Sensitivity, Devaney’s chaos and Li–Yorke $\varepsilon$-chaos

Abstract

We consider the sensitivity from a topological point of view. We show that a continuous, topologically transitive and non-minimal action of a monoid S on an infinite $T_4$ topological space which admits a dense set of almost periodic points is sensitive. We also prove that a uniformly continuous, topologically transitive and non-minimal action of a monoid S on an infinite Hausdorff uniform space which admits a dense set of almost periodic points is thickly syndetically sensitive. We point out that if a continuous action of an Abelian group on a compact metric space is chaotic in the sense of Devaney and has a fixed point, then for every positive integer $n\ge 2$, it is Li–Yorke n–$\varepsilon$-chaotic for some $\varepsilon >0$. Moreover, we show that a continuous and transitive compact action of a semigroup S on a compact metric space is Li–Yorke sensitive and Li–Yorke $\varepsilon$-chaotic for some $\varepsilon >0$.

Appendix

The authors would like to thank the reviewer for providing this appendix.

A uniform space $(X,{\mathscr {U}})$ is totally bounded if for every entourage $\Theta \in {\mathscr {U}}$ there is a finite subset $F\subset X$ such that $\Theta [F]=X$.

Theorem 6.1

For a uniform space $(X,{\mathscr {U}})$ the following are equivalent:

• (1) X is totally bounded.

• (2) For every entourage$\Theta \in {\mathscr {U}}$there exists a finite cover$\{A_1,A_2,\ldots , A_n\}$such that$\Theta _1=\bigcup _{i=1}^nA_i\times A_i\subset \Theta$.

• (3) For every entourage$\Theta \in {\mathscr {U}}$ there exists a finite open cover$\{U_1,U_2,\ldots , U_n\}$ such that$\Theta _1=\bigcup _{i=1}^nU_i\times U_i\subset \Theta$ and$\Theta _1\in {\mathscr {U}}$.

Proof

$(3)\Rightarrow (2)$ is obvious.

$(2)\Rightarrow (1)$. Choose $x_i\in A_i$ where $i=1,2, \ldots ,n$, so that $A_i\subset \Theta _1[x_i]\subset \Theta [x_i]$, and so $\{\Theta [x_i]:i=1,2, \ldots ,n\}$ is a finite cover of X.

$(1)\Rightarrow (3)$. Choose open, symmetric entourages $\Theta _2$, $\Theta _3$ such that $\Theta _3\circ \Theta _3\subset \Theta _2$, $\Theta _2\circ \Theta _2\subset \Theta$. By total boundedness we can choose a finite set $\{x_1,x_2,\ldots ,x_n\}\subset X$ such that $\{\Theta _3[x_i]:i=1,2,\ldots ,n\}$ covers X. Let $U_i=\Theta _2[x_i]$ where $i=1,2,\ldots ,n$. If $y_1,y_2\in U_i$ for some i, then $(y_1,y_2)\in \Theta _2\circ \Theta _2\subset \Theta$ and so $\Theta _1\subset \Theta$. If $(y_1,y_2)\in \Theta _3$ then there exists $x_i$ such that $y_1\in \Theta _3[x_i]\subset \Theta _2[x_i]$ and so $y_2\in \Theta _3\circ \Theta _3[x_i]\subset \Theta _2[x_i]$. Thus, $y_1,y_2\in U_i$ and so $(y_1,y_2)\in \Theta _1$. That is, $\Theta _3\subset \Theta _1$ and so $\Theta _1\in {\mathscr {U}}$.

$\square$

Remark 6.2

By Theorem 6.1, we have that the two notions of sensitivity which do not depend on the specific metric coincide in totally bounded uniform spaces.

Theorem 6.3

For a topological space X the following are equivalent:

• (1) X is normal.

• (2) Every finite open cover has a locally finite closed cover refinement.

• (3) For any finite open cover$\{U_1,U_2,\ldots , U_n\}$ there is a finite closed cover$\{A_1,A_2,\ldots , A_n\}$ with$A_i\subset U_i$ for$i=1,2,\ldots ,n$.

• (4) For any finite open cover$\{U_1,U_2,\ldots , U_n\}$ there is a finite open cover$\{V_1,V_2,\ldots , V_n\}$ with$\overline{V_i}\subset U_i$ for$i=1,2,\ldots ,n$.

Proof

$(4)\Rightarrow (3)\Rightarrow (2)$ is obvious.

$(2)\Rightarrow (1)$. Let A be closed, U be open and $A\subset U$. Let ${\mathscr {B}}$ be a locally finite closed cover which refines $\{U, X{\setminus } A\}$. Let $B_1=\cup \{B \in {\mathscr {B}}: B\subset U\},B_2=\cup \{B\in {\mathscr {B}}: B\subset X{\setminus } A\}$. Since a union of a locally finite collection of closed sets is closed, $B_1$ and $B_2$ are closed. Since $B_2\subset X{\setminus } A$, $A\subset X{\setminus } B_2$ as both are equivalent to $A\cap B_2=\emptyset$. Since $\{B_1,B_2\}$ is a cover, then $A\subset X {\setminus } B_2 \subset B_1 \subset U$. Hence, $B_1$ is a closed neighborhood of A contained in U.

$(1)\Rightarrow (4)$. Let $A_1= X{\setminus } \cup \{U_2,U_3,\ldots ,U_n\}$. Then $A_1\subset U_1$. By normality there exists an open set $V_1$ with $A_1\subset V_1$ and $\overline{V_1}\subset U_1$. Inductively, assume that $V_1,V_2,\ldots ,V_k$ are open sets with $\overline{V_i} \subset U_i$ for $i=1,2,\ldots ,k$ and such that $\{V_1,\ldots ,V_k,U_{k+1},\ldots ,U_n\}$ is cover. Let $A_{k+1}=X{\setminus } \cup \{V_1,\ldots ,V_k$, $U_{k+2},\ldots ,U_n\}$ and let $V_{k+1}$ be open with $A_{k+1}\subset V_{k+1}$ and $\overline{V_{k+1}}\subset U_{k+1}$. $\square$

Theorem 6.4

If X is a Hausdorff normal space, then letting$\{U_1,U_2,\ldots , U_n\}$ vary over all finite open covers ofX, the sets$\bigcup _{i=1}^nU_i\times U_i$ generate the maximum totally bounded uniformity which is the one pulled back from the Stone-$\check{C}$ech compactification $\beta X$.

Proof

Because $\beta X$ is compact its unique uniformity, consisting of all neighborhoods of the diagonal, is totally bounded as is its restriction to X. Hence, every entourage of the induced uniformity contains $\theta _{1}=\bigcup _{U\in {\mathscr {A}}}U\times U$ for some finite open cover ${\mathscr {A}}$.

Conversely, let $\{U_{1},U_{2},\ldots ,U_n\}$ be a finite open cover of X. By Theorem 6.3 we can choose a closed cover $\{A_{1},A_2,\ldots , A_n\}$ with $A_{i}\subset U_{i}$ for $i=1,2,\ldots , n$. By Urysohn’s Lemma there exists a continuous $u_{i}(x)~:~X\rightarrow ~[0,1]$ which is 1 on $A_{i}$ and 0 on $X{\setminus } U_{i}$ for $i=1,2,\ldots , n$. Let $v_{i}(x)=u_{i}(x)/\sum _{j=1}^nu_{j}(x)$ so that $\sum _{i=1}^nv_{i}(x)=1.$ For the Stone-$\check{C}$ech compactification, each $v_{i}(x)$ extends to a continuous $\overline{v_{i}}(x)$ on $\beta X$, and $\sum _{i=1}^n\overline{v_{i}}(x)=1$ by $Cl_{\beta X}X=\beta X$. Let $V_{i}=\{x\in \beta X:\overline{v_{i}}(x)>0\}$ which forms an open cover of $\beta X$. Clearly, $W_{i}=V_{i}\bigcap X\subset U_{i}.$$\bigcup _{i=1}^nV_{i}\times V_{i}$ is a neighborhood of the diagonal in $\beta X \times \beta X$ and so is an entourage of the uniformity on $\beta X$. It follows that $\bigcup _{i=1}^nW_{i}\times W_{i}\subset \bigcup _{i=1}^nU_{i}\times U_{i}$ is an entourage of the induced uniformity on X, so is $\bigcup _{i=1}^nU_{i}\times U_{i}$.

$\square$