Specification properties on uniform spaces

TL;DR

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Contribution

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Abstract

In the following text we introduce specification property (stroboscopical property) for dynamical systems on uniform space. We focus on two classes of dynamical systems: generalized shifts and dynamical systems with Alexandroff compactification of a discrete space as phase space. We prove that for a discrete finite topological space $X$ with at least two elements, a nonempty set $\Gamma$ and a self--map $\varphi:\Gamma\to\Gamma$ the generalized shift dynamical system $(X^\Gamma,\sigma_\varphi)$: \begin{itemize} \item has (almost) weak specification property if and only if $\varphi:\Gamma\to\Gamma$ does not have any periodic point, \item has (uniform) stroboscopical property if and only if $\varphi:\Gamma\to\Gamma$ is one-to-one. \end{itemize}

Equations

$$(h^i(x),h^i(y_j))\in \alpha \forall j\in \{1,\dots ,n\}\forall i\in \{l_j,l_j+1,\dots ,k_j\}.$$

$${\alpha }_H:=\{((x_{\alpha }{)}_{\alpha \in \Gamma },(y_{\alpha }{)}_{\alpha \in \Gamma })\in X^{\Gamma }\times X^{\Gamma }:\forall \alpha \in H(x_{\alpha }=y_{\alpha })\},$$

$$\{U\subseteq F:b\notin U\vee (F\setminus U\mathrm{is}\mathrm{finite})\},$$

$${\gamma }_H:=((F\setminus H)\times (F\setminus H))\cup {\Delta }_H$$

$$({\sigma }_{\phi }^i(x),{\sigma }_{\phi }^i(y_j))\in \alpha \forall j\in \{1,2\}\forall i\in \{l_j,l_j+1,\dots ,k_j\},$$

$$((x_{\phi (\theta )}{)}_{\theta \in \Gamma },(p{)}_{\theta \in \Gamma })\in \alpha ,$$

$$((x_{{\phi }^i(\theta )}{)}_{\theta \in \Gamma },(q{)}_{\theta \in \Gamma })\in \alpha \forall i=N_{\alpha }(m)+2,\dots ,N_{\alpha }(m)+2+m,$$

$$z_{\theta}=\left\{\begin{array}[]{lc}x^{s}_{\varphi^{t}(\gamma)}&1\leq s\leq k\wedge i_{s}\leq t\leq j_{s}\wedge\gamma\in F\wedge\theta=\varphi^{t}(\gamma)\\ p&{\rm otherwise}\end{array}\right.$$

$$({\sigma }_{\phi }^i(x),{\sigma }_{\phi }^i(y_j))\in \beta \forall j\in \{1,\dots ,n\}\forall i\in \{l_j,l_j+1,\dots ,k_j\}.$$

$$(u,v)\in \mu \Rightarrow ({\sigma }_{\phi }^i(u),{\sigma }_{\phi }^i(v))\in \beta (\forall i\in \{0,1,\dots ,k_n\}.$$

$$({\mathfrak{h}}^i(x),{\mathfrak{h}}^i(y_j))\in {\gamma }_{\{z\}}\forall j\in \{1,\dots ,n\}\forall i\in \{l_j,l_j+1,\dots ,k_j\}.$$

$$l_1=k_1=0,l_2=k_2\ge N_{{\gamma }_{\{z\}}}(0),$$

$$(x_1,z)=({\mathfrak{h}}^{l_1}(x_1),{\mathfrak{h}}^{l_1}(z)),({\mathfrak{h}}^{l_2}(x_1),z)=({\mathfrak{h}}^{l_2}(x_1),{\mathfrak{h}}^{l_2}(\mathfrak{h}(w)))\in {\gamma }_{\{z\}},$$

$$(x_2,z)=({\mathfrak{h}}^{l_1}(x_2),{\mathfrak{h}}^{l_1}(z)),({\mathfrak{h}}^{l_2}(x_2),{\mathfrak{h}}^{l_2}(w))\in {\gamma }_{\{z\}}.$$

$$(x_3,z)=({\mathfrak{h}}^{l_1}(x_3),{\mathfrak{h}}^{l_1}(z)),({\mathfrak{h}}^{l_2}(x_3),y)=({\mathfrak{h}}^{l_2}(x_3),{\mathfrak{h}}^{l_2}(v))\in {\gamma }_{\{z\}},$$

$$({\mathfrak{h}}^i(y_1),{\mathfrak{h}}^i(y_j))\in \alpha \forall j\in \{1,\dots ,n\}\forall i\in \{l_j,l_j+1,\dots ,k_j\},$$

$$\forall n\ge m({\mathfrak{h}}^n(F)=\{c\}).$$

$$F\setminus \{b\}\subseteq Per(\mathfrak{h})\subseteq \bigcap \{{\mathfrak{h}}^n(F):n\ge 1\},$$

$$z_{\alpha}=\left\{\begin{array}[]{lc}p&\alpha=\beta\>,\\ q&{\rm otherwise}\>.\end{array}\right.$$

$$z_{\alpha }^{\ast }=z_{{\phi }^{k_{\alpha }-f(k_{\alpha })}(\alpha )}(\alpha \in \Gamma ).$$

$$\forall \alpha \in \Gamma \forall i\ge f(k_{\alpha })({\phi }^{k_{\alpha }-f(k_{\alpha })+n_i}(\alpha )=\alpha .)$$

$${\Lambda }_1:=\{\alpha \in \Lambda :\frac{\alpha }{{\sim }_{\phi }}=\{{\phi }^n(\alpha ):n\ge 0\}\}$$

$${\Lambda }_2:=\{\alpha \in \Lambda :\forall n\ge 0({\phi }^{-n}(\alpha )\ne \varnothing )\}.$$

$$z^{*}_{\alpha}=\left\{\begin{array}[]{lc}&\\ z_{\varphi^{j}(\lambda_{\alpha})}&0\leq j<n_{i+1}-n_{i},\alpha=\varphi^{n_{i}+j}(\lambda_{\alpha}),\lambda_{\alpha}\in\Lambda_{1}\>,\\ &\\ z_{\varphi^{j}(\lambda_{\alpha})}&-i<j<i,\alpha=\varphi^{n_{i}+j}(\lambda_{\alpha}),\lambda_{\alpha}\in\Lambda_{2}\>,\\ &\\ p&{\rm otherwise}\>.\\ &\end{array}\right.$$

$$\forall \mu \in {\Lambda }_1\forall j\ge 0\forall i\ge j+2(z_{{\phi }^{n_i+j}(\mu )}^{\ast }=z_{{\phi }^j(\mu )}),$$

$$\forall \mu \in {\Lambda }_2\forall j\in \mathbb{Z}\forall i>|j|(z_{{\phi }^{n_i+j}(\mu )}^{\ast }=z_{{\phi }^j(\mu )}).$$

$$\forall \theta \in H\forall i\ge N+2(z_{{\phi }^{n_i}(\theta )}^{\ast }=z_{\theta }),$$

$$\forall k\ge N({\mathfrak{h}}^{n_k}(x)=z),$$

$$\forall t\ge 1\forall k\ge t(n_k^k\stackrel{\mathrm{mod}t}{\equiv }{r}_t).$$

$$\begin{array}[]{c}\mathfrak{h}_{1}(x)=\left\{\begin{array}[]{lc}\frac{x}{x-1}&x\neq 1\>,\\ 0&x=1\>,\end{array}\right.\>\>\>\>\>\>\mathfrak{h}_{2}(x)=\left\{\begin{array}[]{lc}\frac{x}{x-1}&x\neq 1\>,\\ -1&x=1\>,\end{array}\right.\\ \\ \mathfrak{h}_{3}(x)=-x\>(x\in\{\frac{\pm 1}{n}:n\in{\mathbb{Z}}\setminus\{0\}\}\cup\{0\})\>.\end{array}$$

Full text

• •

has (almost) weak specification property if and only if $\varphi:\Gamma\to\Gamma$ does not have any periodic point,

• •

has (uniform) stroboscopical property if and only if $\varphi:\Gamma\to\Gamma$ is one-to-one.

1. Introduction

The main purpose of this paper is to generalize tracing properties to uniform spaces and in particular to compact Hausdorff spaces (one may find similar ideas in [12, 4]).

The action of a group $G$ on metric space $(Z,d)$ is sensitive if there exists $\epsilon>0$ such that for all $x\in Z$ and any open neighbourhood $U$ of $x$ there exists $y\in U$ and $g\in G$ with $d(gx,gy)\geq\epsilon$. T. Ceccherini–Silberstein and M. Coornaert [4] indicate that the action of a group $G$ on a uniform space $(Y,{\mathcal{F}})$ is sensitive if there exists $\alpha\in{\mathcal{F}}$ such that for all $x\in Y$ and any open neighbourhood $U$ of $x$ there exists $y\in U$ and $g\in G$ with $(gx,gy)\notin\alpha$. In this paper we introduce specification properties for dynamical systems on uniform space .

We realize our generalizations in uniform spaces through two classes of dynamical systems “generalized shift dynamical systems” and “dynamical systems with Alexandroff compactification of a discrete space as phase space” on various specification and stroboscopical properties.

2. Preliminaries

Let’s recall that for an arbitrary set $Y$ we call the collection $\mathcal{F}$ of subsets of $Y\times Y$ a uniform structure on $Y$ if :

• •

$\forall\alpha\in{\mathcal{F}}\>(\Delta_{Y}\subseteq\alpha)$;

• •

$\forall\alpha,\beta\in{\mathcal{F}}\>(\alpha\cap\beta\in{\mathcal{F}})$;

• •

$\forall\alpha\in{\mathcal{F}}\>\exists\beta\in{\mathcal{F}}\>(\beta\circ\beta^{-1}\subseteq\alpha)$;

• •

$\forall\alpha\in{\mathcal{F}}\>\forall\beta\subseteq Y\times Y\>(\alpha\subseteq\beta\Rightarrow\beta\in{\mathcal{F}})$.

where for $\alpha,\beta\subseteq X\times X$ we have $\alpha^{-1}=\{(y,x):(x,y)\in\alpha\}$ and $\alpha\circ\beta=\{(x,z):\exists y\>((x,y)\in\beta\wedge(y,z)\in\alpha)\}$ also $\Delta_{Y}=\{(x,x):x\in Y\}$) [6].

Moreover, for all $\alpha\in\mathcal{F}$ and $x\in Y$ let $\alpha[x]=\{y\in Y:(x,y)\in\alpha\}$. For uniform structure $\mathcal{F}$ on $Y$, the collection $\{U\subseteq Y:\forall x\in Y\>\exists\alpha\in{\mathcal{F}}\>(\alpha[x]\subseteq U)\}$ is a topology on $Y$, we call it the uniform topology induced from $\mathcal{F}$, so accordingly we denote this topological space by uniform topological space $(Y,{\mathcal{F}})$ whenever we want to emphasize $\mathcal{F}$. We call the topological space $Z$ uniformizable if there exists a uniform structure $\mathcal{G}$ in $Z$ such that uniform topology induced from $\mathcal{G}$ coincides with original topology on $Z$, and in this case we call $\mathcal{G}$ compatible uniform structure on $Z$. Every compact Hausdorff (and in particular, compact metric) space is uniformizable and has a unique compatible uniform structure.

Note to the fact that compact metric space$(Z,d)$ has a unique compatible uniform structure $\{D\subseteq Z\times Z:\exists\kappa>0\>(\{(x,y)\in Z\times Z:d(x,y)<\kappa\}\subseteq D)\}$.

In this text by a dynamical system $(Y,h)$ we mean a topological space $Y$ and continuous map $h:Y\to Y$. Similar to the definition of specification properties in metric spaces [5] we have the following definition.

Definition 2.1**.**

For a uniform topological space $(Y,\mathcal{F})$, we say that the dynamical system $((Y,\mathcal{F}),h)$ (or briefly $(Y,h)$) has almost weak specification property if for all $\alpha\in\mathcal{F}$ there exists a map $N_{\alpha}:{\mathbb{Z}}_{+}\to{\mathbb{Z}}_{+}$ with ${\displaystyle\lim_{n\to\infty}\dfrac{N_{\alpha}(n)}{n}}=0$ such that for all $y_{1},\ldots,y_{n}\in Y$ ($n\geq 2$) and $0\leq l_{1}\leq k_{1}<l_{2}\leq k_{2}<\ldots<l_{n}\leq k_{n}$ with $l_{i+1}-k_{i}\geq N_{\alpha}(k_{i+1}-l_{i+1})$ (for all $i=1,\ldots,n-1$) there exists $x\in Y$ such that

[TABLE]

If in the above definition we suppose in addition that $N_{\alpha}$ is a constant map (resp. and $x$ is a periodic point of $h$), then we say that $((Y,\mathcal{F}),h)$ has weak specification property (resp. specification property).

Let’s recall that in the dynamical system $(Z,f)$, for $x\in Z$ and strictly increasing sequence $A=\{n_{i}\}_{i\geq 0}$ in $\mathbb{N}$ we set $\omega_{f}(A,x)=\{y\in Z:\{f^{n_{i}}(x)\}_{i\geq 0}$ has a subsequence converging to $y\}$. Now for definition of various stroboscopical properties see [8].

Definition 2.2**.**

We say that $(Z,f)$ has stroboscopical property (resp. strongly stroboscopical property) if for all $z\in Z$ and every strictly increasing sequence $A=\{n_{i}\}_{i\geq 0}$ in $\mathbb{N}$, $\{x\in Z:z\in\omega_{f}(A,x)\}$ is a nonempty (resp. dense) subset of $Z$. Moreover if $(Z,\mathcal{F})$ is a uniform space and for all strictly increasing sequence $A=\{n_{i}\}_{i\geq 0}$ in $\mathbb{N}$ there exists a subsequence $B=\{k_{i}\}_{i\geq 0}$ of $A$ and $\varrho:Z\to Z$ such that the sequence $\{f^{k_{i}}\circ\varrho\}_{i\geq 0}$ converges uniformly to $id_{Z}$, then we say that $((Z,\mathcal{F}),f)$ has uniform stroboscopical property.

Generalized shifts

The one sided shift $\mathop{\{1,\ldots,k\}^{\mathbb{N}}\to\{1,\ldots,k\}^{\mathbb{N}}}\limits_{(x_{n})_{n\geq 1}\mapsto(x_{n+1})_{n\geq 1}}$ and two sided shift $\mathop{\{1,\ldots,k\}^{\mathbb{Z}}\to\{1,\ldots,k\}^{\mathbb{Z}}}\limits_{(x_{n})_{n\in{\mathbb{Z}}}\mapsto(x_{n+1})_{n\in{\mathbb{Z}}}}$ due to P. Walters [11] have been investigated very well from the ergodic theory and dynamical systems point of view. For nonempty sets $X$ and $\Gamma$ and arbitrary map $\varphi:\Gamma\to\Gamma$, we call $\sigma_{\varphi}:X^{\Gamma}\to X^{\Gamma}$ with $\sigma_{\varphi}((x_{\alpha})_{\alpha\in\Gamma})=(x_{\varphi(\alpha)})_{\alpha\in\Gamma}$ (for $(x_{\alpha})_{\alpha\in\Gamma}\in X^{\Gamma}$), a generalized shift. Whenever $X$ is a topological space and $X^{\Gamma}$ is equiped with product (pointwise convergence) topology, it’s evident that $\sigma_{\varphi}:X^{\Gamma}\to X^{\Gamma}$ is continuous and one may consider dynamical system $(X^{\Gamma},\sigma_{\varphi})$. Generalized shifts has been introduced for the first time in [2], moreover their dynamical properties have been studied in several texts, like [3] and [7].

Convention 2.3**.**

Suppose that $X$ is a finite discrete topological space with at least two elements, $\Gamma$ is a nonempty set, and $\varphi:\Gamma\to\Gamma$ is an arbitrary map. For $H\subseteq\Gamma$ let:

[TABLE]

so

${\mathcal{U}}:=\{D\subseteq X^{\Gamma}\times X^{\Gamma}:$ there exists a finite subset $H$ of $\Gamma$ with $\alpha_{H}\subseteq D\}$

is the unique compatible uniform structure on $X^{\Gamma}$ (which is equipped with product topology). Note that, the compact Hausdorff topological space $X^{\Gamma}$ is metrizable if and only if $\Gamma$ is countable.

Alexandroff compactification and Fort spaces

For a topological space $Y$ choose a point $b\notin Y$ and consider the set $A(Y):=Y\cup\{b\}$ with the topology $\{U\subseteq Y:U$ is an open subset of $Y\}\cup\{V\subseteq A(Y):Y\setminus V$ is a closed compact subset of $Y\}$, then we call $A(Y)$ the Alexandroff compactification or one point compactification of $Y$ [9, 10]. Obviously one may study a dynamical system with Alexandroff compactification of a topological space as phase space. Now suppose $b\in F$ and consider $F$ with the topology:

[TABLE]

then we say that $F$ is a Fort space (with particular point $b$) [10, Counterexample 24]. It’s evident that the class of Fort spaces is just the class of Alexandroff compactification of discrete spaces.

Convention 2.4**.**

From now on, consider $F$ is a Fort space with the particular point $b$ and $\mathfrak{h}:F\to F$ is continuous. For $H\subseteq F$ let:

[TABLE]

so

${\mathcal{K}}:=\{D\subseteq F\times F:$ there exists a finite subset $H$ of $F\setminus\{b\}$ with $\gamma_{H}\subseteq D\}$

is the unique compatible uniform structure on $F$. Note that, the compact Hausdorff topological space $F$ is metrizable if and only if $F$ is countable.

Remark 2.5**.**

In Fort spaces $Y_{1}$ with particular point $b_{1}$ and $Y_{2}$ with particular point $b_{2}$, $f:Y_{1}\to Y_{2}$ is continuous if and only if one of the following conditions holds true [1, Lemma 5.1]:

• •

$f(b_{1})=b_{2}$ and $f^{-1}(y)$ is finite, for all $y\in Y_{2}\setminus\{b_{2}\}$,

• •

$f(b_{1})\neq b_{2}$ and $Y_{1}\setminus f^{-1}(f(b_{1}))$ is finite.

3. Weak specification property

In this section we show that $(X^{\Gamma},\sigma_{\varphi})$ has weak specification property (almost weak specification property) if and only if $\varphi:\Gamma\to\Gamma$ does not have any periodic point. Moreover $(X^{\Gamma},\sigma_{\varphi})$ has specification property, if and only if $\varphi:\Gamma\to\Gamma$ is one–to–one without any periodic point. On the other hand $(F,\mathfrak{h})$ has weak specification property (almost weak specification property) if and only if $\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}$ is singleton. Finally, $(F,\mathfrak{h})$ has specification property if and only if $F$ is singleton.

Lemma 3.1**.**

If $\varphi:\Gamma\to\Gamma$ has a periodic point, then the dynamical system $(X^{\Gamma},\sigma_{\varphi})$ does not have almost weak specification property.

Proof.

Suppose that $\lambda$ is a periodic point of $\Gamma$, then there exists $m\geq 1$ with $\varphi^{m}(\lambda)=\lambda$. Choose distinct $p,q\in X$ and consider $\alpha:=\alpha_{\{\lambda\}}\in\mathcal{U}$. If $(X^{\Gamma},\sigma_{\varphi})$ has almost weak specification property, then there exists $N_{\alpha}:{\mathbb{Z}}_{+}\to{\mathbb{Z}}_{+}$ such that for all $0\leq l_{1}\leq k_{1}<l_{2}\leq k_{2}$ with $l_{2}-k_{1}\geq N_{\alpha}(k_{2}-l_{2})$ there exists $x\in X^{\Gamma}$ with the property:

[TABLE]

for $y_{1}=(p)_{\theta\in\Gamma}$ and $y_{2}=(q)_{\theta\in\Gamma}$. In particular for $l_{1}=k_{1}=1,l_{2}=N_{\alpha}(m)+2,k_{2}=N_{\alpha}(m)+2+m$ there exists $(x_{\theta})_{\theta\in\Gamma}$ such that:

[TABLE]

and

[TABLE]

which imply that $x_{\varphi(\lambda)}=p$ and $x_{\varphi^{i}(\lambda)}=q$ for all $i=N_{\alpha}(m)+2,\ldots,N_{\alpha}(m)+2+m$. Choose $j\in\{N_{\alpha}(m)+2,\ldots,N_{\alpha}(m)+2+m\}$ such that there exists $n\geq 1$ with $j=nm+1$, thus $\varphi^{j}(\lambda)=\varphi(\lambda)$ and $q=x_{\varphi^{j}(\lambda)}=x_{\varphi(\lambda)}=p$ which is a contradiction with $p\neq q$. Therefore $(X^{\Gamma},\sigma_{\varphi})$ does not have almost weak specification property. ∎

Note 3.2**.**

For a map $\varphi:\Gamma\to\Gamma$ and $\alpha,\beta\in\Gamma$ we say $\alpha\thicksim_{\varphi}\beta$ if there exists $i,j\geq 0$ with $\varphi^{i}(\alpha)=\varphi^{j}(\beta)$. Then $\thicksim_{\varphi}$ is an equivalence relation on $\Gamma$.

Lemma 3.3**.**

If $\varphi:\Gamma\to\Gamma$ does not have any periodic point, then the dynamical system $(X^{\Gamma},\sigma_{\varphi})$ has weak specification property.

Proof.

Suppose $\varphi:\Gamma\to\Gamma$ does not have any periodic. For $\alpha\in\mathcal{U}$ there exists a finite subset $F$ of $\Gamma$ with $\alpha_{F}\subseteq\alpha$. Also, there exist $\beta_{1},\ldots,\beta_{m}\in\Gamma$ and $N\geq 1$ such that $F\subseteq\bigcup\{\varphi^{-i}(\beta_{j}):j=1,\ldots,m,i=0,\ldots,N\}$ and for all distinct $s,t\in\{1,\ldots,m\}$, $\beta_{s}\not\thicksim_{\varphi}\beta_{t}$. Now consider $x_{1}=(x^{1}_{\theta})_{\theta\in\Gamma},\ldots,x_{k}=(x^{k}_{\theta})_{\theta\in\Gamma}\in X^{\Gamma}$ and $0\leq i_{1}\leq j_{1}<i_{2}\leq j_{2}<\cdots<i_{k}\leq j_{k}$ with $i_{s+1}-j_{s}\geq N+1$ for all $s\in\{1,\ldots,k-1\}$. Choose $p\in X$ and define $z=(z_{\theta})_{\theta\in\Gamma}\in X^{\Gamma}$ in the following way:

[TABLE]

First of all note that $z_{\theta}$s are well-defined, otherwise there exist $\gamma_{1},\gamma_{2}\in F$, $s_{1}<s_{2}$, and $t_{1}<t_{2}$ with $i_{s_{1}}\leq t_{1}\leq j_{s_{1}}<i_{s_{2}}\leq t_{2}\leq j_{s_{2}}$ with $\varphi^{t_{1}}(\gamma_{1})=\varphi^{t_{2}}(\gamma_{2})$, thus $\gamma_{1}\thicksim_{\varphi}\gamma_{2}$ and there exists $i\in\{1,\ldots,m\}$ and $r_{1},r_{2}\in\{0,\ldots,N\}$ with $\varphi^{r_{1}}(\gamma_{1})=\varphi^{r_{2}}(\gamma_{2})=\beta_{i}$, so by $i_{s_{2}}-j_{s_{1}}\geq N+1$, $t_{2}>N$ and $\varphi^{t_{1}}(\gamma_{1})=\varphi^{t_{2}}(\gamma_{2})=\varphi^{t_{2}-r_{2}}(\beta_{i})=\varphi^{t_{2}-r_{2}+r_{1}}(\gamma_{1})$. By $t_{2}-r_{2}+r_{1}\geq t_{2}-N\geq i_{s_{2}}-N>j_{s_{1}}\geq t_{1}$, we have $t_{2}-r_{2}+r_{1}>t_{1}$ and $\varphi^{t_{1}}(\gamma_{1})$ is a periodic point of $\varphi$, which is a contradiction, so $z_{\theta}$s are well-defined.

It is easy to see that for all $s\in\{1,\ldots,k\}$, $\gamma\in F$ and $i_{s}\leq t\leq j_{s}$ we have $x^{s}_{\varphi^{t}(\gamma)}=z_{\varphi^{t}(\gamma)}$, i.e. $(\sigma_{\varphi}^{t}(z),\sigma_{\varphi}^{t}(x^{s}))\in\alpha_{F}\subseteq\alpha$, which completes the proof. ∎

Theorem 3.4**.**

The following statements are equivalent:

• •

$(X^{\Gamma},\sigma_{\varphi})$ has weak specification property,

• •

$(X^{\Gamma},\sigma_{\varphi})$ has almost weak specification property,

• •

$\varphi:\Gamma\to\Gamma$ does not have any periodic point.

Proof.

Use Lemmas 3.1 and  3.3. ∎

Corollary 3.5**.**

The following statements are equivalent:

$(X^{\Gamma},\sigma_{\varphi})$ has specification property,

• 2.

$\varphi:\Gamma\to\Gamma$ is one–to–one without any periodic point.

Proof.

“(1) $\Rightarrow$ (2)” Suppose $(X^{\Gamma},\sigma_{\varphi})$ has specification property, then for all $x\in X^{\Gamma}$ and open neighbourhood $U$ of $x$ there exists $\alpha\in\mathcal{U}$ with $\alpha[x]\subseteq U$. Since $(X^{\Gamma},\sigma_{\varphi})$ has specification property, there exists a periodic point $z$ of $\sigma_{\varphi}$ with $(x,z)\in\alpha$, i.e. $z\in\alpha[x]\subseteq U$ and the collection of all periodic points of $\sigma_{\varphi}$ is dense in $X^{\Gamma}$. By [3, Theorem 2.6] $\varphi:\Gamma\to\Gamma$ is one–to–one and using Lemma 3.1 $\varphi:\Gamma\to\Gamma$ does not have any periodic point.

“(2) $\Rightarrow$ (1)” Suppose $\varphi:\Gamma\to\Gamma$ is one–to–one without any periodic point. By using Theorem 3.4, we get that $(X^{\Gamma},\sigma_{\varphi})$ has weak specification property. Consider $\alpha\in\mathcal{U}$, there exists $\beta\in\mathcal{U}$ with $\beta^{-1}\circ\beta\subseteq\alpha$, moreover since $(X^{\Gamma},\sigma_{\varphi})$ has weak specification property there exists $N\geq 1$ such that for all $y_{1},\ldots,y_{n}\in X^{\Gamma}$ ($n\geq 2$) and $0\leq l_{1}\leq k_{1}<l_{2}\leq k_{2}<\ldots<l_{n}\leq k_{n}$ with $l_{i+1}-k_{i}\geq N$ (for all $i=1,\ldots,n-1$) there exists $x\in X^{\Gamma}$ such that

[TABLE]

Since $\sigma_{\varphi}:X^{\Gamma}\to X^{\Gamma}$ is uniformly continuous, there exists $\mu\in\mathcal{U}$ with:

[TABLE]

Since $\varphi:\Gamma\to\Gamma$ is one–to–one, the collection of all periodic points of $\sigma_{\varphi}$ is dense in $X^{\Gamma}$ by [3, Theorem 2.6]. Hence there exists a periodic point $z$ of $\sigma_{\varphi}$ with $(x,z)\in\mu$, thus $(\sigma_{\varphi}^{i}(x),\sigma_{\varphi}^{i}(z))\in\beta$ for all $i\in\{0,1,\ldots,k_{n}\}$. Therefore $(\sigma_{\varphi}^{i}(z),\sigma_{\varphi}^{i}(y_{j}))\in\beta^{-1}\circ\beta\subseteq\alpha$ for all $j\in\{1,\ldots,n\}$ and $i\in\{l_{j},l_{j}+1,\ldots,k_{j}\}$, which completes the proof. ∎

Theorem 3.6**.**

In the dynamical system $(F,\mathfrak{h})$ the following statements are equivalent:

• a.

the dynamical system $(F,\mathfrak{h})$ has weak specification property,

• b.

the dynamical system $(F,\mathfrak{h})$ has almost weak specification property,

• c.

$\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}$ is singleton.

Proof.

It’s evident that (a) implies (b).

(b) $\Rightarrow$ (c): Suppose $(F,\mathfrak{h})$ has almost weak specification property. Note that $\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}\neq\varnothing$ since $F$ is a nonempty compact Hausdorff and $\mathfrak{h}:F\to F$ is continuous. If $\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}$ is not singleton, then there exist distinct points $z,y\in\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}$. We may suppose $z\neq b$. There exists $N_{\gamma_{\{z\}}}:{\mathbb{Z}}_{+}\to{\mathbb{Z}}_{+}$ such that ${\displaystyle\lim_{n\to\infty}\frac{N_{\gamma_{\{z\}}}(n)}{n}}=0$ and for all $y_{1},\ldots,y_{n}\in F$ and $0\leq l_{1}\leq k_{1}<l_{2}\leq k_{2}<\ldots<l_{n}\leq k_{n}$ with $l_{i+1}-k_{i}\geq N_{\gamma_{\{z\}}}(k_{i+1}-l_{i+1})$ (for all $i=1,\ldots,n-1$) there exists $x\in Z$ such that

[TABLE]

Let:

[TABLE]

Choose $w\in\mathfrak{h}^{-(l_{2}+1)}(z)$, hence there exists $x_{1},x_{2}\in F$ such that:

[TABLE]

and:

[TABLE]

By (3.1), $x_{1}=z=\mathfrak{h}^{l_{2}}(x_{1})$, thus $z=\mathfrak{h}^{l_{2}}(z)$. Considering (3.2) we have $x_{2}=z$ and therefore $(z,\mathfrak{h}^{l_{2}}(w))=(\mathfrak{h}^{l_{2}}(z),\mathfrak{h}^{l_{2}}(w))=(\mathfrak{h}^{l_{2}}(x_{2}),\mathfrak{h}^{l_{2}}(w))\in\gamma_{\{z\}}$, $z=\mathfrak{h}^{l_{2}}(w)$, which leads to $\mathfrak{h}(z)=\mathfrak{h}^{l_{2}+1}(w)=z$.

Now for $y\in\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}\setminus\{z\}$ and choose $v\in\mathfrak{h}^{-l_{2}}(y)$, there exists $x_{3}\in F$ such that:

[TABLE]

thus $x_{3}=z$ and $(z,y)=(\mathfrak{h}^{l_{2}}(z),y)=(\mathfrak{h}^{l_{2}}(x_{3}),\mathfrak{h}^{l_{2}}(v))\in\gamma_{\{z\}}$ which shows that $y=z$. Therefore $\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}=\{z\}$ which is in contradiction with $y\neq z$. Thus $\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}$ is singleton.

(c) $\Rightarrow$ (a): Suppose $\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}$ is singleton:

Case I: $\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}=\{b\}$. Let $\alpha\in{\mathcal{K}}$. There exists finite subset $A$ of $F\setminus\{b\}$ with $\gamma_{A}\subseteq\alpha$. For all $y\in A$ there exists $m_{y}\geq 1$ with $y\notin\mathfrak{h}^{m_{y}}(F)$, thus $\mathfrak{h}^{-m_{y}}(y)=\varnothing$. Let $m=\mathop{\Sigma}\limits_{y\in A}m_{y}$, then $\mathfrak{h}^{-m}(A)=\varnothing$. So for all $n\geq m$ , $\mathfrak{h}^{n}(F)\cap A=\varnothing$. Let $N_{\alpha}=m$, then for all $y_{1},\ldots,y_{n}\in F$ and $0\leq l_{1}\leq k_{1}<l_{2}\leq k_{2}<\ldots<l_{n}\leq k_{n}$ with $l_{i+1}-k_{i}\geq m$ (for all $i=1,\ldots,n-1$), $m\leq l_{2}$. Thus for all $i\geq l_{2}(\geq m)$ and $j\geq 2$, $\mathfrak{h}^{i}(y_{1}),\mathfrak{h}^{i}(y_{j})\notin A$, so $(\mathfrak{h}^{i}(y_{1}),\mathfrak{h}^{i}(y_{j}))\in\gamma_{A}\subseteq\alpha$. On the other hand for $0\leq i\leq k_{1}$, $(\mathfrak{h}^{i}(y_{1}),\mathfrak{h}^{i}(y_{1}))\in\gamma_{A}\subseteq\alpha$. Hence

[TABLE]

which completes the proof.

Case II: $\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}=\{c\}$ and $c\neq b$. In this case $\mathfrak{h}(b)\neq b$ and $\mathfrak{h}(F)$ is finite, so by $\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}=\{c\}$, there exists $m\geq 1$ such that:

[TABLE]

Now for all $y_{1},\ldots,y_{n}\in F$ and $0\leq l_{1}\leq k_{1}<l_{2}\leq k_{2}<\ldots<l_{n}\leq k_{n}$ with $l_{i+1}-k_{i}\geq m$ (for all $i=1,\ldots,n-1$), $l_{2}\geq m$ and $\mathfrak{h}^{i}(y_{j})=\mathfrak{h}^{i}(y_{1})$ for all $i\in\{l_{j},\ldots,k_{j}\}$ and $j\in\{1,\ldots,n\}$. Thus $(F,\mathfrak{h})$ has weak specification property. ∎

Theorem 3.7**.**

The dynamical system $(F,\mathfrak{h})$ has specification property, if and only if $F=\{b\}$.

Proof.

It’s clear that for $F=\{b\}$, $(F,\mathfrak{h})$ has specification property.

Conversely, suppose $(F,\mathfrak{h})$ has specification property, then it has weak specification property. By Theorem 3.6, $\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}$ is singleton. For $z\in F\setminus\{b\}$ there exists a periodic point $x\in F$ such that for $l_{1}=0$, $(x,z)=(\mathfrak{h}^{l_{1}}(x),\mathfrak{h}^{l_{1}}(z))\in\gamma_{\{z\}}$ so $z=x$ and $z$ is a periodic point of $\mathfrak{h}$. Thus

[TABLE]

hence $F\setminus\{b\}$ has at most one element. Suppose that $F=\{x,b\}$. We may consider the following cases, for $x\neq b$:

Case I: $\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}=\{x\}\neq\{b\}$. In this case $\mathfrak{h}(x)=\mathfrak{h}(b)=x$. So $Per(f)=\{x\}$, thus for $l_{1}=0$ , $(x,b)=(\mathfrak{h}^{l_{1}}(x),\mathfrak{h}^{l_{1}}(b))\in\gamma_{\{x\}}$ which leads to the contradiction $x=b$, hence this case does not occur.

Case II: $\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}=\{b\}\neq\{x\}$. In this case we have the contradiction $x\in F\setminus\{b\}\subseteq Per(\mathfrak{h})\subseteq\bigcap\{\mathfrak{h}^{n}(F):n\geq 1\}=\{b\}$ hence this case does not occur too.

Considering the above cases, we get $x=b$ and $F=\{b\}$. ∎

4. Stroboscopical property

In this section we show that $(X^{\Gamma},\sigma_{\varphi})$ has uniform stroboscopical property, (stroboscopical property) if and only if $\varphi:\Gamma\to\Gamma$ is one–to–one. Also $(F,\mathfrak{h})$ has uniform stroboscopical property (stroboscopical property) if and only if all points of $F$ are periodic points of $\mathfrak{h}$.

Lemma 4.1**.**

If $(X^{\Gamma},\sigma_{\varphi})$ has stroboscopical property, then $\varphi:\Gamma\to\Gamma$ is one–to–one.

Proof.

Suppose $\varphi:\Gamma\to\Gamma$ is not one–to–one, then there are distinct points $\beta,\lambda\in\Gamma$ such that $\varphi(\beta)=\varphi(\lambda)$. Choose distinct $p,q\in X$. Consider $z=(z_{\alpha})_{\alpha\in\Gamma}\in X^{\Gamma}$ with:

[TABLE]

then for all $(x_{\alpha})_{\alpha\in\Gamma}\in X^{\Gamma}$ and $n\geq 1$ , $x_{\varphi^{n}(\beta)}=x_{\varphi^{n}(\lambda)}$. Thus there is not any sequence $\{n_{i}\}_{i\geq 0}$ in $\mathbb{N}$ with ${\displaystyle\lim_{i\to\infty}\sigma_{\varphi}^{n_{i}}((x_{\alpha})_{\alpha\in\Gamma})}=z$. Therefore $(X^{\Gamma},\sigma_{\varphi})$ does not have stroboscopical property. ∎

Lemma 4.2**.**

If $A=\{m^{1}_{i}\}_{i\geq 0}$ is a strictly increasing sequence in $\mathbb{N}$, then:

there exists a map $f:{\mathbb{N}}\to{\mathbb{N}}$, and $A$ has a subsequence $\{n_{i}\}_{i\geq 0}$ such that for all $m\geq 1$ and $i\geq m$, $n_{i}\equiv f(m)\>({\rm mod}\>m)$ and $f(m)\leq m$;

• 2.

$A$ has a subsequence $\{m_{i}\}_{i\geq 0}$ such that $m_{i+1}-m_{i}>2i$ for all $i\geq 0$.

Proof.

1. Suppose $A=\{m^{1}_{i}\}_{i\geq 0}=:A_{1}$ and $k_{1}=1$, then there exists $k_{2}\in\{1,2\}$ and $A$ has a subsequence $A_{2}=\{m^{2}_{i}\}_{i\geq 0}$ such that for all $i\geq 0$, $m^{2}_{i}\equiv k_{2}({\rm mod}\>2)$. For $s\geq 2$, suppose that we have chosen subsequences $A_{1}=\{m^{1}_{i}\}_{i\geq 0},\ldots,A_{s}=\{m^{s}_{i}\}_{i\geq 0}$ and natural numbers $k_{1},\ldots,k_{s}$. Then there exists $k_{s+1}\in\{1,\ldots,s+1\}$, and $A_{s}$ has a subsequence $A_{s+1}=\{m^{s+1}_{i}\}_{i\geq 0}$ such that for all $i\geq 0$, $m^{s+1}_{i}\equiv k_{s+1}({\rm mod}\>s+1)$. Let $f:\mathbb{N}\to\mathbb{N}$ with $f(i)=k_{i}$ ($i\geq 1$) and subsequence $\{n_{i}\}_{i\geq 0}$ of $A$ with $n_{0}=m_{0}^{1}$ and $n_{i}=m_{i}^{i}$ for $i\geq 1$, then the map $f:\mathbb{N}\to\mathbb{N}$ and subsequence $\{n_{i}\}_{i\geq 0}$ of $A$ satisfies (1).

2. We construct $\{m_{i}\}_{i\geq 0}$ inductively. Let $m_{0}:=m^{1}_{0}$, for $i\geq 0$ suppose we have chosen $m_{i}=m^{1}_{n_{i}}$ let $m_{i+1}:=m^{1}_{n_{i}+2i+1}$. The the subsequence $\{m_{i}\}_{i\geq 0}$ of $A$ satisfies (2). ∎

Lemma 4.3**.**

If all points of $\Gamma$ are periodic points of $\varphi:\Gamma\to\Gamma$, then $(X^{\Gamma},\sigma_{\varphi})$ has uniform stroboscopical property.

Proof.

Let $A$ be a strictly increasing sequence in $\mathbb{N}$. By Lemma 4.2 there exists a map $f:{\mathbb{N}}\to{\mathbb{N}}$ and $A$ has a subsequence $\{n_{i}\}_{i\geq 0}$ such that for all $m\geq 1$ and $i\geq m$, $n_{i}\equiv f(m)\>({\rm mod}\>m)$ and $f(m)\leq m$. For all $\alpha\in\Gamma$ let $k_{\alpha}=\min\{n\geq 1:\varphi^{n}(\alpha)=\alpha\}$. For $z=(z_{\alpha})_{\alpha\in\Gamma}\in X^{\Gamma}$ define $z^{*}=(z^{*}_{\alpha})_{\alpha\in\Gamma}\in X^{\Gamma}$ with:

[TABLE]

Then for all $\alpha\in\Gamma$ and $i\geq f(k_{\alpha})$, $k_{\alpha}-f(k_{\alpha})+n_{i}\equiv 0\>({\rm mod}\>k_{\alpha})$, so

[TABLE]

For $D\in\mathcal{U}$ there exists finite subset $H=\{\beta_{1},\ldots,\beta_{s}\}$ of $\Gamma$ with $\alpha_{H}\subseteq D$. For all $i\geq\max\{f(k_{\beta_{1}}),\ldots,f(k_{\beta_{m}})\}$ and for all $z=(z_{\alpha})_{\alpha\in\Gamma}\in X^{\Gamma}$ we have $(z,\sigma_{\varphi}^{n_{i}}(z^{*}))\in\alpha_{H}\subseteq D$. Thus for $\mathop{\varrho:X^{\Gamma}\to X^{\Gamma}}\limits_{\>\>\>\>\>\>\>\>\>\>z\mapsto z^{*}}$ the sequence $\{\sigma_{\varphi}^{n_{i}}\circ\varrho\}_{i\geq 1}$ converges uniformly to $id_{X^{\Gamma}}$. ∎

Remark 4.4**.**

If $\varphi:\Gamma\to\Gamma$ is one–to–one, then each $D\in\frac{\Gamma}{\thicksim_{\varphi}}(=\{\frac{\alpha}{\thicksim_{\varphi}}:\alpha\in\Gamma\})$ satisfies exactly one of the following conditions:

• •

$D$ has finite elements like $m$ and for all $\alpha\in D$, $D=\{\varphi^{i}(\alpha):1\leq i\leq m\}$,

• •

$D$ is infinite and there exists $\alpha\in\Gamma$ with $D=\{\varphi^{n}(\alpha):n\geq 0\}$,

• •

$D$ is infinite and we may suppose that $D=\{\beta_{n}:n\in\mathbb{Z}\}$ with $\varphi(\beta_{n})=\beta_{n+1}$ for all $n\in\mathbb{Z}$.

Lemma 4.5**.**

If $\varphi:\Gamma\to\Gamma$ is one–to–one and does not have any periodic point, then $(X^{\Gamma},\sigma_{\varphi})$ has uniform stroboscopical property.

Proof.

Suppose $\varphi:\Gamma\to\Gamma$ is a one–to–one map without any periodic point. Using axiom of choice there exists a subset of $\Gamma$ like $\Lambda$ such that $\bigcup\{\frac{\alpha}{\thicksim_{\varphi}}:\alpha\in\Lambda\}=\Gamma$ and for all distinct $\alpha,\beta\in\Lambda$, $\frac{\alpha}{\thicksim_{\varphi}}\cap\frac{\beta}{\thicksim_{\varphi}}=\varnothing$, also we may suppose each $\alpha\in\Lambda$ provides exactly one of the following conditions (use Remark 4.4 too):

• •

$\frac{\alpha}{\thicksim_{\varphi}}=\{\varphi^{n}(\alpha):n\geq 0\}$,

• •

$\forall n\geq 0\exists\beta\in\Gamma\>(\varphi^{n}(\beta))=\alpha)$.

Let

[TABLE]

and

[TABLE]

Moreover for all $\alpha\in\Gamma$, choose $\lambda_{\alpha}\in\Lambda\cap\frac{\alpha}{\thicksim_{\varphi}}$. Suppose $A$ is a strictly increasing sequence in $\mathbb{N}$. By Lemma 4.2, $A$ has a subsequence like $\{n_{i}\}_{i\geq 0}$ such that $n_{i+1}-n_{i}>2i$ for all $i\geq 0$. Consider $p\in X$ and for $z=(z_{\alpha})_{\alpha\in\Gamma}\in X^{\Gamma}$ define $z^{*}=(z^{*}_{\alpha})_{\alpha\in\Gamma}\in X^{\Gamma}$ with:

[TABLE]

Note that using $0<n_{1}<n_{2}<n_{3}<\cdots$, for $k\geq n_{1}$ there exists unique $i\geq 1$ and $j\in\{0,\ldots,n_{i+1}-n_{i}-1\}$ with $k=n_{i}+j$ thus for $\mu\in\Lambda_{1}$ and $\alpha\in\{\varphi^{n}(\mu):n\geq n_{1}\}$, $z^{*}_{\alpha}$ is well-defined. Moreover for $\mu\in\Lambda_{2}$ if there exist $i_{1},i_{2}\geq 1$, $j_{1}\in\{-(i_{1}-1),\ldots,i_{1}-1\}$ and $j_{2}\in\{-(i_{2}-1),\ldots,i_{2}-1\}$ with $\varphi^{n_{i_{1}}+j_{1}}(\mu)=\varphi^{n_{i_{2}}+j_{2}}(\mu)$ we have $n_{i_{1}}+j_{1}=n_{i_{2}}+j_{2}$. If $i_{1}\neq i_{2}$ we may suppose $i_{1}<i_{2}$ thus $2(i_{2}-1)<n_{i_{2}}-n_{i_{2}-1}\leq n_{i_{2}}-n_{i_{1}}=j_{1}-j_{2}\leq i_{1}-1+i_{2}-1$ which leads to contradiction $i_{2}<i_{1}$, thus $i_{2}=i_{1}$ and $j_{2}=j_{1}$. Therefore $z^{*}_{\alpha}$s are well-defined. We have:

[TABLE]

since for $\mu\in\Lambda_{1}$, $j\geq 0$ and $i\geq j+2$ the relation $\lambda_{\varphi^{n_{i}+j}(\mu)}=\mu$ and inequalities $0\leq j<2(j+2)\leq 2i<n_{i+1}-n_{i}$ lead to $z^{*}_{\varphi^{n_{i}+j}(\mu)}=z_{\varphi^{j}(\mu)}$.

Also

[TABLE]

For $D\in\mathcal{U}$ there exists finite subset $H=\{\beta_{1},\ldots,\beta_{s}\}$ of $\Gamma$ with $\alpha_{H}\subseteq D$. There exists $N\geq 1$, finite subset $H_{1}$ of $\Lambda_{1}$ and finite subset $H_{2}$ of $\Lambda_{2}$ with $H\subseteq\bigcup\{\varphi^{t}(H_{1}\cup H_{2}):|t|\leq N\}(=(\bigcup\{\varphi^{t}(H_{1}):0\leq t\leq N\})\cup(\bigcup\{\varphi^{t}(H_{2}):-t\leq t\leq N\}))$. Using (i) and (ii) for all $z=(z_{\alpha})_{\alpha\in\Gamma}\in X^{\Gamma}$ we have:

[TABLE]

thus $(z,\sigma_{\varphi}^{n_{i}}(z^{*}))\in\alpha_{H}\subseteq D$ for all $z=(z_{\alpha})_{\alpha\in\Gamma}\in X^{\Gamma}$ and $i\geq N+2$. Hence for $\mathop{\varrho:X^{\Gamma}\to X^{\Gamma}}\limits_{\>\>\>\>\>\>\>\>\>\>z\mapsto z^{*}}$ the sequence $\{\sigma_{\varphi}^{n_{i}}\circ\varrho\}_{i\geq 1}$ converges uniformly to $id_{X^{\Gamma}}$. ∎

Theorem 4.6**.**

The dynamical system $(X^{\Gamma},\sigma_{\varphi})$ has uniform stroboscopical property if and only if $\varphi:\Gamma\to\Gamma$ is one-to-one.

Proof.

If $(X^{\Gamma},\sigma_{\varphi})$ has uniform stroboscopical property, then it has stroboscopical property and by Lemma 4.1 the map $\varphi:\Gamma\to\Gamma$ is one–to–one.

Now suppose $\varphi:\Gamma\to\Gamma$ is one–to–one. If either all points of $\Gamma$ are periodic points of $\varphi$ or $\varphi$ does not have any periodic point, then we are done by Lemmas 4.3 and 4.5. Otherwise $Per(\varphi)$ (the collection of all periodic points of $\varphi$) and $\Gamma\setminus Per(\varphi)$ are two nonempty subsets of $\Gamma$. By Lemmas 4.3 and 4.5 both dynamical systems $(X^{Per(\varphi)},\sigma_{\varphi\restriction_{Per(\varphi)}})$ and $(X^{\Gamma\setminus Per(\varphi)},\sigma_{\varphi\restriction_{\Gamma\setminus Per(\varphi)}})$ have uniform stroboscopical property. In order to complete the proof use the fact that if both dynamical systems $(Z,f)$ and $(Y,h)$ have uniform stroboscopical property, then $(Z\times Y,f\times h)$ (where $(f\times h)(u,v)=(f(u),h(v))$ for $(u,v)\in Z\times Y$) has uniform stroboscopical property. ∎

Note 4.7**.**

Using Lemmas 4.1 and Theorem 4.6, for finite discrete topological space $X$ with at least two elements, nonempty set $\Gamma$, and arbitrary map $\varphi:\Gamma\to\Gamma$ the following statements are equivalent:

• •

$(X^{\Gamma},\sigma_{\varphi})$ has uniform stroboscopical property,

• •

$(X^{\Gamma},\sigma_{\varphi})$ has stroboscopical property,

• •

$\varphi:\Gamma\to\Gamma$ is one–to–one.

Note 4.8**.**

For countable $\Gamma$, by [8, Proposition 5] and [3], $(X^{\Gamma},\sigma_{\varphi})$ has the strong stroboscopical property if and only if $\varphi:\Gamma\to\Gamma$ is one–to–one without periodic points.

Theorem 4.9**.**

The following statements are equivalent:

• a.

$(F,\mathfrak{h})$ has uniform stroboscopical property,

• b.

$(F,\mathfrak{h})$ has stroboscopical property,

• c.

$Per(\mathfrak{h})=F$.

Proof.

(b) $\Rightarrow$ (c). Suppose $(F,\mathfrak{h})$ has stroboscopical property and $z$ is an isolated point of $F$, then there exists $x\in F$ and $\{n\}_{n\geq 0}$ has a subsequence like $\{n_{k}\}_{k\geq 0}$ such that ${\displaystyle\lim_{k\to\infty}\mathfrak{h}^{n_{k}}(x)}=z$, hence there exists $N\geq 1$ such that:

[TABLE]

in particular $\mathfrak{h}^{n_{N}}(x)=z=\mathfrak{h}^{n_{N+1}}(x)$, thus $\mathfrak{h}^{n_{N+1}-n_{N}}(z)=z$ and $z\in Per(\mathfrak{h})$.

On the other hand, if $z$ is limit point of $F$, then $F$ is infinite and $z=b$. Since $F\setminus\{b\}\subseteq Per(\mathfrak{h})$, $\mathfrak{h}\restriction_{F\setminus\{b\}}:F\setminus\{b\}\to F$ is one-to-one which leads to $\mathfrak{h}(b)=b$ by Remark 2.5.

(c) $\Rightarrow$ (a). Suppose $Per(\mathfrak{h})=F$ and $A$ is an arbitrary increasing sequence in $\mathbb{N}$. The subsequences $A_{1},A_{2},\ldots$ of of $A$ defined inductively as follows:

• •

let $A_{1}:=A=\{n^{1}_{k}\}_{k\geq 0}$ and $r_{1}=0$,

• •

suppose for $t\geq 1$, $A_{t}=\{n^{t}_{k}\}_{k\geq 0}$ is a subsequence of $A$ and $r_{t}\in\{0,\ldots,t-1\}$ such that for all $k\geq 0$, $n^{t}_{k}\mathop{\equiv}\limits^{\mod t}r_{t}$; then there exists $r_{t+1}\in\{0,\ldots,t\}$ and subsequence $A_{t+1}=\{n^{t+1}_{k}\}_{k\geq 0}$ of $A_{t}$ such $n^{t+1}_{k}\mathop{\equiv}\limits^{\mod t+1}r_{t+1}$.

Then $B:=\{n_{k}^{k}\}_{k\geq 1}$ is a subsequence of $A$ with

[TABLE]

For $z\in F$ let $m_{z}:=\min\{m\geq 1:\mathfrak{h}^{m}(z)=z\}$. Consider $\varrho:F\to F$ with $\varrho(z)=\mathfrak{h}^{m_{z}-r_{m_{z}}}(z)$ ($z\in F$). Then for all $z\in Z$ and $k\geq m_{z}$, $\mathfrak{h}^{n_{k}^{k}}(\varrho(z))=z$. For $\alpha\in\mathcal{K}$ there exists a finite subset $A$ of $F\setminus\{b\}$ with $\gamma_{A}\subseteq\alpha$. Let $N=\max(\{m_{z}:z\in A\}\cup\{m_{b}\})$, for all $k\geq N$ and $z\in F$.

• •

$m_{z}\leq N$: in this case $\mathfrak{h}^{n_{k}^{k}}(\varrho(z))=z$, and $(\mathfrak{h}^{n_{k}^{k}}(\varrho(z)),z)=(z,z)\in\alpha$.

• •

$m_{z}>N$: note that for all $i\geq 0$, $m_{z}=m_{\mathfrak{h}^{i}(z)}$, thus for all $n\geq N$ we have $m_{\mathfrak{h}^{n_{k}^{k}}(\varrho(z))}=m_{\mathfrak{h}^{n_{k}^{k}+m_{z}-r_{m_{z}}}(z)}=m_{z}>N$ therefore $z,\mathfrak{h}^{n_{k}^{k}}(\varrho(z))\notin A$ and $(\mathfrak{h}^{n_{k}^{k}}(\varrho(z)),z)\in\gamma_{A}\subseteq\alpha$.

Regarding the above two cases we have $(\mathfrak{h}^{n_{k}^{k}}(\varrho(z)),z)=(\mathfrak{h}^{n_{k}^{k}}(\varrho(z)),id_{F}(z))\in\alpha$ for all $k\geq N$. Thus $\{\mathfrak{h}^{n_{k}^{k}}\circ\varrho\}_{k\geq 1}$ converges uniformly to $id_{F}:F\to F$ and $(F,\mathfrak{h})$ has uniform stroboscopical property. ∎

Theorem 4.10**.**

The dynamical system $(F,\mathfrak{h})$ has strongly stroboscopical property if and only if $F=\{b\}$.

Proof.

Suppose $(F,\mathfrak{h})$ has strongly stroboscopical property, if $F\neq\{b\}$, then $F$ has at least two (distinct) isolated points like $x,y$. There exists $m\geq 1$ with $\mathfrak{h}^{m}(x)=x$, by Theorem 4.9, since $(F,\mathfrak{h})$ has strongly stroboscopical property and $x$ is an isolated point of $F$, $y\in\omega_{\mathfrak{h}}(\{mn\}_{n\geq 0},x)=\{x\}$ which is in contradiction with $y\neq x$, thus $F=\{b\}$. ∎

5. Two diagrams

Let’s recall that inthe paper, $X$ is a finite discrete space with at least two elements, $\Gamma$ is nonempty, $\varphi:\Gamma\to\Gamma$ is an arbitrary map, $F$ is a Fort space with particular point $b$, and $\mathfrak{h}:F\to F$ is continuous. So according to the previous sections, we get the following table:

[TABLE]

Table A

Where the dynamical system $(Z,f)$ has $\rho$ property if and only if it has the phrase occurs in the corresponding case.

Using Table A, we have the following diagrams:

$\>\>\>(X^{\Gamma},\sigma_{\varphi})$has (almost) weak specification property$(X^{\Gamma},\sigma_{\varphi})$has (uniform) stroboscopical property(C3)(C1)(C2)$(X^{\Gamma},\sigma_{\varphi})$has specification property(strongly stroboscopical property)

Where (Ci) denotes the counterexample $(X^{\mathbb{Z}},\sigma_{\varphi_{i}})$ for:

• •

$\varphi_{1}:{\mathbb{Z}}\to{\mathbb{Z}}$ with $\varphi_{1}(n)=n^{2}+1$ ($n\in{\mathbb{Z}}$),

• •

$\varphi_{2}:{\mathbb{Z}}\to{\mathbb{Z}}$ with $\varphi_{2}(n)=-n$ ($n\in{\mathbb{Z}}$),

• •

$\varphi_{3}:{\mathbb{Z}}\to{\mathbb{Z}}$ with $\varphi_{3}(n)=n+1$ ($n\in{\mathbb{Z}}$).

And:

$\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>(F,\mathfrak{h})$has (almost) weak specification property$(F,\mathfrak{h})$has (uniform) stroboscopical property$(F,\mathfrak{h})$has specification property(strongly stroboscopical property)$F=\{b\}$(D1)(D2)(D3)

Where (Di) denotes the counterexample $(\{\frac{\pm 1}{n}:n\in{\mathbb{Z}}\setminus\{0\}\}\cup\{0\},\mathfrak{h}_{i})$ for (consider $\{\frac{\pm 1}{n}:n\in{\mathbb{Z}}\setminus\{0\}\}\cup\{0\}$ with induced topology from Euclidean line $\mathbb{R}$):

[TABLE]