A note on the sensitivity of semiows

TL;DR

This paper investigates the properties of syndetic sensitivity in dynamical systems, providing counterexamples that challenge previous assumptions and answering open questions in the field.

Contribution

It constructs specific examples of dynamical systems that demonstrate unexpected behaviors of syndetic sensitivity, addressing open questions from prior research.

Findings

Existence of non-syndetically sensitive cascades with syndetically sensitive products.

Existence of syndetically sensitive semiflows with non-sensitive submonoids.

Abstract

In this note, it is shown that there exist two non-syndetically sensitive cascades defined on complete metric spaces whose product is syndetically sensitive, answering negatively the Question 9.2 posed in [12, Miller, A., Money, C., Turk. J. Math., 41 (2017): 1323{1336]. Moreover, it is shown that there exists a syndetically sensitive semiflow (G;X) defined on a complete metric space X such that (G1;X) is not sensitive for some syndetic closed submonoid G1 of G, answering negatively the Open question 3 posed in [13, Money, C., PhD thesis, University of Louisville, 2015] and Question 43 posed in [8, Miller, A., Real Anal. Exchange, 42 (2017): 9{24].

Equations

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

$${\pi }_t:X$$

$$x$$

$$\mathrm{N}(A,\epsilon )=\left\{t\in G:\mathrm{diam}(t.A)>\epsilon \right\},\text{ where }\mathrm{diam}(t.A)\text{ denotes the diameter of }t.A.$$

$$X=[0,1]$$

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

$$Y=[0,1]$$

$$\bigcup \left(\bigcup _{n=1}^{+\infty }\bigcup _{i=0}^{2^{{\mathcal{L}}_{2n-1}}-4n+2}\left[{\mathcal{L}}_{2n}+i,{\mathcal{L}}_{2n}+i+\frac{1}{2n}\right]\right).$$

$${\mathcal{A}}_n=\left[{\mathcal{L}}_{2n-1}+2^{{\mathcal{L}}_{2n-2}},{\mathcal{L}}_{2n-1}+2^{{\mathcal{L}}_{2n-2}}+\frac{1}{2n-1}\right],$$

$${\mathcal{B}}_n=\left[{\mathcal{L}}_{2n}+(2n+1)\cdot 2^{{\mathcal{L}}_{2n-1}},{\mathcal{L}}_{2n}+(2n+1)\cdot 2^{{\mathcal{L}}_{2n-1}}+2n\right],$$

$${\mathcal{C}}_n=\left[{\mathcal{L}}_{2n-1}+2n\cdot \left(2^{{\mathcal{L}}_{2n-2}}+4n-4\right),{\mathcal{L}}_{2n-1}+2n\cdot \left(2^{{\mathcal{L}}_{2n-2}}+4n-4\right)+(2n-1)\right],$$

$${\mathcal{D}}_n=\left[{\mathcal{L}}_{2n}+2^{{\mathcal{L}}_{2n-1}}-4n+2,{\mathcal{L}}_{2n}+2^{{\mathcal{L}}_{2n-1}}-4n+2+\frac{1}{2n}\right].$$

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

$$g(x)=\{\begin{array}{rl}& x+2,\text{ if }x\in [0,1],\\ & x+1,\text{ if }x\in \left[{\mathcal{L}}_{2n}+i,{\mathcal{L}}_{2n}+i+\frac{1}{2n}\right]\text{ for some }0\le i<2^{{\mathcal{L}}_{2n-1}}-2n,n\in \mathbb{N},\\ & x+2n,\text{ if }x\in \left[{\mathcal{L}}_{2n-1}+2ni,{\mathcal{L}}_{2n-1}+2ni+(2n-1)\right]\text{ for some }0\le i<2^{{\mathcal{L}}_{2n-2}}+2n-2,n\in \mathbb{N},\\ & \frac{1}{2n(2n-1)}\left(x-{\mathcal{L}}_{2n-1}-2n\cdot \left(2^{{\mathcal{L}}_{2n-2}}+4n-4\right)\right)+{\mathcal{L}}_{2n},\text{ if }x\in {\mathcal{C}}_n,n\in \mathbb{N},\\ & 2n(2n+1)\left(x-{\mathcal{L}}_{2n}-2^{{\mathcal{L}}_{2n-1}}+4n-2\right)+{\mathcal{L}}_{2n+1},\text{ if }x\in {\mathcal{D}}_n,n\in \mathbb{N}.\end{array}$$

$$f^n([0,1])=\left[{\mathcal{L}}_{2k+1}+n-a_k,{\mathcal{L}}_{2k+1}+n-a_k+\frac{1}{2k+2}\right],$$

$$f^n([0,1])=\left[{\mathcal{L}}_{2k+2}+(n-b_k)(2k+3),{\mathcal{L}}_{2k+2}+(n-b_k)(2k+3)+2k+2\right],$$

$$g^n([0,1])=\left[{\mathcal{L}}_{2k+1}+(n-c_k)(2k+2),{\mathcal{L}}_{2k+1}+(n-c_k)(2k+2)+2k+1\right],$$

$$g^n([0,1])=\left[{\mathcal{L}}_{2k+2}+n-d_k,{\mathcal{L}}_{2k+2}+n-d_k+\frac{1}{2k+2}\right],$$

$$\mathrm{diam}(f^n([{\alpha }_1,{\beta }_1]))=({\beta }_1-{\alpha }_1)\cdot \mathrm{diam}(f^n([0,1])),$$

$$\mathrm{diam}(g^n([{\alpha }_2,{\beta }_2]))=({\beta }_2-{\alpha }_2)\cdot \mathrm{diam}(g^n([0,1])).$$

$$\mathcal{P}:=\bigcup _{k=K}^{+\infty }\left[2^{{\mathcal{L}}_1}+\cdots +2^{{\mathcal{L}}_{2k}}+2k+3-p,2^{{\mathcal{L}}_1}+\cdots +2^{{\mathcal{L}}_{2k+1}}+2k+3-p\right]\subset \mathrm{N}(U,1),$$

$$\mathcal{Q}:=\bigcup _{k=K}^{+\infty }\left[2^{{\mathcal{L}}_1}+\cdots +2^{{\mathcal{L}}_{2k-1}}+2-q,2^{{\mathcal{L}}_1}+\cdots +2^{{\mathcal{L}}_{2k}}+4k+2-q\right]\subset \mathrm{N}(V,1),$$

$$\mathrm{N}(W,1)\supset \mathrm{N}(U\times V,1)\supset \mathrm{N}(U,1)\cup \mathrm{N}(V,1)\supset \mathcal{P}\cup \mathcal{Q}.$$

$$\left[2^{{\mathcal{L}}_1}+\cdots +2^{{\mathcal{L}}_{2K-1}}+2-q,+\infty \right)=\mathcal{P}\cup \mathcal{Q},$$

$$\mathrm{diam}(f^n(U))\ge \mathrm{diam}(f^n(f^p([\alpha ,\beta ])))=\mathrm{diam}(f^{n+p}([\alpha ,\beta ]))=(\beta -\alpha )\cdot \mathrm{diam}(f^{n+p}([0,1])).$$

$$\bigcup _{k=K}^{+\infty }\left[2^{{\mathcal{L}}_1}+\cdots +2^{{\mathcal{L}}_{2k}}+2k+3-p,2^{{\mathcal{L}}_1}+\cdots +2^{{\mathcal{L}}_{2k+1}}+2k+3-p\right]\subset \mathrm{N}(U,1),$$

$$f(x)=\{\begin{array}{rl}& x+2,\text{ if }x\in [0,1],\\ & \frac{1}{2n(2n-1)}\left(x-{\mathcal{L}}_{2n-1}\right)+{\mathcal{L}}_{2n},\text{ if }x\in \left[{\mathcal{L}}_{2n-1},{\mathcal{L}}_{2n-1}+2n-1\right]\text{ for some }n\in \mathbb{N},\\ & 2n(2n+1)\left(x-{\mathcal{L}}_{2n}\right)+{\mathcal{L}}_{2n+1},\text{ if }x\in \left[{\mathcal{L}}_{2n},{\mathcal{L}}_{2n}+\frac{1}{2n}\right]\text{ for some }n\in \mathbb{N}.\end{array}$$

$$\mathrm{diam}(f^{2n+1-k}(U))\ge \mathrm{diam}(f^{2n+1}(I))=(2n+1)\cdot |I|⟶+\infty (n⟶+\infty ).$$

Full text

††journal: Topology and its Applications

Let $\mathbb{N}=\left\{1,2,3,\ldots\right\}$, $\mathbb{N}_{0}=\left\{0,1,2,\ldots\right\}$ and $G$ denote a noncompact abelian topological monoid with the identity element [math]. A subset $A$ of $G$ is

1. (1)

syndetic if there exists a compact subset $K$ of $G$ such that, for every $t\in G$, $(t+K)\cap A\neq\O$, 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 $t+K\subset A$.

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

A jointly continuous monoid action $\pi:G\times X\longrightarrow X$ of $G$ on a metric space $(X,d)$ is called a semiflow and denoted by $(G,X,\pi)$ or $(G,X)$. In particular, if the acting topological monoid is a topological group, a semiflow is called a flow. The element $\pi(t,x)$ will be denoted by $t.x$ or $tx$, so that the defining conditions for a semiflow 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 semiflow $(G,X)$ 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 $t.A=\{t.x:x\in A\}$. Now, let $X$ be a metric space and let $f:X\longrightarrow X$ be a continuous map, which leads to a natural semiflow, where $G=\mathbb{N}_{0}$ (with the discrete topology), and for any $x\in X$ and any $n\in\mathbb{N}_{0}$, $n.x=f^{n}(x)$. This type of semiflow is called a cascade and is often denoted by $(X,f)$ instead of $(\mathbb{N}_{0},X)$.

Let $(G,X)$ be a semiflow. For any $\varepsilon>0$ and any subset $A$ of $X$, let

[TABLE]

A semiflow $(G,X)$ is

1. (1)

sensitive if there exists $\varepsilon>0$ such that, for every nonempty open subset $U$ of $X$, $\mathrm{N}(U,\varepsilon)\neq\O$; 2. (2)

syndetically sensitive if there exists $\varepsilon>0$ such that, for every nonempty open subset $U$ of $X$, $\mathrm{N}(U,\varepsilon)$ is syndetic; 3. (3)

thickly sensitive if there exists $\varepsilon>0$ such that, for every nonempty open subset $U$ of $X$, $\mathrm{N}(U,\varepsilon)$ is thick; 4. (4)

cofinitely sensitive if there exists $\varepsilon>0$ such that, for every nonempty open subset $U$ of $X$, $\mathrm{N}(U,\varepsilon)$ is cofinite; 5. (5)

multi-sensitive if there exists $\varepsilon>0$ such that, for any nonempty open subsets $U_{1},\ldots,U_{k}$ of $X$, $\bigcap_{i=1}^{k}\mathrm{N}(U_{i},\varepsilon)\neq\O$.

Recently, Miller and Money [11] proved that a non-minimal syndetically transitive semiflow is syndetically sensitive, generalizing some results in [1, 3, 14, 16]. Then, they [12] generalized some results on chaotic properties of cascades to the product of semiflows and asked the following question. (For more recent results on sensitivity, refer to [2, 4, 5, 7, 8, 9, 10, 17, 18, 19, 21, 22] and some references therein.)

We proved [20] that $(X\times Y,f\times g)$ is multi-sensitive if and only if $(X,f)$ or $(Y,g)$ is multi-sensitive. Using the analogous method in [20], Thakur and Das [15] showed that this also holds for semiflows. Money [13] then conducted a systemic investigation on chaotic properties for semiflows and asked the following question which also can be found in [8, Question 43]:

Recently, Miller [10] discussed relations between various types of sensitivity in general semiflows and asked the following question:

This paper constructs two examples (Examples 4 and 6 below), answering negatively Questions 1, 2, and 3 above.

[TABLE]

and

[TABLE]

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

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Define $f:X\longrightarrow X$ and $g:Y\longrightarrow Y$ respectively by

[TABLE]

and

[TABLE]

Clearly, $f$ and $g$ are continuous. Arrange all closed intervals $[0,1],[\mathscr{L}_{1},\mathscr{L}_{1}+\frac{1}{2}],\ldots,[\mathscr{L}_{1}+2^{\mathscr{L}_{0}},\mathscr{L}_{1}+2^{\mathscr{L}_{0}}+\frac{1}{2}],[\mathscr{L}_{2},\mathscr{L}_{2}+2],\ldots,[\mathscr{L}_{2}+2\cdot 2^{\mathscr{L}_{1}},\mathscr{L}_{2}+2\cdot 2^{\mathscr{L}_{1}}+2],\ldots,[\mathscr{L}_{2n-1},\mathscr{L}_{2n-1}+\frac{1}{2n}],\ldots,[\mathscr{L}_{2n-1}+2^{\mathscr{L}_{2n-2}},\mathscr{L}_{2n-1}+2^{\mathscr{L}_{2n-2}}+\frac{1}{2n}],[\mathscr{L}_{2n},\mathscr{L}_{2n}+2n],\ldots,[\mathscr{L}_{2n}+(2n+1)2^{\mathscr{L}_{2n-1}},\mathscr{L}_{2n}+(2n+1)2^{\mathscr{L}_{2n-1}}+2n],\ldots$ of $X$ by this natural order and denote them as $I_{0},I_{1},I_{2},\ldots$. It is easy to see that $I_{2k+2+2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k-1}}}=[\mathscr{L}_{2k+1},\mathscr{L}_{2k+1}+\frac{1}{2(k+1)}]$, $\ldots$, $I_{2k+2+2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k}}}=[\mathscr{L}_{2k+1}+2^{\mathscr{L}_{2k}},\mathscr{L}_{2k+1}+2^{\mathscr{L}_{2k}}+\frac{1}{2(k+1)}]$, $I_{2k+3+2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k}}}=[\mathscr{L}_{2k+2},\mathscr{L}_{2k+2}+2(k+1)]$, $\ldots$, $I_{2k+3+2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k+1}}}=[\mathscr{L}_{2k+2}+(2k+3)\cdot 2^{\mathscr{L}_{2k+1}},\mathscr{L}_{2k+2}+(2k+3)\cdot 2^{\mathscr{L}_{2k+1}}+2(k+1)]$. Similarly, arrange all closed intervals of $Y$ by this natural order and denote them as $J_{0},J_{1},J_{2},\ldots$. It is easy to see that $J_{2+2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k-1}}}=[\mathscr{L}_{2k+1},\mathscr{L}_{2k+1}+(2k+1)]$, $\cdots$, $J_{2+4k+2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k}}}=[\mathscr{L}_{2k+1}+2(k+1)\cdot(2^{\mathscr{L}_{2k}}+4k),\mathscr{L}_{2k+1}+2(k+1)\cdot(2^{\mathscr{L}_{2k}}+4k)+(2k+1)]$, $J_{3+4k+2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k}}}=[\mathscr{L}_{2k+2},\mathscr{L}_{2k+2}+\frac{1}{2(k+1)}]$, $\cdots$, $J_{1+2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k+1}}}=[\mathscr{L}_{2k+2}+2^{\mathscr{L}_{2k+1}}-4k-2,\mathscr{L}_{2k+2}+2^{\mathscr{L}_{2k+1}}-4k-2+\frac{1}{2(k+1)}]$. 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 $J_{n+1}$, for all $n\in\mathbb{N}_{0}$. According to the constructions of $f$ and $g$, it can be verified that

1. (i)

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

$X$ and $Y$ are complete metric subspaces of $\mathbb{R}^{1}$, implying that $X\times Y$ is a complete subspace of $\mathbb{R}^{2}$; 3. (iii)

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]

where $a_{k}=2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k-1}}+2k+2$, implying that $\mathrm{diam}(f^{n}([0,1]))=\frac{1}{2k+2}$; 4. (iv)

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]

where $b_{k}=2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k}}+2k+3$, implying that $\mathrm{diam}(f^{n}([0,1]))=2k+2$; 5. (v)

for any $n\in\left[2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k-1}}+2,2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k}}+4k+2\right]$ ($k\in\mathbb{N}$),

[TABLE]

where $c_{k}=2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k-1}}+2$, implying that $\mathrm{diam}(g^{n}([0,1]))=2k+1$; 6. (vi)

for any $n\in\left[2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k}}+4k+3,2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k+1}}+1\right]$ ($k\in\mathbb{N}$),

[TABLE]

where $b_{k}=2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{4k}}+2k+3$, implying that $\mathrm{diam}(g^{n}([0,1]))=\frac{1}{2k+2}$.

Claim 1. $(X,f)$ is not syndetically sensitive.

Fix an open subset $U=[0,1)$ of $X$. For any $\varepsilon>0$, applying (iii) yields that $\{n\in\mathbb{N}_{0}:\mathrm{diam}(f^{n}(U))\leq\varepsilon\}\supset\bigcup_{n=K}^{+\infty}\left[2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k-1}}+2k+2,2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k}}+2k+2\right]$ for some $K\in\mathbb{N}$, i.e., $\{n\in\mathbb{N}_{0}:\mathrm{diam}(f^{n}(U))\leq\varepsilon\}$ is a thick set. This implies that $\mathrm{N}_{f}(U,\varepsilon)$ is not syndetic. Therefore, $(X,f)$ is not syndetically sensitive.

Claim 2. $(Y,g)$ is not syndetically sensitive.

Similarly to the proof of Claim 1, applying (vi) follows that this is true.

Claim 3. $(X\times Y,f\times g)$ is cofinitely sensitive.

Given any nonempty open subset $W$ of $X\times Y$, there exist open subsets $U\subset X$ and $V\subset Y$ such that $U\times V\subset W$. From (iii)–(vi), it follows that there exist non-degenerate closed intervals $[\alpha_{1},\beta_{1}],[\alpha_{2},\beta_{2}]\subset[0,1]$ and $p,q\in\mathbb{N}_{0}$ such that $f^{p}([\alpha_{1},\beta_{1}])\subset U$ and $g^{q}([\alpha_{2},\beta_{2}])\subset V$. As $f$ and $g$ are piecewise linear mappings, for any $n\in\mathbb{N}_{0}$, one has

[TABLE]

and

[TABLE]

This, together with (iv) and (v), implies that there exists $K>2(p+q)$ such that

[TABLE]

and

[TABLE]

implying that

[TABLE]

Noting that $2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k}}+4k+2-q>2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k}}+2k+3-p$ and $2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k+1}}+2k+3-p>2^{\mathscr{L}_{1}}+\cdots+2^{\mathscr{L}_{2k+1}}+2-q$ hold for all $k\geq K$, it can be verified that

[TABLE]

i.e., $\mathscr{P}\cup\mathscr{Q}$ is a cofinite set. This implies that $f\times g$ is cofinitely sensitive.

Claim 4. $(X,f)$ is thickly sensitive.

Given any nonempty open subset $U$ of $X$, similarly to the proof of Claim 3, one has

1. (i)

there exist non-degenerate closed interval $[\alpha,\beta]\subset[0,1]$ and $p\in\mathbb{N}_{0}$ such that $f^{p}([\alpha,\beta])\subset U$; 2. (ii)

for any $n\geq p$,

[TABLE]

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

[TABLE]

implying that $(X,f)$ is thickly sensitive.

**

1. (1)

Example 4 shows that the answer to Question 1 for syndetic sensitivity and cofinite sensitivity is negative. Combining this with [15, Theorem 3.6, Proposition 3.7] completely solves Question 1. 2. (2)

Example 4, together with [20, Theorem 10], also gives a complete answer to [6, Remark 3.4]. 3. (3)

From Claims 1 and 4 of Example 4, it follows that there exists a thickly sensitive semiflow which is not syndetically sensitive, answering negatively Question 3.

Similarly to the constructed semiflow in Example 4, the following example gives an negative answer to Question 2, showing that there exists a syndetically sensitive semiflow $(\mathbb{N}_{0},X)$ defined on a complete metric space $X$ such that $(2\mathbb{N}_{0},X)$ is not sensitive.

[TABLE]

For any nonempty open subset $U$ of $X$, there exist a non-degenerate closed interval $I\subset[0,1]$ and $k\in\mathbb{N}_{0}$ such that $f^{k}(I)\subset U$. Thus, for any $n>k$, one has

[TABLE]

This implies that $(X,f)$ is syndetically sensitive.

Meanwhile, it is easy to see that, for any $n\in\mathbb{N}$, $\mathrm{diam}(f^{2n}([0,1]))=\frac{1}{2n}$, implying that $(X,f^{2})=(2\mathbb{N}_{0},X)$ is not sensitive. **