Shifts, rotations and distributional chaos

Abstract

Let $R_{r_{0}}, R_{r_{1}}: \mathbb{S}^{1}\longrightarrow \mathbb{S} ^{1}$ be rotations on the unit circle $\mathbb{S}^{1}$ and define $f: \varSigma _{2}\times \mathbb{S}^{1}\longrightarrow \varSigma _{2}\times \mathbb{S}^{1}$ as

$$ f(x, t)=\bigl(\sigma (x), R_{r_{x_{1}}}(t)\bigr), $$

for $x=x_{1}x_{2}\cdots \in \varSigma _{2}:=\{0, 1\}^{\mathbb{N}}$, $t\in \mathbb{S}^{1}$, where $\sigma: \varSigma _{2}\longrightarrow \varSigma _{2}$ is the shift, and $r_{0}$ and $r_{1}$ are rotational angles. It is first proved that the system $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ exhibits maximal distributional chaos for any $r_{0}, r_{1}\in \mathbb{R}$ (no assumption of $r_{0}, r_{1}\in \mathbb{R}\setminus \mathbb{Q}$), generalizing Theorem 1 in Wu and Chen (Topol. Appl. 162:91–99, 2014). It is also obtained that $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is cofinitely sensitive and $(\hat{\mathscr{M}} ^{1}, \hat{\mathscr{M}}^{1})$-sensitive and that $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is densely chaotic if and only if $r_{1}-r_{0} \in \mathbb{R}\setminus \mathbb{Q}$.

1 Introduction and preliminaries

A discrete dynamical system (briefly, dynamical system) is a pair $(X, g)$, where X is a compact metric space and $g: X\longrightarrow X$ is a continuous map. Let $\mathbb{N}=\{1, 2, 3, \ldots \}$ and $\mathbb{Z}^{+}=\{0, 1, 2, \ldots \}$. Sharkovsky’s amazing discovery [10], as well as Li and Yorke’s famous work which introduced the concept of “chaos” known as Li–Yorke chaos today in a mathematically rigorous way [4], have activated sustained interest and provoked the recent rapid advancement of the frontier research on discrete chaos theory. In their study, Li and Yorke suggested considering “divergent pairs” $(x, y)$, which are proximal but not asymptotic and called Li–Yorke pairs, i.e.,

$$ \liminf_{n\rightarrow \infty }d\bigl(g^{n}(x), g^{n}(y)\bigr)=0\quad \text{and}\quad \limsup_{n\rightarrow \infty }d \bigl(g^{n}(x), g^{n}(y)\bigr)>0. $$

A pair $(x, y)$ is called a Li–Yorke pair of modulus δ if

$$ \liminf_{n\rightarrow \infty }d\bigl(g^{n}(x), g^{n}(y)\bigr)=0 \quad\text{and}\quad \limsup_{n\rightarrow \infty }d \bigl(g^{n}(x), g^{n}(y)\bigr)> \delta . $$

Clearly, $(x, y)$ is a Li–Yorke pair if and only if it is a Li–Yorke pair of modulus δ for some $\delta >0$. The set of Li–Yorke pairs of modulus δ is denoted by $\mathrm{LY}(g, \delta )$ and the set of Li–Yorke pairs by $\mathrm{LY}(g)$. According to the idea of Li and Yorke [4], a subset D of X is a δ-scrambled set of g, if $D\times D\setminus \Delta \subset \mathrm{LY}(g, \delta )$, where $\Delta =\{(x, x)\in X\times X: x \in X\}$. In particular, if $D\times D\setminus \Delta \subset \mathrm{LY}(g)$, then D is a scrambled set of g. If a scrambled set D of g is also uncountable, it is called a Li–Yorke scrambled set for g, and g is said to be chaotic in the sense of Li–Yorke, or Li–Yorke chaotic.

In 1985, Piórek [8] introduced the concept of generic chaos. Inspired by this, Snoha [11, 12] defined generic δ-chaos, dense chaos, and dense δ-chaos in 1990. The notion of Li–Yorke sensitivity was firstly introduced by Akin and Kolyada [1] in 2003. More recent results on chaos can be found in [2, 3, 7, 17, 18, 20,21,22,23,24,25,26,27,28,29,30,31].

Definition 1

([1, 8, 11, 12])

A dynamical system $(X, g)$ is

• (1) sensitive if there exists $\varepsilon >0$ such that for any $x\in X$ and any $\delta >0$, there exist $y\in B(x, \delta ):=\{z\in X: d(z, x)<\delta \}$ and $n\in \mathbb{N}$, such that $d(g^{n}(x), g^{n}(y))\geq \varepsilon$;

• (2) Li–Yorke sensitive if there exists $\varepsilon >0$ such that for any $x\in X$ and any $\delta >0$, there exists $y\in B(x, \delta )$ such that $(x, y)\in \mathrm{LY}(g, \varepsilon )$;

• (3) densely chaotic if $\mathrm{LY}(g)$ is dense in $X\times X$;

• (4) densely δ-chaotic for some $\delta >0$ if $\mathrm{LY}(g, \delta )$ is dense in $X\times X$;

• (5) generically chaotic if $\mathrm{LY}(g)$ is residual in $X\times X$;

• (6) generically δ-chaotic for some $\delta >0$ if $\mathrm{LY}(g, \delta )$ is residual in $X\times X$.

Distributional chaos. The notion of distributional chaos was first introduced in [9], where it was called “strong chaos”, which is characterized by a distributional function of distances between trajectories of two points. It is described as follows.

Let $(X, g)$ be a dynamical system. For any pair $(x, y)\in X\times X$, define the lower and upper distributional functions as

$$ F_{x, y}(t, g)=\liminf_{n\rightarrow \infty } \frac{1}{n} \bigl\vert \bigl\{ i: d\bigl(g^{i}(x), g^{i}(y)\bigr)< t, 0\leq i< n \bigr\} \bigr\vert $$

and

$$ F^{*}_{x, y}(t, g)=\limsup_{n\rightarrow \infty } \frac{1}{n} \bigl\vert \bigl\{ i: d\bigl(g^{i}(x), g^{i}(y)\bigr)< t, 0\leq i< n \bigr\} \bigr\vert , $$

respectively, where $|A|$ denotes the cardinality of set A. Both functions $F_{x, y}$ and $F_{x, y}^{*}$ are non-decreasing and $F_{x, y}\leq F_{x, y}^{*}$.

According to Schweizer and Smítal [9], a dynamical system $(X, g)$ is distributionally ε-chaotic for some $\varepsilon >0$ if there exists an uncountable subset $\mathcal{S}\subset X$ such that for any pair of distinct points $x, y\in \mathcal{S}$, one has that $F_{x, y}^{*}(t, g)=1$ for all $t>0$ and $F_{x, y}(\varepsilon, g)=0$. The set $\mathcal{S}$ is called a distributionally ε-scrambled set and the pair $(x, y)$ a distributionally ε-chaotic pair. If $(X, g)$ is distributionally ε-chaotic for any $0<\varepsilon <\mathrm{diam}(X)$, then $(X, g)$ is said to exhibit maximal distributional chaos.

Let $\mathcal{P}$ be the collection of all subsets of $\mathbb{Z}^{+}$. A collection $\mathscr{F}\subset \mathcal{P}$ is called a Furstenberg family if it is hereditary upwards, i.e., $F_{1}\subset F_{2}$ and $F_{1}\in \mathscr{F}$ imply $F_{2}\in \mathscr{F}$. A family $\mathscr{F}$ is proper if it is a proper subset of $\mathcal{P}$, i.e., neither empty nor the whole $\mathcal{P}$. It is easy to see that $\mathscr{F}$ is proper if and only if $\mathbb{Z}^{+}\in \mathscr{F}$ and $\emptyset \notin \mathscr{F}$. Let $\mathscr{F}_{\mathrm{inf}}$ be the collection of all infinite subsets of $\mathbb{Z}^{+}$ and $\mathscr{F}_{cf}$ the family of cofinite subset, i.e., the collection of subsets of $\mathbb{Z}^{+}$ with finite complements.

For $A\subset \mathbb{Z}^{+}$, define

$$ \overline{d}(A)=\limsup_{n\rightarrow \infty }\frac{1}{n} \bigl\vert A \cap [0, n-1] \bigr\vert \quad\text{and}\quad \underline{d}(A)= \liminf _{n\rightarrow \infty }\frac{1}{n} \bigl\vert A\cap [0, n-1] \bigr\vert . $$

Then, $\overline{d}(A)$ and $\underline{d}(A)$ are the upper density and the lower density of A, respectively. Fix any $\alpha \in [0,1]$ and denote by $\mathscr{\hat{M}}_{\alpha }$ (resp. $\mathscr{\hat{M}}^{\alpha }$) the family consisting of sets $A\subset \mathbb{Z}^{+}$ with $\underline{d}(A)\geq \alpha$ (resp. $\overline{d}(A)\geq \alpha$).

Using Furstenberg family, Wang et al. [6, 13, 14] introduced the notions of $\mathscr{F}$-sensitivity, $(\mathscr{F} _{1}, \mathscr{F}_{2})$-sensitivity, and $(\mathscr{F}_{1}, \mathscr{F}_{2})$-chaos for generalizing sensitivity, Li–Yorke sensitivity, and Li–Yorke chaos, respectively.

Let $\mathscr{F}$ be a Furstenberg family. According to Moothathu [6], a dynamical system $(X, g)$ is

• (1) $\mathscr{F}$-sensitive if there exists $\varepsilon >0$ such that for any nonempty open subset U of X, $\{n\in \mathbb{Z}^{+}: \mathrm{diam}(g^{n}(U))\geq \delta \} \in \mathscr{F}$.

• (2) cofinitely sensitive if there exists $\varepsilon >0$ such that for any nonempty open subset U of X, $\{n\in \mathbb{Z}^{+}: \mathrm{diam}(g^{n}(U))\geq \delta \} \in \mathscr{F} _{cf}$.

Definition 2

([13, 14])

Let $\mathscr{F}_{1}$, $\mathscr{F}_{2}$ be Furstenberg families. A dynamical system $(X, g)$ is

• (1) $(\mathscr{F}_{1}, \mathscr{F}_{2})$-sensitive if there exists some $\varepsilon >0$ such that for any $x\in X$ and any $\delta >0$, there exists $y\in B(x, \delta )$ such that $\{n\in \mathbb{Z}^{+}: d(g^{n}(x), g^{n}(y))< t\}\in \mathscr{F}_{1}$ for any $t>0$ and $\{n\in \mathbb{Z}^{+}: d(g^{n}(x), g^{n}(y))>\varepsilon \}\in \mathscr{F}_{2}$.

• (2) $(\mathscr{F}_{1}, \mathscr{F}_{2})$-chaotic if there exists an uncountable subset $D\subset X$ such that for any $(x, y)\in D\times D\setminus \Delta$, there exists $\delta >0$ such that $\{n\in \mathbb{Z}^{+}: d(g^{n}(x), g^{n}(y))< t\}\in \mathscr{F} _{1}$ for any $t>0$ and $\{n\in \mathbb{Z}^{+}: d(g^{n}(x), g^{n}(y))> \delta \}\in \mathscr{F}_{2}$.

From Definition 2, it can be verified that a dynamical system is Li–Yorke chaotic (resp., distributionally chaotic, Li–Yorke sensitive) if and only if it is $(\mathscr{F}_{\mathrm{inf}}, \mathscr{F}_{\mathrm{inf}})$-chaotic (resp., $(\mathscr{\hat{M}}^{1}, \mathscr{\hat{M}}^{1})$-chaotic, $(\mathscr{F}_{\mathrm{inf}}, \mathscr{F}_{\mathrm{inf}})$-sensitive).

Shift and rotation. Let $\varSigma =\{0,1\}$ and consider a product space $\varSigma _{2}= \varSigma ^{\mathbb{N}}$ with the product topology which is compact and metrizable. Let $\varSigma _{2}$ endow with the following metric:

$$ d_{1}(x,y)= \textstyle\begin{cases} 0, & x=y, \\ \frac{1}{\min \{m\geq 1: x_{m}\neq y_{m}\}}, & x\neq y, \end{cases} $$

for any $x=x_{1}x_{2}\cdots$ , $y=y_{1}y_{2}\cdots \in \varSigma _{2}$.

Define the shift $\sigma :\varSigma _{2}\longrightarrow \varSigma _{2}$ by $\sigma (x)=x_{2}x_{3}\cdots$ for any $x=x_{1}x_{2}\cdots \in \varSigma _{2}$. Clearly, σ is continuous. If X is a closed and invariant subset of $\varSigma _{2}$, then $(X,\sigma |_{X})$ is called a shift space or subshift.

Any element A of the set $\varSigma ^{n}$ is called an n-word over Σ and the length of A is n, denoted by $|A|$. A word over Σ is an element of the set $\bigcup_{n\in \mathbb{N}}\varSigma ^{n}$. Let $A=a_{1}\cdots a_{n}\in \varSigma ^{n}$ and $B=b_{1}\cdots b_{m}\in \varSigma ^{m}$. Denote $AB=a_{1}\cdots a_{n}b_{1}\cdots b_{m}$ and $\overline{A}=\overline{a_{1}}\cdots \overline{a _{n}}$, where

$$ \overline{a_{i}}= \textstyle\begin{cases} 0, & a_{i}=1, \\ 1, & a_{i}=0. \end{cases} $$

Clearly, $AB\in \varSigma ^{n+m}$ and $\overline{A}\in \varSigma ^{n}$. For any $a\in \varSigma$, denote $a^{n}$ as an n-length concatenation of a (for example, $0^{3}=000$), and $a^{\infty }=aa\cdots$ as an infinite concatenation. If $x=x_{1}x_{2}\cdots \in \varSigma _{2}$ and $i\leq j \in \mathbb{N}$, then let $x_{[i, j]} =x_{i}x_{i+1}\cdots x_{j}$ and $x_{(i, j]}=x_{[i+1, j]}$. For any $B=b_{1}\cdots b_{n} \in \bigcup_{n\in \mathbb{N}}\varSigma ^{n}$, the set $[B]= \{x_{1}x_{2} \cdots \in \varSigma _{2}: x_{i}=b_{i}, 1\leq i\leq n \}$ is called the cylinder generated by B. For any $n\in \mathbb{N}$, let $\mathcal{B}_{n}= \{[b_{1}\cdots b_{n}]: b_{i}\in \varSigma, 1 \leq i\leq n \}$.

The function $\wp : \bigcup_{n\in \mathbb{N}}\varSigma ^{n} \longrightarrow \mathbb{Z}^{+}$ is defined by

$$ \wp (A)= \vert A \vert -\sum_{j=1}^{ \vert A \vert }a_{j}= \bigl\vert \bigl\{ j: 1\leq j\leq \vert A \vert , a_{j}=0 \bigr\} \bigr\vert , $$

for any $A=a_{1}\cdots a_{|A|}\in \bigcup_{n\in \mathbb{N}}\varSigma ^{n}$.

Consider the unit circle $\mathbb{S}^{1}$ defined by

$$ \mathbb{S}^{1}= \bigl\{ (x, y): x^{2}+y^{2}=1 \bigr\} \subset \mathbb{R}^{2}. $$

Identifying $\mathbb{R}^{2}$ with the complex plane $\mathbb{C}$, one can write

$$ \mathbb{S}^{1}= \bigl\{ e^{2\pi i\theta }: 0\leq \theta < 1 \bigr\} \subset \mathbb{C}. $$

Consider a rotation $R_{r}$ of angle $2\pi r$ on the circle, given by

$$ R_{r} \bigl(e^{2\pi i\theta } \bigr)=e^{2\pi i(\theta + r)}. $$

There is a natural distance $d_{2}(z_{1}, z_{2})$ between points $z_{1}$ and $z_{2}$ on $\mathbb{S}^{1}$, given by the arc length distance. It is normalized by dividing with 2π.

This paper considers the following dynamical system.

Let $R_{r_{0}}, R_{r_{1}}: \mathbb{S}^{1}\longrightarrow \mathbb{S} ^{1}$ be rotations and define $f: \varSigma _{2}\times \mathbb{S}^{1} \longrightarrow \varSigma _{2}\times \mathbb{S}^{1}$ by

$$ f(x, t)=\bigl(\sigma (x), R_{r_{x_{1}}}(t)\bigr), $$

for any $x=x_{1}x_{2}\cdots \in \varSigma _{2}$, $t\in \mathbb{S}^{1}$ (so, $r_{x_{1}}=r_{0}$ or $r_{1}$). Note that the nth iteration of f at the point $(x, t)\in \varSigma _{2}\times \mathbb{S}^{1}$ is given by

$$ f^{n}(x, t)=\bigl(\sigma ^{n}(x), R_{r_{x_{n}}} \circ \cdots \circ R_{r_{x _{2}}}\circ R_{r_{x_{1}}}(t)\bigr). $$

Let d be the product metric on the product space $\varSigma _{2}\times \mathbb{S}^{1}$, i.e.,

$$ d\bigl((x, t_{1}), (y, t_{2})\bigr)=\max \bigl\{ d_{1}(x, y), d_{2}(t_{1}, t _{2}) \bigr\} , $$

for any $(x, t_{1})$, $(y, t_{2})\in \varSigma _{2}\times \mathbb{S}^{1}$.

Wang et al. [15] proved that a dynamical system having a regular shift-invariant set is distributionally chaotic and posed the following question:

Question 3

([15])

Is the system $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ distributionally chaotic when $r_{0}, r_{1}\in \mathbb{R}\setminus \mathbb{Q}$ with $r_{0}\neq \pm r_{1}$?

Recently, Wu and Chen [19] gave a positive answer to this question and proved that $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is Li–Yorke sensitive. By further discussing $(\varSigma _{2}\times \mathbb{S}^{1}, f)$, we in this paper investigate the chaos of $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ for any $r_{0}, r_{1}\in \mathbb{R}$ (not assuming $r_{0}, r_{1}\in \mathbb{R}\setminus \mathbb{Q}$). More precisely, we first prove that $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is distributionally chaotic for any $r_{0}, r_{1} \in \mathbb{R}$. Meanwhile, we obtain that (1) it is cofinitely sensitive and $(\hat{\mathscr{M}}^{1}, \hat{\mathscr{M}}^{1})$-sensitive, and (2) it is densely chaotic if and only if $r_{1}-r_{0}\in \mathbb{R}\setminus \mathbb{Q}$.

2 Distributional chaos for $(\varSigma _{2}\times \mathbb{S}^{1}, f)$

The following lemma will be the key to prove the distributional chaoticity for $(\varSigma _{2}\times \mathbb{S}^{1}, f)$.

Lemma 4

([5, 16])

There exists an uncountable subset E of $\varSigma _{2}$ such that for any distinct points $x=x_{1}x_{2}\cdots$ , $y=y_{1}y_{2}\cdots$ in E, they satisfy $x_{n}=y_{n}$ for infinitely many n and $x_{m}\neq y _{m}$ for infinitely many m.

Theorem 5

There exists an uncountable subset $\mathcal{T}\subset \varSigma _{2} \times \mathbb{S}^{1}$ which is a distributionally β-scrambled set of f for any $0<\beta \leq \mathrm{diam} (\varSigma _{2}\times \mathbb{S}^{1})=1$, i.e., $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ exhibits maximal distributional chaos.

Proof

Let $\mathscr{L}_{1}=L_{1}=2$ and $L_{k+1}=2^{\mathscr{L}_{k}}+2 \mathscr{L}_{k}$, $\mathscr{L}_{k+1}=\mathscr{L}_{k}+L_{k+1}$ for any $k\in \mathbb{N}$. From Lemma 4, it follows that there exists an uncountable subset $E\subset \varSigma _{2}$ such that for any two distinct points $x=x_{1}x_{2}\cdots$ , $y=y_{1}y_{2}\cdots \in E$, $x_{n}=y_{n}$ holds for infinitely many n and $x_{m}\neq y_{m}$ holds for infinitely many m. For any $x=x_{1}x_{2}\cdots \in E$, take $\widetilde{x}=\widetilde{x}_{1}\widetilde{x}_{2}\widetilde{x}_{3} \cdots$ as

$$ \widetilde{x}=x_{1}^{\mathscr{L}_{1}}\overline{\widetilde{x}_{[1, \mathscr{L}_{1}]}}x_{2}^{2^{\mathscr{L}_{1}}+\mathscr{L}_{1}} \cdots \overline{\widetilde{x}_{[1, \mathscr{L}_{k}]}}x_{k+1}^{2^{ \mathscr{L}_{k}}+\mathscr{L}_{k}} \cdots. $$

Set $\mathcal{T}=\{\widetilde{x}: x\in E\}\times \{1\}$. Clearly, $\mathcal{T}$ is uncountable.

Now, we claim that $\mathcal{T}$ is a distributionally β-scrambled set of f for any $0<\beta \leq 1$.

Fixing any pair of distinct points $(x, 1)$, $(y, 1)\in \mathcal{T}$, it follows from the construction of of $\mathcal{T}$ that there exist two different points $u=u_{1}u_{2}\cdots$ , $v=v_{1}v_{2}\cdots \in E$ and two increasing sequences $\{p_{k}\}_{k=1}^{\infty }$, $\{q_{k}\}_{k=1}^{\infty }$ of $\mathbb{N}$ such that for any $k\in \mathbb{N}$,

• (a) $\widetilde{u}=x$, $\widetilde{v}=y$,

• (b) $u_{p_{k}}=v_{p_{k}}$, $u_{q_{k}}=\overline{v_{q_{k}}}$.

(1) Note that $\wp (\widetilde{u}_{[1, 2\mathscr{L}_{p_{k}}]})=\wp ( \widetilde{v}_{[1, 2\mathscr{L}_{p_{k}}]})$ for any $k\in \mathbb{N}$. This implies that

$$ R_{r_{x_{2\mathscr{L}_{p_{k}}}}}\circ \cdots \circ R_{r_{x_{2}}} \circ R_{r_{x_{1}}}= R_{r_{y_{2\mathscr{L}_{p_{k}}}}}\circ \cdots \circ R_{r_{y_{2}}}\circ R_{r_{y_{1}}}. $$

(2.1)

For any $2\mathscr{L}_{p_{k}}\leq j\leq \mathscr{L}_{p_{k}+1}- \mathscr{L}_{p_{k}}$, the first $\mathscr{L}_{p_{k}}$ respective symbols of $\sigma ^{j}(x)$ and $\sigma ^{j}(y)$ coincide, implying that

$$ d_{1}\bigl(\sigma ^{j}(x), \sigma ^{j}(y) \bigr)\leq \frac{1}{\mathscr{L}_{p_{k}}} \rightarrow 0 \quad (k\rightarrow \infty ). $$

Meanwhile, applying (2.1) yields that for any $2\mathscr{L}_{p _{k}}\leq j\leq \mathscr{L}_{p_{k}+1}-\mathscr{L}_{p_{k}}$,

$$ R_{r_{x_{j}}}\circ \cdots \circ R_{r_{x_{2}}}\circ R_{r_{x_{1}}}(1)= R _{r_{y_{j}}}\circ \cdots \circ R_{r_{y_{2}}} \circ R_{r_{y_{1}}}(1). $$

Then, for any $t>0$, there exists some $K\in \mathbb{N}$ such that for any $k\geq K$ and any $2\mathscr{L}_{p_{k}}\leq j\leq \mathscr{L}_{p _{k}+1}-\mathscr{L}_{p_{k}}$,

$$ d \bigl(f^{j}(x, 1), f^{j}(y, 1) \bigr)< t. $$

Consequently,

$$\begin{aligned} &F_{(x, 1), (y, 1)}^{*}(t, f) \\ &\quad = \limsup_{n\rightarrow \infty }\frac{1}{n} \bigl\vert \bigl\{ k: d\bigl(f^{k}(x, 1), f^{k}(y, 1)\bigr)< t, 0\leq k< n \bigr\} \bigr\vert \\ &\quad \geq \limsup_{k\rightarrow \infty }\frac{1}{\mathscr{L} _{p_{k}+1}-\mathscr{L}_{p_{k}}} \bigl\vert \bigl\{ n: d\bigl(f^{j}(x, 1), f ^{j}(y, 1)\bigr)< t, 0 \leq j< \mathscr{L}_{p_{k}+1}-\mathscr{L}_{p_{k}} \bigr\} \bigr\vert \\ &\quad \geq \limsup_{k\rightarrow \infty }\frac{\mathscr{L}_{p _{k}+1}-3\mathscr{L}_{p_{k}}}{\mathscr{L}_{p_{k}+1}-\mathscr{L}_{p _{k}}} \\ & \quad= \limsup_{k\rightarrow \infty }\frac{2^{\mathscr{L}_{p _{k}}}}{2^{\mathscr{L}_{p_{k}}}+2\mathscr{L}_{p_{k}}} \\ &\quad = 1. \end{aligned}$$

(2) It is easy to see that for any $2\mathscr{L}_{q_{k}}\leq j\leq \mathscr{L}_{q_{k}+1}-1$, the first respective symbols of $\sigma ^{j}(x)$ and $\sigma ^{j}(y)$ are distinct, implying that $d_{1}(\sigma ^{j}(x), \sigma ^{j}(y))=1$. Then, for any $2\mathscr{L} _{q_{k}}\leq j\leq \mathscr{L}_{q_{k}+1}-1$ and any $0<\beta \leq 1$,

$$ d\bigl(f^{j}(x, 1), f^{j}(y, 1)\bigr)\geq \beta. $$

Therefore,

$$\begin{aligned} &F_{(x, 1), (y, 1)}(\beta, f) \\ &\quad = \liminf_{n\rightarrow \infty }\frac{1}{n} \bigl\vert \bigl\{ k: d\bigl(f^{k}(x, 1), f^{k}(y, 1)\bigr)< \beta, 0\leq k< n \bigr\} \bigr\vert \\ &\quad \leq \liminf_{k\rightarrow \infty }\frac{1}{\mathscr{L} _{q_{k}+1}-1} \bigl\vert \bigl\{ j: d\bigl(f^{kj}(x, 1), f^{j}(y, 1)\bigr)< \beta, 0 \leq j\leq \mathscr{L}_{q_{k}+1}-1 \bigr\} \bigr\vert \\ &\quad \leq \liminf_{k\rightarrow \infty }\frac{2\mathscr{L}_{q _{k}}}{\mathscr{L}_{q_{k}+1}-1} \\ &\quad = \liminf_{k\rightarrow \infty }\frac{2\mathscr{L}_{q_{k}}}{3 \mathscr{L}_{q_{k}}+2^{\mathscr{L}_{q_{k}}}} \\ &\quad = 0. \end{aligned}$$

Hence, $\mathcal{T}$ is a distributionally β-scrambled set for any $0<\beta \leq 1$, i.e., $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ exhibits maximal distributional chaos. □

3 Other chaos for $(\varSigma _{2}\times \mathbb{S}^{1}, f)$

This section shall show that $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is cofinitely sensitive and $(\hat{\mathscr{M}}^{1}, \hat{\mathscr{M}} ^{1})$-sensitive and obtain a sufficient and necessary condition for a dense chaos.

Theorem 6

The system $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is cofinitely sensitive.

Proof

Given any point $(x, t)\in \varSigma _{2}\times \mathbb{S}^{1}$ and any $\delta >0$, take $K=[\frac{1}{\delta }]+2$ and assume that $x=x_{1}x_{2}x_{3}\cdots$ . Choose $y=y_{1}y_{2}y_{3}\cdots \in \varSigma _{2}$ by the following formula:

$$ y_{n}= \textstyle\begin{cases} x_{n}, & 1\leq n\leq K, \\ \overline{x_{n}}, & n>K. \end{cases} $$

Clearly, $d((x, t), (y, t))<\delta$. For any $n>K$, it can be verified that

$$ d\bigl(f^{n}(x, t), f^{n}(y, t)\bigr)\geq d_{1}\bigl(\sigma ^{n}(x), \sigma ^{n}(y) \bigr)=1, $$

implying that $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is cofinitely sensitive, due to the arbitrariness of δ and $(x, t)$ □

Theorem 7

The system $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is $( \hat{\mathscr{M}}^{1}, \hat{\mathscr{M}}^{1})$-sensitive. In particular, it is Li–Yorke sensitive.

Proof

Given any $(x, t)\in \varSigma _{2}\times \mathbb{S}^{1}$ and any $\delta >0$, choose $L_{1}=\mathscr{L}_{1}=[\frac{1}{\delta }]+2$ and $L_{k+1}=2^{\mathscr{L}_{k}}+2\mathscr{L}_{k}$, $\mathscr{L}_{k+1}= \mathscr{L}_{k}+L_{k+1}$ for any $k\in \mathbb{N}$. Take $y=y_{1}y _{2}y_{3}\cdots \in \varSigma _{2}$ as the following inductive method:

• (1) $y_{[1, \mathscr{L}_{2}]}=x_{[1, \mathscr{L}_{1}]}\overline{x _{[1, \mathscr{L}_{1}]}}x_{[2\mathscr{L}_{1}+1, \mathscr{L}_{2}]}$;

• (2) $y_{[\mathscr{L}_{2n}+1, \mathscr{L}_{2n+1}]}=\overline{y _{[1, \mathscr{L}_{2n}]}}\overline{x_{[2\mathscr{L}_{2n}+1, \mathscr{L}_{2n+1}]}}$ for any $n\in \mathbb{N}$;

• (3) $y_{[\mathscr{L}_{2n+1}+1, \mathscr{L}_{2n+2}]}=\overline{y _{[1, \mathscr{L}_{2n+1}]}}x_{[2\mathscr{L}_{2n+1}+1, \mathscr{L}_{2n+2}]}$ for any $n\in \mathbb{N}$.

Clearly, $d((x, t), (y, t))=d_{1}(x, y)=\frac{1}{\mathscr{L}_{1}+1}< \delta$. Similarly as in the proof of Theorem 5, it can be verified that

• (1) for any $\varepsilon >0$, $\{n\in \mathbb{Z}^{+}: d(f^{n}(x, t), f^{n}(y, t))< t\}\in \hat{\mathscr{M}^{1}}$;

• (2) for any $0<\beta \leq 1$, $\{n\in \mathbb{Z}^{+}: d(f^{n}(x, t), f^{n}(y, t))\geq \beta \}\in \hat{\mathscr{M}^{1}}$,

implying that $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is $( \hat{\mathscr{M}^{1}}, \hat{\mathscr{M}^{1}})$-sensitive, due to the arbitrariness of δ and $(x, t)$. □

Theorem 8

The system $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is densely chaotic if and only if $r_{1}-r_{0}\in \mathbb{R}\setminus \mathbb{Q}$.

Proof

$(\Longrightarrow )$ Suppose on the contrary that $r_{1}-r_{0}\in \mathbb{Q}$. Then there exist $t_{0}\in \mathbb{S}^{1}$ and $\eta >0$ such that

$$ \xi :=\inf \biggl\{ d_{2}(z_{1}, z_{2}): z_{1}\in B(t_{0}, \eta ), z _{2}\in \bigcup _{n\in \mathbb{Z}}R_{r_{1}-r_{0}}^{n}\bigl(B(1, \eta )\bigr) \biggr\} >0. $$

Fix any $x\in \varSigma _{2}$ and set $U=B(x, \eta )\times B(1, \eta )$ and $V=B(x, \eta )\times B(t_{0}, \eta )$. Clearly, both U and V are nonempty open subsets of $\varSigma _{2}\times \mathbb{S}^{1}$.

Now, we claim that $(U\times V) \cap \mathrm{LY}(f)=\emptyset$.

In fact, for any $((y, t_{1}), (z, t_{2}))\in U\times V$ and any $n\in \mathbb{Z}^{+}$, it can be verified that for any $n\in \mathbb{Z}^{+}$,

$$\begin{aligned} &R_{r_{y_{n}}}\circ \cdots \circ R_{r_{y_{2}}}\circ R_{r_{y_{1}}}(t _{1})\\ &\quad= e^{2\pi i (\sum _{k=1}^{n}r_{y_{k}})}t_{1}=e^{2\pi i [r _{0}\wp (y_{[1, n]})+r_{1}(n-\wp (y_{[1, n]})) ]}t_{1} \\ &\quad=e^{2\pi i [r_{0}\wp (z_{[1, n]})+r_{1}(n-\wp (z_{[1, n]})) ]} \cdot e^{2\pi i [(r_{1}-r_{0})(\wp (z_{[1, n]})-\wp (y_{[1, n]})) ]}t _{1} \end{aligned}$$

and

$$ R_{r_{z_{n}}}\circ \cdots \circ R_{r_{z_{2}}}\circ R_{r_{z_{1}}}(t _{2})= e^{2\pi i (\sum _{k=1}^{n}r_{z_{k}})}t_{2}=e^{2\pi i [r _{0}\wp (z_{[1, n]})+r_{1}(n-\wp (z_{[1, n]})) ]}t_{2}, $$

implying that

$$\begin{aligned} &d_{2}\bigl(R_{r_{y_{n}}}\circ \cdots \circ R_{r_{y_{2}}}\circ R_{r_{y _{1}}}(t_{1}), R_{r_{z_{n}}} \circ \cdots \circ R_{r_{z_{2}}}\circ R _{r_{z_{1}}}(t_{2}) \bigr) \\ &\quad=d_{2}\bigl(e^{2\pi i [(r_{1}-r_{0})(\wp (z_{[1, n]})-\wp (y_{[1, n]})) ]}t _{1}, t_{2}\bigr) \\ &\quad=d_{2}\bigl(R_{r_{1}-r_{0}}^{\wp (z_{[1, n]})-\wp (y_{[1, n]})}(t_{1}), t _{2}\bigr)\geq \xi. \end{aligned}$$

Therefore,

$$\begin{aligned} &\liminf_{n\rightarrow \infty }d\bigl(f^{n}(y, t_{1}), f^{n}(z, t_{2})\bigr) \\ &\quad \geq \liminf_{k\rightarrow \infty } d_{2} \bigl(R_{r_{y_{n}}}\circ \cdots \circ R_{r_{y_{2}}}\circ R_{r_{y_{1}}}(t_{1}), R_{r_{z_{n}}} \circ \cdots \circ R_{r_{z_{2}}}\circ R_{r_{z_{1}}}(t_{2})\bigr) \geq \xi >0. \end{aligned}$$

This means that $(U\times V) \cap \mathrm{LY}(f)=\emptyset$. Hence, f is not densely chaotic, which is a contradiction.

$(\Longleftarrow )$ For any nonempty open subsets $U, V$ of $\varSigma _{2}\times \mathbb{S}^{1}$, choose $(u, t_{1})\in U$, $(v, t_{2})\in V$ and $K\in \mathbb{N}$ such that $B((u, t_{1}), \frac{1}{K})\subset U$ and $B((v, t_{2}), \frac{1}{K})\subset V$. From $r_{1}-r_{0}\in \mathbb{R}\setminus \mathbb{Q}$, it follows that there exists an increasing sequence $\{p_{n}\}_{n=1}^{\infty }\subset \mathbb{N}$ such that $d_{2}(R_{r_{1}-r_{0}}^{p_{n}}(t_{1}), t_{2})< \frac{1}{n}$ for any $n\in \mathbb{N}$. Let $L_{1}=\mathscr{L}_{1}=K$ and $L_{n+1}=2\mathscr{L}_{n}+p_{n}$, $\mathscr{L}_{n+1}=\mathscr{L} _{n}+L_{n}$ for any $n\in \mathbb{N}$. Take $\hat{u}=\hat{u}_{1} \hat{u}_{2}\hat{u}_{3}\cdots$ and $\hat{v}=\hat{v}_{1}\hat{v}_{2} \hat{v}_{3}\cdots \in \varSigma _{2}$ as

$$ \hat{u}=u_{[1, \mathscr{L}_{1}]}\overline{u_{[1, \mathscr{L}_{1}]}}1^{p _{1}}0^{\mathscr{L}_{1}} \cdots \overline{u_{[1, \mathscr{L}_{n}]}}1^{p _{n}}0^{\mathscr{L}_{n}}\cdots $$

and

$$ \hat{v}=v_{[1, \mathscr{L}_{1}]}\overline{v_{[1, \mathscr{L}_{1}]}}0^{p _{1}+\mathscr{L}_{1}}\cdots \overline{v_{[1, \mathscr{L}_{n}]}}0^{p _{n}+\mathscr{L}_{n}}\cdots, $$

respectively. Clearly, $(\hat{u}, t_{1})\in U$ and $(\hat{v}, t_{2}) \in V$.

(1) Noting that $\hat{u}_{2\mathscr{L}_{n}+1}=1$ and $\hat{v}_{2 \mathscr{L}_{n}+1}=0$, it follows that

$$ d\bigl(f^{2\mathscr{L}_{n}}(\hat{u}, t_{1}), f^{2\mathscr{L}_{n}}( \hat{v}, t_{2})\bigr) \geq d_{1}\bigl(\sigma ^{2\mathscr{L}_{n}}(\hat{u}), \sigma ^{2\mathscr{L}_{n}}(\hat{v})\bigr)=1, $$

implying that

$$ \limsup_{n\rightarrow \infty }d\bigl(f^{n}(\hat{u}, t_{1}), f^{n}(\hat{v}, t_{2})\bigr)\geq \limsup_{n\rightarrow \infty }d\bigl(f^{2\mathscr{L}_{n}}( \hat{u}, t_{1}), f^{2\mathscr{L}_{n}}(\hat{v}, t_{2})\bigr) \geq 1. $$

(2) From the fact that the first $\mathscr{L}_{n}$ respective symbols of $\sigma ^{2\mathscr{L}_{n}+p_{n}}(\hat{u})$ and $\sigma ^{2\mathscr{L}_{n}+p_{n}}(\hat{v})$ coincide, it follows that

$$ d_{1}\bigl(\sigma ^{2\mathscr{L}_{n}+p_{n}}(\hat{u}), \sigma ^{2\mathscr{L}_{n}+p_{n}}(\hat{v})\bigr)\leq \frac{1}{\mathscr{L} _{n}}< \frac{1}{n}. $$

(3.1)

Meanwhile, it can verified that

$$ d_{2}\bigl(R_{\hat{u}_{2\mathscr{L}_{n}+p_{n}}}\circ \cdots \circ R_{ \hat{u}_{2}}\circ R_{\hat{u}_{1}}(t_{1}), R_{\hat{v}_{2\mathscr{L} _{n}+p_{n}}}\circ \cdots \circ R_{\hat{v}_{2}}\circ R_{\hat{v}_{1}}(t _{2})\bigr)= d_{2} \bigl(R_{r_{1}-r_{0}}^{p_{n}}(t_{1}), t_{2} \bigr)< \frac{1}{n}. $$

This, together with (3.1), implies that

$$ d\bigl(f^{2\mathscr{L}_{n}+p_{n}}(\hat{u}, t_{1}), f^{2\mathscr{L}_{n}+p _{n}}( \hat{v}, t_{2})\bigr)< \frac{1}{n}. $$

Then,

$$ \liminf_{n\rightarrow \infty }d\bigl(f^{n}(\hat{u}, t_{1}), f^{n}(\hat{v}, t_{2})\bigr)\leq \liminf_{n\rightarrow \infty }d\bigl(f^{2\mathscr{L}_{n}+p_{n}}( \hat{u}, t_{1}), f^{2\mathscr{L}_{n}+p_{n}}(\hat{v}, t_{2})\bigr)=0. $$

Therefore, $((\hat{u}, t_{1}), (\hat{v}, t_{2}))\in (U\times V)\cap \mathrm{LY}(f)$. Hence, $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is densely chaotic. □

Applying Theorem 8, the following can be verified.

Corollary 9

The following statements are equivalent:

• (1) $r_{1}-r_{0}\in \mathbb{R}\setminus \mathbb{Q}$;

• (2) $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is densely chaotic;

• (3) $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is densely δ-chaotic for any $0<\delta <1$;

• (4) $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is generically chaotic;

• (5) $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is generically δ-chaotic for any $0<\delta <1$.