted by $I^{\prime}_{i}$. The map $T_{i}^{\prime}$ defined on $I^{\prime}_{i}$ can be induced from $T_{i}$, that is, $T_{i}^{\prime}$ keeps the relative position.

Let $X=\overline{\bigcup_{i=1}^{\infty}I^{\prime}_{i}}$ and $T$ be the map from $X$ to itself such that $T\mid_{I^{\prime}_{i}}=T^{\prime}_{i}$ and $T\mid_{\{0\}\times[-1,1]}$ is identity.

Put $A=\lim_{i\rightarrow\infty}A^{\prime}_{i}(=\lim_{i\rightarrow\infty}D^{\prime}_{i})$ , $B=\lim_{i\rightarrow\infty}B^{\prime}_{i}(=\lim_{i\rightarrow\infty}C^{\prime}_{i})$ and $I=\{0\}\times[-1,1]$ where $A^{\prime}_{i}(\frac{2i+1}{2i(i+1)},1),D^{\prime}_{i}(\frac{1}{i},-1)$.

See the following figure for the construction.

y$$x$$C^{\prime}_{1}$$1$$D^{\prime}_{1}$$B^{\prime}_{1}$$A^{\prime}_{1}$$C^{\prime}_{2}$$D^{\prime}_{2}$$B^{\prime}_{2}$$A^{\prime}_{2}$$C^{\prime}_{3}$$D^{\prime}_{3}$$B^{\prime}_{3}$$A^{\prime}_{3}

Claim 1: For each $d\geq 1$, we have $(A,B)\in\mathbf{RP}^{[d]}(X,T)$, hence $(A,B)\in\mathbf{RP}^{[\infty]}(X,T).$

Proof.

Fix $d\geq 1$, for every $\varepsilon>0$, there is an $i\in\mathbb{N}$ with $\rho(B^{\prime}_{i},B)=\rho(A^{\prime}_{i},A)=\frac{2i+1}{2i(i+1)}<\varepsilon$ and $(1+d)di<2^{i}$.

Put $\bar{n}=(2i,\ldots,2di)\in\mathbb{Z}_{+}^{d}$. For each $\alpha\in\{0,1\}^{d}\backslash\{\mathbf{0}\}$, we have $2i\leq\bar{n}\cdot\alpha\leq(1+d)di<2^{i}$. It follows the construction that $T^{2^{i+1}}B^{\prime}_{i}=C^{\prime}_{i},T^{2i}A^{\prime}_{i}=B^{\prime}_{i}$, which implies

[TABLE]

Thus

[TABLE]

which means $(A,B)\in\mathbf{RP}^{[d]}(X,T)$, hence $(A,B)\in\mathbf{RP}^{[\infty]}(X,T).$ ∎

Remark 5.2**.**

Actually, by the similar argument, we can show that for every $C\in I$ with $C\neq B$, $(C,B)\in\mathbf{RP}^{[\infty]}(X,T).$

Claim 2: $\text{Ind}_{fip}(X,T)=\emptyset.$

Proof.

If there is a pair $(C,D)\in\text{Ind}_{fip}(X,T)$, then $C,D\in I.$ Indeed, as $\text{Ind}_{fip}(I_{i}^{\prime},T)=\emptyset$ for every $i\in\mathbb{N}$, we obtain that $C,D$ cannot belong to the same $I_{i}^{\prime}$. Now if $C$ belongs to $I_{k}^{\prime}$ for some $k\in\mathbb{N}$, we can choose the neighborhoods $U,V$ of $C,D$ respectively which are small enough such that $(\bigcup_{n\in\mathbb{N}}T^{-n}U)\cap(\bigcup_{n\in\mathbb{N}}T^{-n}V)=\emptyset$. Therefore $(C,D)\notin\text{Ind}_{fip}(X,T)$, contracting the assumption.

Without loss of generality, assume that $C=(0,c),D=(0,d)$ with $0\leq d<c\leq 1.$

Choose $k\in\mathbb{N}$ such that $c>\frac{2}{k}$. Let $U_{1},U_{2}$ be $C,D$ neighbourhoods respectively with $\operatorname{diam}(\overline{U_{1}})<\min\{\frac{1}{4}(c-d),\frac{2k+1}{2k(k+1)}\}$ and $\rho(\overline{U_{1}},\overline{U_{2}})>\frac{3}{4}(c-d)$.

As $(C,D)\in\text{Ind}_{fip}(X,T)$, there are $n_{1},n_{2}\in{\mathbb{Z}_{+}}$ and $P\in X$ with

[TABLE]

Assume that $P\in I^{\prime}_{i}$ and the period of $P$ is $m_{i}$. As $P\in U_{1}$ and $\operatorname{diam}(\overline{U_{1}})<\frac{2k+1}{2k(k+1)}$, we obtain $x_{P}<\frac{2k+1}{2k(k+1)}$ which implies $i\geq k$ and

[TABLE]

For $j=1,2$, we have $n_{j}=k_{j}m_{i}+r_{j}$ where $k_{j}\in{\mathbb{Z}_{+}}$ and $0\leq r_{j}<m_{i}$ with

[TABLE]

As $\frac{c}{4}>\operatorname{diam}(\overline{U_{1}})\geq d(C,P)>|c-y_{P}|$, we obtain $y_{P}>\frac{3}{4}c>\frac{1}{k}>\frac{1}{2i}$. Similarly, we also have $y_{T^{r_{2}}P}>\frac{1}{2i}$. It follows the definition of $I_{i}$ that the maximal distance of two adjacent points is $\frac{1}{2i}$, that is, $\rho(TQ,Q)\leq\frac{1}{2i}$, for every $Q\in I_{i}$. We have that $\rho(T^{r_{2}}P,P)+\frac{1}{2i}\geq\rho(T^{r_{2}+s}P,T^{s}P)$ for any $s\in\mathbb{N}$.

Hence

[TABLE]

As $\rho(\overline{U_{1}},\overline{U_{2}})>\frac{3}{4}(c-d)$, it follows that $T^{n_{1}+n_{2}}P\notin U_{2}.$ It is a contradiction which implies the claim. ∎

6. Some open questions

Some questions concerning null systems are open. Here we state two of them. The first one appears in [15].

Question 6.1**.**

Is a null transitive system minimal?

We conjecture that Question 6.1 has a negative answer. A question we can not solve in this paper is:

Question 6.2**.**

Is a null transitive system mean equicontinuous?