Relationship among several types of sensitivity in general semi-flows

TL;DR

This paper explores the relationships among different types of sensitivity in semi-flows, demonstrating the existence of specific monoids and semi-flows with particular sensitivity properties, thereby answering open questions in the field.

Contribution

It provides counterexamples showing certain sensitivity properties do not imply others, advancing understanding of sensitivity in semi-flows and answering open questions from prior research.

Findings

Existence of a monoid without syndetic or dual syndetic property.

Existence of a strongly mixing semi-flow lacking various sensitivities.

A thickly sensitive cascade that is not multi-sensitive.

Abstract

In this paper, we show that there exists a monoid, on which neither the syndetic property nor the dual syndetic property holds, and there exists a strongly mixing semi-flow with this monoid action which does not have thick sensitivity, syndetic sensitivity, thickly syndetic sensitivity, or thickly periodical sensitivity. Meanwhile, we show that there exists a thickly sensitive cascade which is not multi-sensitive. The first result answers positively Question 2, and the first and the second results answer negatively Question 3 in [A. Miller, A note about various types of sensitivity in general semiflows, Appl. Gen. Topol., 2018].

Equations

$$s.(t.x)=(s+t).x\text{ and }0.x=x,\text{ for all }s,t\in G\text{ and all }x\in X.$$

$${\pi }_t:X$$

$$x$$

$$D(U,V)=\{t\in G:tU\cap V\ne \varnothing \}$$

$$D(U_1,V_1)\cap D(U_2,V_2)\ne \varnothing ;$$

$$D(U,c)=\left\{t\in G:\mathrm{diam}(tU)>\epsilon \right\}\ne \varnothing ;$$

$$x+y=\{\begin{array}{ll}x+y,& \text{if }x,y\in {\mathbb{N}}_0,\\ \infty ,& \text{if }x=\infty \text{ or }y=\infty .\end{array}$$

$$(\infty +K_1)\cap (G\setminus K)=\{\infty \}\cap (G\setminus \{\infty \})=\varnothing .$$

$$G\setminus \{g\in G:gU\cap V\ne \varnothing \}\subset \{\infty \}\cup \left({\mathbb{N}}_0\setminus \{n\in {\mathbb{N}}_0:f^n(U)\cap V\ne \varnothing \}\right),$$

$$L_0={\mathcal{L}}_0=0,L_1={\mathcal{L}}_1=2,$$

$$L_n=2^{L_1+\cdots +L_{n-1}}\cdot (2n),{\mathcal{L}}_n=L_1+L_2+\cdots +L_n,n\ge 2,$$

$$X=\left[0.5,1.5\right]$$

$$\bigcup \left(\bigcup _{n=1}^{+\infty }\bigcup _{i=0}^{2^{{\mathcal{L}}_{2n-1}}}\left[{\mathcal{L}}_{2n}+(2n+1)i,{\mathcal{L}}_{2n}+(2n+1)i+2n\right]\right),$$

$$Y=\left[0.5,1.5\right]$$

$$\bigcup \left(\bigcup _{n=1}^{+\infty }\bigcup _{i=0}^{2^{{\mathcal{L}}_{2n-1}}}\left[{\mathcal{L}}_{2n}+i,{\mathcal{L}}_{2n}+i+\frac{1}{2n+1}\right]\right).$$

$${\mathcal{A}}_n$$

$${\mathcal{B}}_n$$

$${\mathcal{C}}_n$$

$${\mathcal{D}}_n$$

$$f(x)=\{\begin{array}{rl}& 0.5x+1.75,\text{ if }x\in [0.5,1.5],\\ & x+1,\text{ if }x\in \left[{\mathcal{L}}_{2n-1}+i,{\mathcal{L}}_{2n-1}+i+\frac{1}{2n}\right]\text{ for some }0\le i<2^{{\mathcal{L}}_{2n-2}},n\in \mathbb{N},\\ & x+2n+1,\text{ if }x\in \left[{\mathcal{L}}_{2n}+(2n+1)i,{\mathcal{L}}_{2n}+(2n+1)i+2n\right]\\ & \text{ for some }0\le i<2^{{\mathcal{L}}_{2n-1}},n\in \mathbb{N},\\ & 2n(2n-1)\left(x-{\mathcal{L}}_{2n-1}-2^{{\mathcal{L}}_{2n-2}}\right)+{\mathcal{L}}_{2n},\text{ if }x\in {\mathcal{A}}_n,n\in \mathbb{N},\\ & \frac{1}{2n(2n+1)}\left(x-{\mathcal{L}}_{2n}-(2n+1)\cdot 2^{{\mathcal{L}}_{2n-1}}\right)+{\mathcal{L}}_{2n+1},\text{ if }x\in {\mathcal{B}}_n,n\in \mathbb{N},\end{array}$$

$$g(x)=\{\begin{array}{rl}& x+1.5,\text{ if }x\in [0.5,1.5],\\ & x+1,\text{ if }x\in \left[{\mathcal{L}}_{2n}+i,{\mathcal{L}}_{2n}+i+\frac{1}{2n}\right]\text{ for some }0\le i<2^{{\mathcal{L}}_{2n-1}},n\in \mathbb{N},\\ & x+2n,\text{ if }x\in \left[{\mathcal{L}}_{2n-1}+2ni,{\mathcal{L}}_{2n-1}+2ni+(2n-1)\right]\\ & \text{ for some }0\le i<2^{{\mathcal{L}}_{2n-2}},n\in \mathbb{N},\\ & \frac{1}{(2n+1)\cdot (2n-1)}\left(x-{\mathcal{L}}_{2n-1}-2n\cdot 2^{{\mathcal{L}}_{2n-2}}\right)+{\mathcal{L}}_{2n},\text{ if }x\in {\mathcal{C}}_n,n\in \mathbb{N},\\ & (2n+1{)}^2\left(x-{\mathcal{L}}_{2n}-2^{{\mathcal{L}}_{2n-1}}\right)+{\mathcal{L}}_{2n+1},\text{ if }x\in {\mathcal{D}}_n,n\in \mathbb{N},\end{array}$$

$$[0.5,1.5],[{\mathcal{L}}_1,{\mathcal{L}}_1+\frac{1}{2}],\dots ,[{\mathcal{L}}_1+2^{{\mathcal{L}}_0},{\mathcal{L}}_1+2^{{\mathcal{L}}_0}+\frac{1}{2}],$$

$$[{\mathcal{L}}_2,{\mathcal{L}}_2+2],\dots ,[{\mathcal{L}}_2+2\cdot 2^{{\mathcal{L}}_1},{\mathcal{L}}_2+2\cdot 2^{{\mathcal{L}}_1}+2],$$

$$⋮$$

$$[{\mathcal{L}}_{2n-1},{\mathcal{L}}_{2n-1}+\frac{1}{2n}],\dots ,[{\mathcal{L}}_{2n-1}+2^{{\mathcal{L}}_{2n-2}},{\mathcal{L}}_{2n-1}+2^{{\mathcal{L}}_{2n-2}}+\frac{1}{2n}],$$

$$[{\mathcal{L}}_{2n},{\mathcal{L}}_{2n}+2n],\dots ,[{\mathcal{L}}_{2n}+(2n+1)2^{{\mathcal{L}}_{2n-1}},{\mathcal{L}}_{2n}+(2n+1)2^{{\mathcal{L}}_{2n-1}}+2n],\dots$$

$$I_0,I_1,I_2,\dots .$$

$$I_{2k+2+2^{{\mathcal{L}}_1}+\cdots +2^{{\mathcal{L}}_{2k-1}}}=[{\mathcal{L}}_{2k+1},{\mathcal{L}}_{2k+1}+\frac{1}{2(k+1)}],$$

$$⋮$$

$$I_{2k+2+2^{{\mathcal{L}}_1}+\cdots +2^{{\mathcal{L}}_{2k}}}=[{\mathcal{L}}_{2k+1}+2^{{\mathcal{L}}_{2k}},{\mathcal{L}}_{2k+1}+2^{{\mathcal{L}}_{2k}}+\frac{1}{2(k+1)}],$$

$$I_{2k+3+2^{{\mathcal{L}}_1}+\cdots +2^{{\mathcal{L}}_{2k}}}=[{\mathcal{L}}_{2k+2},{\mathcal{L}}_{2k+2}+2(k+1)],$$

$$⋮$$

$$I_{2k+3+2^{{\mathcal{L}}_1}+\cdots +2^{{\mathcal{L}}_{2k+1}}}=[{\mathcal{L}}_{2k+2}+(2k+3)\cdot 2^{{\mathcal{L}}_{2k+1}},{\mathcal{L}}_{2k+2}+(2k+3)\cdot 2^{{\mathcal{L}}_{2k+1}}+2(k+1)].$$

Full text

1. Introduction

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. The binary operation of a semigroup is most often denoted addition. A monoid is an algebraic structure intermediate between groups and semigroups, and is a semigroup having an identity element, thus obeying all but one of the axioms of a group; the existence of inverses is not required of a monoid.

Let $(X,d)$ be a metric space and $G$ be a non-compact abelian (commutative) topological monoid with the identity element [math]. A jointly continuous monoid action $\pi:G\times X\longrightarrow X$ of $G$ on the metric space $X$ is called a semi-flow and denoted by $(G,X,\pi)$ or $(G,X)$. In particular, if the acting topological monoid is a topological group, a semi-flow is called a flow. The element $\pi(t,x)$ will be denoted by $t.x$ or $tx$, so that the defining conditions for a semi-flow have the following form:

[TABLE]

The maps

[TABLE]

are called transition maps. For any $x\in X$, the set $Gx:=\{t.x:t\in G\}$ is called the orbit of $x$. A semi-flow $(G,X)$ is minimal if the orbit of every point $x\in X$ is dense in $X$, i.e., $\overline{Gx}=X$. Otherwise, it is called non-minimal. For any subset $A$ of $X$ and any $t\in G$, let $t.A=\{t.x:x\in A\}$, denoted by $tA$ for convenience.

In particular, let $\mathbb{N}=\left\{1,2,3,\ldots\right\}$ and $\mathbb{N}_{0}=\left\{0,1,2,\ldots\right\}$. Consider a continuous map $f:X\longrightarrow X$ and a monoid $G=\mathbb{N}_{0}$ (with the discrete topology), which leads to a natural semi-flow, that is, for any $x\in X$ and any $n\in\mathbb{N}_{0}$, $n.x=f^{n}(x)$. This type of semi-flow is called a cascade and is often denoted by $(X,f)$ instead of $(\mathbb{N}_{0},X)$.

Let $(X,d)$ be a metric space. A non-empty open subset of $X$ is said to be a nopen, or a nopen subset. For any non-empty subset $U$ of $X$, the diameter $\mbox{diam}(U)$ of $U$ is $\mbox{diam}(U)=\sup\{d(x,y):x,y\in U\}$.

Definition 1**.**

[10] Let $G$ be a monoid. A subset $B$ of $G$ is

1. (1)

syndetic if there exists a compact subset $K$ of $G$ such that, for every $t\in G$, $(t+K)\cap B\neq\emptyset$, where $t+K=\left\{t+a:a\in K\right\}$; 2. (2)

thick if, for every compact subset $K$ of $G$, there exists some $t\in G$ such that $t+K\subset B$; 3. (3)

thickly syndetic if, for every compact $K$ of $G$, there exists a syndetic subset $S$ of $G$ such that $S+K\subset B$; 4. (4)

periodic if there exist a closed syndetic sub-monoid $S$ of $G$ and $t\in G$ such that $t+S\subset B$; 5. (5)

thickly periodic if, for every compact subset $K$ of $G$, there exists a periodic subset $P$ of $G$ such that $P+K\subset B$.

Clearly, a subset $B$ of $G$ is syndetic (thick) if and only if $G\setminus B$ is not thick (not syndetic).

Definition 2**.**

[10, Definition 1.1] A monoid $G$ satisfies the syndetic property (sp property), or $G$ is an sp monoid, if all syndetic subsets of $G$ are non-compact; the dual syndetic property (dsp property), or a dsp monoid can be defined similarly: for every compact subset $K$ of $G$, the set $G\setminus K$ is a syndetic subset of $G$.

Definition 3**.**

[10, Definition 1.4] A semi-flow $(G,X)$ is

1. (1)

strongly mixing (StrM) if, for any two nopens $U,V$ of $X$, the set

[TABLE]

contains $G\setminus K$ for some compact subset $K$ of $G$; 2. (2)

weak mixing (WM) if, for any nopens $U_{1},V_{1},U_{2},V_{2}$ of $X$,

[TABLE] 3. (3)

sensitive (S) if there exists $\varepsilon>0$ such that for any nopen $U$ of $X$,

[TABLE] 4. (4)

strongly sensitive (StrS) if there exists $\varepsilon>0$ such that for any nopen $U$ of $X$, $D(U,c)$ contains $G\setminus K$ for some compact subset $K$ of $G$; 5. (5)

multi-sensitive (MulS) if there exists $\varepsilon>0$ such that for any $n\in\mathbb{N}$ and any nopens $U_{1},\ldots,U_{n}$ of $X$, $\cap_{i=1}^{n}D(U_{i},\varepsilon)\neq\emptyset$; 6. (6)

strongly multi-sensitive (StrMulS) if there exists $\varepsilon>0$ such that for any $n\in\mathbb{N}$ and any nopens $U_{1},\ldots,U_{n}$ of $X$, $\cap_{i=1}^{n}D(U_{i},\varepsilon)$ contains $G\setminus K$ for some compact subsets $K$ of $G$; 7. (7)

thickly sensitive (TS) if there exists $\varepsilon>0$ such that for any nopen $U$ of $X$, $D(U,\varepsilon)$ is a thick subset of $G$; 8. (8)

syndetically sensitive (SyndS) if there exists $\varepsilon>0$ such that for any nopen $U$ of $X$, $D(U,\varepsilon)$ is a syndetic subset of $G$; 9. (9)

thickly syndetically sensitive (TSynS) if there exists $\varepsilon>0$ such that for any nopen $U$ of $X$, $D(U,\varepsilon)$ is a thickly syndetic subset of $G$; 10. (10)

periodically sensitive (PerS) if there exists $\varepsilon>0$ such that for any nopen $U$ of $X$, $D(U,\varepsilon)$ is a periodic subset of $G$; 11. (11)

thickly periodically sensitive (TPerS) if there exists $\varepsilon>0$ such that for any nopen $U$ of $X$, $D(U,\varepsilon)$ is a thickly periodic subset of $G$.

Ceccherini-Silberstein and Coornaert proved that every transitive semi-flow on an infinite Hausdorff uniform space admitting a dense set of periodic points is sensitive [2]. Miller obtained that every non-minimal syndetically transitive semi-flow is syndetically sensitive [7], generalizing the main results in [1, 13, 15]. Further, Miller gave a summary on the sensitivity of semi-flows with monoid actions [8]. Money conducted a systemic investigation on chaotic properties for semi-flows [12]. Wang showed that every $M$-action on a compact Hausdorff uniform space having at least two disjoint compact invariant subsets is thickly syndetically sensitive [14]. Recently, Miller studied the relationships among various types of sensitivity in general semi-flows and proposed several open questions [10]. For more recent results on sensitivity, refer to [3, 4, 5, 6, 9, 16, 17, 19, 20, 21] and some references therein.

Question 4**.**

[10, Question 2] Find examples showing that in the implications StrS $\Rightarrow$ TS and StrS $\Rightarrow$ TSyndS the condition (sp) is indeed needed, and that in the implication SM $\Rightarrow$ SyndS the condition (dsp) is indeed needed.

Question 5**.**

[10, Question 3] Investigate if, in general, StrS implies TPerS, TS implies MulS, and TS implies SyndS.

Remark 6*.*

Recently, we [18] proved that

1. (1)

there exist two non-syndetically sensitive cascades defined on complete metric spaces whose product is cofinitely sensitive; 2. (2)

there exists a syndetically sensitive semi-flow $(G,X)$ defined on a complete metric space $X$ such that $(G_{1},X)$ is not sensitive for some syndetic closed sub-monoid $G_{1}$ of $G$, which gives a negative answer to [8, Question 43]; 3. (3)

there exists a thickly sensitive cascade which is not syndetically sensitive, i.e., (TS) $\nRightarrow$ (SyndS).

This paper proves that there exists a monoid, on which neither the syndetic property nor the dual syndetic property holds, and there exists a semi-flow given by this monoid satisfying that the strong mixing property can not imply the thick sensitivity, the syndetic sensitivity, the thickly syndetic sensitivity, or the thickly periodical sensitivity (See Example 7). Meanwhile, it is proved that there exists a thickly sensitive cascade which is not multi-sensitive (See Example 9). These results, together with (3) of Remark 6, give complete answers to Questions 4 and 5.

2. Relationship among several types of sensitivity

2.1. A monoid without the sp or dsp property

In this subsection, a monoid is provided, on which neither the syndetic property nor the dual syndetic property holds, and there exists a semi-flow given by this monoid satisfying that the strong mixing property can not imply the thick sensitivity, the syndetic sensitivity, the thickly syndetic sensitivity, or the thickly periodical sensitivity.

Example 7**.**

Consider the set $G=\mathbb{N}_{0}\cup\{\infty\}$ with the discrete topology, which can be thought of as the one-point compactification of $\mathbb{N}_{0}$. The addition ‘$+$’ defined on $G$ is given as follows:

[TABLE]

First, we verify the following properties.

• (1)

It is clear that $G$ is a monoid with the identity element [math].

• (2)

$G$ does not satisfy the dsp or the sp property.

Take a compact subset $K=\{\infty\}$ of $G$. It suffices to show that $G\setminus K$ is not syndetic, implying that $G$ does not satisfy the dsp property. It can be verified that for any non-empty compact subset $K_{1}$ of $G$, $\infty+K_{1}=\{\infty\}$. Then,

[TABLE]

Thus, $G\setminus K$ is not syndetic.

This, together with [10, Proposition 1.3], implies that $G$ does not satisfy the sp property.

• (2${}^{{}^{\prime}}$)

Every thick subset of $G$ contains $\infty$.

Take any thick subset $K$ of $G$, from the definition, it follows that for the compact subset $\{\infty\}$, there exists $g\in G$ such that $g+\{\infty\}=\{\infty\}\subset K$.

• (2${}^{{}^{\prime\prime}}$)

Every thickly periodical subset of $G$ contains $\infty$.

Take any thickly periodical subset $K$, for the compact subset $\{\infty\}$, there exists a periodic subset $P$ of $G$ such that $P+\{\infty\}=\{\infty\}\subset K$.

• (2${}^{{}^{\prime\prime\prime}}$)

Every thickly syndetic subset of $G$ contains $\infty$.

Take any thickly syndetic subset $K$, for the compact subset $\{\infty\}$, there exists a syndetic subset $S\subset G$ such that $S+\{\infty\}=\{\infty\}\subset K$.

Now, we construct a semi-flow. Let $I=[0,1]$ and $f(x)=1-|1-2x|$ be the tent map from $I$ to $I$. It is easy to see that $(I,f)$ is strongly mixing. Define the map $\pi:G\times X\longrightarrow X$ as follows:

• (3)

$\pi(n,x)=f^{n}(x)$, $\forall n\in\mathbb{N}_{0}$ and $\forall x\in I$;

• (4)

$\pi(\infty,x)=0$, $\forall x\in I$.

It can be verified that

• (5)

$(G,I,\pi)$ is a semi-flow;

• (6)

$(G,I,\pi)$ is strongly mixing.

For any nopen subsets $U,V$ of $I$, since $(I,f)$ is strongly mixing, noting that

[TABLE]

we have that $G\setminus\{g\in G:\ gU\cap V\neq\emptyset\}$ is a compact subset of $G$, that is, $(G,I,\pi)$ is strongly mixing.

Next, we verify the following properties.

• (7)

From (6) and [10, Proposition 2.1], it follows that $(G,I,\pi)$ is strongly sensitive.

• (8)

$(G,I,\pi)$ is not thickly sensitive.

For any $0<\varepsilon<1$, it is easy to see that $\{g\in G:\ \mbox{diam}(gI)<\varepsilon\}=\{\infty\}$. This, together with (2${}^{{}^{\prime}}$), implies that $\{g\in G:\ \mbox{diam}(gI)\geq\varepsilon\}=G\setminus\{\infty\}$ is not thick. Thus, $(G,I,\pi)$ is not thickly sensitive.

• (9)

$(G,I,\pi)$ is not syndetically sensitive.

Applying similar arguments as in (8), and the fact that $G\setminus\{\infty\}$ is not syndetic in (2) yields that $(G,I,\pi)$ is not syndetically sensitive.

• (10)

$(G,I,\pi)$ is not thickly periodically sensitive.

Applying similar discussions as in (8), this together with (2${}^{{}^{\prime\prime}}$), yields that $(G,I,\pi)$ is not thickly periodically sensitive.

• (11)

$(G,I,\pi)$ is not thickly syndetically sensitive.

Applying similar discussions as in (8), this together with (2${}^{{}^{\prime\prime\prime}}$), yields that $(G,I,\pi)$ is not thickly syndetically sensitive.

Remark 8*.*

1. (1)

Example 7 gives a positive answer to Question 4. 2. (2)

Miller [11] obtained a weakly mixing semi-flow defined on a non-compact metric space which is not thickly sensitive. Example 7 shows that there exists a strongly mixing semi-flow defined on a compact metric space which does not have thick sensitivity, syndetic sensitivity, thickly syndetic sensitivity, or thickly periodical sensitivity. 3. (3)

Clearly, $G_{1}=\{0,\infty\}$ is a syndetic closed sub-monoid of $G$ in Example 7. Meanwhile, it can be verified that $(G_{1},I,\pi)$ is not sensitive. Similarly to (2) of Remark 6, this, together with the strong sensitivity of $(G,I,\pi)$, also gives a negative answer to [8, Question 43].

2.2. A thickly sensitive, but not multi-sensitive, semi-flow

Similarly to the construction of [18, Example 4], a thickly sensitive semi-flow which is not multi-sensitive is constructed in this subsection.

Example 9**.**

Let

[TABLE]

and

[TABLE]

and take

[TABLE]

and

[TABLE]

For $n\in\mathbb{N}$, let

[TABLE]

Define two linear maps $f:X\to X$ and $g:Y\to Y$ defined by

[TABLE]

and

[TABLE]

respectively. Clearly, $f$ and $g$ are continuous. Rearrange all closed intervals as follows:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

of $X$ by this natural order and denote by

[TABLE]

It is easy to see that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Similarly, rearrange all closed intervals of $Y$ by this natural order and denote by

[TABLE]

It is easy to see that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Note that $f$ is a linear homeomorphism from $I_{n}$ to $I_{n+1}$, and $g$ is also a linear homeomorphism from $J_{n}$ to $J_{n+1}$, $n\in\mathbb{N}_{0}$. According to the construction of $f$ and $g$, it can be verified that

1. (i)

$f$ and $g$ are continuous; 2. (ii)

for any $n\in\left[2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k-1}}+2k+2,2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k}}+2k+2\right]$ ($k\in\mathbb{N}$),

[TABLE]

and

[TABLE]

where $a_{k}=2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k-1}}+2k+2$, implying that

[TABLE] 3. (iii)

for any $n\in\left[2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k}}+2k+3,2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k+1}}+2k+3\right]$ ($k\in\mathbb{N}$),

[TABLE]

and

[TABLE]

where $b_{k}=2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k}}+2k+3$, implying that

[TABLE]

Take $Z=\left(X\times\{0\}\right)\cup\left(\{0\}\times Y\right)\subset\mathbb{R}^{2}$. It is easy to see that $Z$ is a complete metric subspaces of $\mathbb{R}^{2}$. Define $F:Z\longrightarrow Z$ by

[TABLE]

Clearly, $F$ is continuous as $f$ and $g$ are continuous.

Claim 1. $(Z,F)$ is thickly sensitive.

Given any nopen subset $W$ of $Z$, there exists a nopen subset $U\subset X$ or $V\subset Y$ such that $U\times\{0\}\subset W$ or $\{0\}\times V\subset W$. Without loss of generality, assume $U\times\{0\}\subset W$. It follows from (ii)–(iii) that there exist non-degenerate closed interval $[\alpha,\beta]\subset[0.5,1.5]$ and $p\in\mathbb{N}_{0}$ such that $f^{p}([\alpha,\beta])\subset U$. Recall that $f$ is a piecewise linear mapping, for any $n\in\mathbb{N}_{0}$, one has

[TABLE]

implying that

[TABLE]

This, together with (iii), implies that there exists $K\in\mathbb{N}$ such that

[TABLE]

i.e., $\mathrm{D}(U,1)$ is a thick set. Therefore, $F$ is thickly sensitive.

Claim 2. $(Z,F)$ is not multi-sensitive.

For any $0<\varepsilon<1$, take two open subsets $U_{1}=[0.5,0.5+\tfrac{\varepsilon}{2})\times\{0\}$ and $U_{2}=\{0\}\times[0.5,0.5+\tfrac{\varepsilon}{2})$ of $Z$. From the proof of Claim 1 and (ii)–(iii), it follows that

1. (1)

for any $n\in\left[2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k-1}}+2k+2,2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k}}+2k+2\right]$,

[TABLE] 2. (2)

for any $n\in\left[2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k}}+2k+3,2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k+1}}+2k+3\right]$,

[TABLE]

This implies that

[TABLE]

Thus, $(Z,F)$ is not multi-sensitive.

Remark 10*.*

From Examples 7 and 9, it follows that (StrS) $\nRightarrow$ (TPerS) and (TS) $\nRightarrow$ (MulS). This, together with (3) of Remark 6, answers negatively Question 5.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 11601449 and 11701328), the National Natural Science Foundation of China (Key Program) (No. 51534006), Science and Technology Innovation Team of Education Department of Sichuan for Dynamical System and its Applications (No. 18TD0013), Youth Science and Technology Innovation Team of Southwest Petroleum University for Nonlinear Systems (No. 2017CXTD02), Scientific Research Starting Project of Southwest Petroleum University (No. 2015QHZ029), Shandong Provincial Natural Science Foundation, China (Grant ZR2017QA006), and Young Scholars Program of Shandong University, Weihai (No. 2017WHWLJH09).