Shadowable Points of Free Semigroup Actions

Abstract

The shadowable points of dynamical systems have been well-studied by Morales in his recent paper (2016). This paper aims to generalize the main results obtained by Morales to free semigroup actions. To this end, we introduce the notion of shadowable points of a free semigroup action. Let G be a free semigroup generated by finite continuous self-maps acting on compact metric space. We will prove the following results for G on compact metric spaces. The set of shadowable points of G is a Borel set. G has the pseudo-orbit tracing property (POTP) if and only if every point is shadowable point of G. The chain recurrent and non-wandering sets of G coincide when every chain recurrent point is shadowable point of G. The space X is totally disconnected at every shadowable point of G under certain condition.

1 Introduction

The shadowing theory is an important component of the qualitative theory of dynamical systems, producing many interesting and profound results. If any pseudo orbit with sufficient accuracy approaches a certain precise trajectory, then the dynamical system has the shadowing property or pseudo-orbit tracing property(POTP). So far, many researchers have studied homeomorphisms with shadowing property, for examples [1, 17]. Recently, Morales [15] introduced the concept of shadowable points by decomposing shadowing property into points and studied the properties of the shadowable points set. Subsequently Kawaguchi [11] extended the notion for continuous maps and gave a quantitative version of a Morales’result in [15]. In addition, Kawaguchi [11] also obtained more and better results. Afterwards, Aponte and Villavicencio [2] transferred the notion of shadowable points for homeomorphisms to the context of continuous time flows. Jung, Lee and Morales [10] introduced the concept of the persistent shadowing property, leading to the derivation of several compelling theorems. Besides the classical theory of dynamical systems (which studies actions of ${\mathbb {Z}}$ and ${\mathbb {R}}$), shadowing for actions of all kinds of groups and semigroups were studied. For instance, Pilyugin and Tikhomirov [18] studied the shadowing property for action of the higher rank groups ${\mathbb {Z}}^p\times {\mathbb {R}}^q$. Afterwards, Osipov and Tikhomirov [16] introduced the notion of shadowing property for actions of finitely generated groups and study its basic properties. Very recently Kim and Lee [12] introduced the notion of shadowable points for finitely generated group actions on compact metric space and extended the theorems of [15] to finitely generated group actions. Jung, Lee and etc. [9] defined robust expansivity and the shadowing property for random group actions and proved the rigidity of actions enjoying these two properties. Barzanouni [5] introduced the notion of persistent shadowing property for finitely generated group actions and studied it via measure theory. Bahabadi [3] introduced the definitions of shadowing and average shadowing properties for iterated function systems, and also proved that an iterated function system has the shadowing property if and only if the skew-product transformation corresponding to the iterated function system has the shadowing property. Hui and Ma [8], Zhu and Ma [21] introduced the definitions of $\delta$-pseudo-orbit for free semigroup actions and the pseudo-orbit tracing property for free semigroup actions, where they considered the $\delta$-pseudo-orbit under free semigroup actions similar to a random route, which is different from the case of finitely generated group in [12]. What’s more, Huang, Wang and Qiu [7] studied the relation between the two different definitions of shadowing property.

   In this paper, we study the shadowing property for free semigroup actions based on [8] and [21] and introduce the notion of shadowable points of free semigroup actions. The main work is to extend Theorem 1.1 and Theorem 1.2 of [15] to free semigroup actions, and improve the proof process of Lemma 2.1, Lemma 2.3 and Theorem 1.2 in [15] using Lemma 2.2 in [11] and new conditions. By studying relationship between free semigroup actions and skew-product transformation corresponding to the free semigroup action, we find out relation between the shadowable points of free semigroup actions and the totally disconnected points. Let X be a compact metric space, $f_0,\cdots ,f_{m-1}$ continuous self-maps on X. Denote G the free semigroup generated by $G_1=\{f_0,\cdots ,f_{m-1}\}$ on X and $F:\Sigma _m^+\times X\rightarrow \Sigma _m^+\times X$ a skew-product transformation, where $\Sigma _m^+$ denotes the one-side symbol space. The main results of this paper are:

Theorem 1.1

Let (X, d) be a compact metric space. If $f_0,f_1,\cdots ,f_{m-1}$ are continuous self-maps on X and G is the free semigroup generated by $\{f_0,f_1,\cdots ,f_{m-1}\}$, then

• (1) G has the POTP if and only if $Sh(G)=X$;

• (2) Sh(G) is a Borel set;

• (3) if $CR(G) \subset Sh(G)$, then $CR(G)=\Omega (G)$,

where POTP is the abbreviation of pseudo-orbit tracing property, Sh(G) denotes the set of shadowable points of G, CR(G) denotes the set of chain recurrent points of G, and $\Omega (G)$ denotes the set of non-wandering points of G.

Theorem 1.2

Let $f_0,\cdots ,f_{m-1}$ be continuous self-maps of a compact metric space X, G the free semigroup generated by $\{f_0,\cdots ,f_{m-1}\}$ and F the skew-product transformation corresponding to $\{f_0,\cdots ,f_{m-1}\}$. If there exists $\delta >0$ satisfying

$B(\gamma (Sh(F)),\delta )\subset R(F)$, then $Sh(G) \subset X^{deg}$,

where $\gamma (Sh(F))$ denotes the union of all connected components of X whose intersection with Sh(F) is non-empty, $X^{deg}$ denotes the set of all totally disconnected points of X, $B(\gamma (Sh(F)),\delta )$ denotes the $\delta$ neighborhood of set $\gamma (Sh(F))$, and R(F) denotes the set of recurrent points of F.

This paper is organized as follows. In Sect. 2, we give some preliminaries. In Sect. 3, we prove our main results. These results extend and improve the work of Morales [15].

2 Preliminaries

We first introduce some basic notions. Let (X, d) be a compact metric space and $f:X \rightarrow X$ be a continuous map. A sequence $\{x_i\}_{i=0}^{\infty }$ is called a $\delta$-pseudo-orbit of f if for each $i\ge 0$,

$$\begin{aligned} d\left( f(x_i),x_{i+1}\right) \le \delta . \end{aligned}$$

The continuous map f is said to have the pseudo-orbit tracing property(POTP) if for each $\varepsilon >0$, there exists $\delta >0$ such that every $\delta$-pseudo-orbit $\{x_i\}_{i=0}^{\infty }$ is $\varepsilon$-shadowed by the orbit of some point $y\in X$, i.e. for all $i\ge 0$,

$$\begin{aligned} d\left( f^i(y),x_i\right) \le \varepsilon . \end{aligned}$$

We say that a point $x\in X$ is shadowable point of f if for every $\varepsilon >0$, there is $\delta (x,\varepsilon )>0$ such that every $\delta$-pseudo-orbit $\{x_i\}_{i=0}^{\infty }$ for f with $x_0=x$ can be $\varepsilon$-shadowed. We denote by Sh(f) the set of shadowable points of f. In [11], the author defined quantitative shadowable points. For $b>0$, a point $x\in X$ is called b-shadowable point of f if there exists $\delta >0$ for which every $\delta$-pseudo-orbit $\{x_i\}_{i=0}^{\infty }$ for f with $x_0=x$ is b-shadowed by some point of X. Denote by $Sh_b^+(f)$ the set of b-shadowable points of f. Then, for $c\ge 0$, defined

$$\begin{aligned} Sh_{c+}^+(f)=\bigcap \limits _{b>c}Sh_b^+(f). \end{aligned}$$

Denote by $F_m^{+}$ the set of all finite words of symbols $0,1,\cdots ,m-1$. For any $w\in F_m^{+}$, |w| stands for the length of w, that is, the number of symbols in w. We write $w\le w'$ if there exists a word $w''\in F_m^{+}$ such that $w'=w''w$.

Denote by $\Sigma _m^+$ the set of all one-side infinite sequences of symbols $0,1,\cdots , m-1$, i.e.

$$\begin{aligned} \Sigma _m^+=\left\{ \omega =(w_0,w_1,\cdots )~|w_i\in \{0,1,\cdots ,m-1\} ~for~all~i\ge 0\right\} . \end{aligned}$$

Let $\omega \in \Sigma _m^+$, $w \in F_m^+$, $a,b\ge 0$, and $a\ge b$. We write $\omega |_{[a,b]}=w$ if $w=w_aw_{a+1}\cdots w_{b-1}w_b$. For $\omega =(w_0,w_1,\cdots )\in \Sigma _m^+$, denote $(\omega )_i=w_i$, $i\ge 0$.

Let $w \in F_m^+, w=w_1w_2\cdots w_k$, where $w_i\in \{0,1,\cdots ,m-1\}$ for all $i\in \{1,\cdots ,k\}$, and denote $\bar{w}=w_kw_{k-1} \cdots w_1$. Let $f_w=f_{w_1}\circ f_{w_2}\circ \cdots \circ f_{w_k}$. Obviously, $f_{ww'}=f_w\circ f_{w'}$, and $f_{\bar{w}}=f_{w_k}\circ f_{w_{k-1}}\circ \cdots \circ f_{w_1}$.

In [3], the author introduced the shadowing property of the semigroup action generated by two continuous maps on compact metric space X. Here we can generalize it to general case. Let (X, d) be a compact metric space and $f_0,f_1,\cdots ,f_{m-1}$ be continuous self maps on X. Denote G the free semigroup acting on the space X generated by $G_1=\{f_0,\cdots ,f_{m-1}\}$. For $\omega =(w_0,w_1,\cdots )\in \Sigma _m^+$, an orbit of $x\in X$ under G for $\omega$ is a sequence $\{f_{\omega }^n(x)\}_{n=0}^{\infty }$, where

$$\begin{aligned} \left\{ \begin{aligned} f_{\omega }^n(x)&=f_{w_{n-1}}\circ f_{w_{n-2}}\circ \cdots \circ f_{w_1}\circ f_{w_0}(x).&n\ge 1\\ f_{\omega }^0(x)&=x.&n=0\\ \end{aligned} \right. \end{aligned}$$

The following definition of pseudo-orbit and pseudo-orbit tracing property for G are refer to [3] and [8].

Definition 2.1

( [8]) Given $\delta >0$, a $\delta$-pseudo-orbit($\delta$-chain) $\{x_i\}_{i=0}^{\infty }$ for G is defined as

$$\begin{aligned} \min \{d(f_0(x_i),x_{i+1}),\cdots ,d(f_{m-1}(x_i),x_{i+1})\}<\delta ~for~every~i\ge 0. \end{aligned}$$

(1)

Also, we can express the above expression as follows: there exists $\omega =(w_0,w_1,\cdots )\in \Sigma _m^+$ such that

$$\begin{aligned} d(f_{w_i}(x_i),x_{i+1})<\delta ~for~every~i\ge 0. \end{aligned}$$

(2)

We say that a $\delta$-pseudo-orbit $\{x_i\}_{i=0}^{\infty }$ for G is $\varepsilon$-shadowed by a point $z\in X$ if for some $\omega$ satisfying (1) or (2), it holds that

$$\begin{aligned} d(f_{\omega }^n(z),x_n)<\varepsilon ~for~all~n\ge 0. \end{aligned}$$

(3)

Definition 2.2

( [8]) The free semigroup action G has the pseudo-orbit tracing property(POTP) if for any $\varepsilon >0$, there is $\delta (\varepsilon )>0$ such that every $\delta$-pseudo-orbit for G can be $\varepsilon$-shadowed by some point of X.

Remark 2.3

The strict inequality in (2) and (3) can be relaxed to an inequality without affecting the proof of the following lemmas and theorems except for Lemma 3.5. So the following default definitions are strict inequalities, except for Lemma 3.5.

Remark 2.4

If G has the POTP, then maps $f_0,f_1,\cdots ,f_{m-1}$ all have POTP by Theorem 1.4 of [3], but the converse is not true. Please refer to Example 1.5 of [3] for specific example.

Example 2.5

( [3] Example 1.2) Define two continuous maps $f_0,f_1$ on $\Sigma _2^+$ as follows:

$$\begin{aligned} f_0(s_0s_1s_2\cdots )=0s_0s_1s_2\cdots , f_1(s_0s_1s_2\cdots )=1s_0s_1s_2\cdots . \end{aligned}$$

Denote G the free semigroup generated by $G_1=\{f_0,f_1\}$, then G has the POTP.

Next, we introduce two new definitions based on [15] in this paper.

Definition 2.6

A point $x\in X$ is called shadowable point of G if for every $\varepsilon >0$, there is $\delta (x,\varepsilon )>0$ such that every $\delta$-pseudo-orbit $\{x_i\}_{i=0}^{\infty }$ for G with $x_0=x$ can be $\varepsilon$-shadowed. We denote by Sh(G) the set of shadowable points of G.

Remark 2.7

Clearly, if G has the POTP, then $Sh(G)=X$(i.e.every point is shadowable point of G). The converse is true on compact metric spaces by Theorem 1.1.

Let X be a compact metric space. We say that a sequence $\{x_i\}_{i= 0}^{\infty }$ is through subset $K\subset X$ if $x_0\in K$.

Definition 2.8

The free semigroup action G has the POTP through K if for every $\varepsilon >0$ there is $\delta >0$ such that every $\delta$-pseudo-orbit for G through K can be $\varepsilon$-shadowed.

For G, we assign the following skew-product transformation $F: \Sigma _m^+ \times X \rightarrow \Sigma _m^+ \times X$ defined as

$$\begin{aligned} F(\omega , x) = \left( \sigma \omega , f_{\omega _0}(x)\right) , \end{aligned}$$

where $\omega = (w_0, w_1, \cdots )\in \Sigma _m^+$ and $\sigma$ is the shift map. Here $f_{w_0}$ stands for $f_0$ if $w_0=0$, and for $f_1$ if $w_0=1$, and so on. Let $\omega = (w_0, w_1, \cdots )\in \Sigma _m^+$, then

$$\begin{aligned} \begin{aligned} F^n(\omega ,x)&=\left( \sigma ^n\omega ,f_{w_{n-1}}\circ f_{w_{n-2}}\circ \cdots \circ f_{w_0}(x)\right) \\&=(\sigma ^n\omega ,f_{\omega }^n(x)). \end{aligned} \end{aligned}$$

The metric on $\Sigma _m^+$ is defined as

$$\begin{aligned} d_1(\omega ,\omega ')=\frac{1}{2^k}, \end{aligned}$$

where $k=\min \{j:w_j\ne w_j'\}$, $\omega =(w_0,w_1,\cdots )$, $\omega '=(w_0',w_1',\cdots )\in \Sigma _m^+$. A metric on $\Sigma _m^+\times X$ is defined as follows:

$$\begin{aligned} D\left( (\omega ,x),(\omega ',x')\right) =\max \{d_1(\omega ,\omega '),d(x,x')\} \end{aligned}$$

for $(\omega ,x),(\omega ',x')\in \Sigma _m^+\times X$, where $d_1$ and d are metrics on $\Sigma _m^+$ and X respectively.

The following definitions refer to [19] and [21].

We say that a point $x\in X$ is a $non-wandering$ point of G if for every neighbourhood U of x there is $\omega \in \Sigma _m^+$ and $k\in {\mathbb {N}}$ such that $\left( f_{\omega }^k\right) ^{-1}(U)\cap U\ne \emptyset$.

For $w=w_0w_1\cdots w_{n-1}\in F_m^+$, a $(w,\varepsilon )$-chain (or $(w,\varepsilon )$-pseudo-orbit) of G from x to y is a sequence $\{x_0=x,x_1,\cdots ,x_n=y\}$ such that $d\left( f_{w_i}(x_i),x_{i+1}\right) <\varepsilon$ for $i\in \{0,\cdots ,n-1\}$. We say that x is a chain recurrent point of G if for every $\varepsilon >0$, there is a $(w,\varepsilon )$-chain from x to itself for some $w\in F_m^+$.

A point $x\in X$ is called a recurrent point of G if there exist $\omega \in \Sigma _m^+$ and an increasing sequence $\{n_i\}_{i\ge 0}$ of positive integers such that $\lim \limits _{i\rightarrow +\infty }f_{\omega }^{n_i}(x)=x$.

Denote by $\Omega (G)$, CR(G) and R(G) the set of non-wandering, chain recurrent and recurrent points of G, respectively. Clearly, we have $R(G)\subset \Omega (G)$, $\Omega (G)\subset CR(G)$.

On the other hand, the space X is $totally~disconnected~at~p\in X$ if the connected component of X containing p is $\{p\}$. As in [4], we denote

$$\begin{aligned} X^{deg}=\{p\in X:X~is~totally~disconnected~at~p\}. \end{aligned}$$

Recall that X is totally disconnected if it is totally disconnected at any point(i.e.$X=X^{deg}$). We denote $\gamma (A)$ is the union of all connected components of X whose intersection A is non-empty.

The following lemmas and theorems are all from references that need to be used in the proof process of this paper.

Lemma 2.9

([11] Lemma 2.2) Let $f: X\rightarrow X$ be a continuous map. If $x\in Sh_{c+}^+(f)$ with $c\ge 0$, then for every $b>c$, there exists $\delta =\delta (x,b)>0$ such that every $\delta$-pseudo orbit $\{x_i\}_{i=0}^{\infty }$ with $d(x,x_0)<\delta$ is b-shadowed by some point of X.

Lemma 2.10

([15] Lemma 2.1) A homeomorphism of a compact metric space has the POTP through a subset K if and only if for every $\varepsilon >0$ there is $\delta >0$ such that every $\delta$-pseudo-orbit of f through K can be $\varepsilon$-shadowed.

Lemma 2.11

([15] Lemma 2.2) Let f be a homeomorphism of a compact metric space. Then, for every $z\in \Omega (f)\cap Sh(f)$ and every $\varepsilon >0$ there are $k\in {\mathbb {N}}^+$ and $y\in X$ such that $f^{pk}(y)\in B[z,\varepsilon ]$ for every $p\in {\mathbb {Z}}$.

Lemma 2.12

([15] Lemma 2.6) If $f: X\rightarrow X$ is a homeomorphism of a compact metric space X, then $Sh(f)=Sh(f^k)$ for every $k\in {\mathbb {Z}}{\setminus }\{0\}$.

Theorem 2.13

([15] Theorem 1.2) If $f: X\rightarrow X$ is a pointwise-recurrent homeomorphism of a compact metric space X, then $Sh(f)\subset X^{deg}$.

Lemma 2.14

([6] Chapter IV Lemma 25) If $f: X\rightarrow X$ is a homeomorphism of a compact metric space X, then for any positive integer m, we have $R(f)=R(f^m)$.

3 The Proof of the Main Result

In this section, we give the proof of Theorem 1.1 and Theorem 1.2. Before our proof, we must give the following lemmas.

Lemma 3.1

If $x\in Sh(G)$, then for every $\varepsilon >0$, there exists $\delta (x,\varepsilon )>0$, such that every $\delta$-pseudo-orbit $\{x_i\}_{i=0}^{\infty }$ for G with $d(x_0,x)<\delta$ is $\varepsilon$-shadowed by some point of X.

Proof

Since $x\in Sh(G)$, then take $\varepsilon >0$. For $\frac{\varepsilon }{2}>0$, there exists $0<\delta _1(x,\varepsilon )<\varepsilon$, such that every $\delta _1$-pseudo-orbit $\{z_i\}_{i=0}^{\infty }$ for G with $z_0=x$ can be $\frac{\varepsilon }{2}$-shadowed. Take $\delta <\frac{\delta _1}{2}$ such that $d(a,b)<\delta$ implies $d\left( f_j(a),f_j(b)\right) <\frac{\delta _1}{2}$ for every $a,b\in X$, $j\in \{0,1,\cdots ,m-1\}$. Given $\delta$-pseudo-orbit $\{x_i\}_{i=0}^{\infty }$ with $d(x_0,x)<\delta$. Define $\{y_i\}_{i=0}^{\infty }$ by $y_0=x$ and $y_i=x_i$ for $i\ge 1$. Assume $d\left( f_{w_i}(x_i),x_{i+1}\right) <\delta$, $\omega =(w_0,w_1,\cdots )\in \Sigma _m^+$, then

$$\begin{aligned} d\left( f_{w_i}(y_i),y_{i+1}\right) = {\left\{ \begin{array}{ll} d\left( f_{w_i}(x_i),x_{i+1}\right) &{}i\ge 1\\ d\left( f_{w_0}(x),x_1\right) &{}i=0 \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned}{} & {} d\left( f_{w_i}(x_i),x_{i+1}\right)<\delta<\delta _1, \\{} & {} \quad d\left( f_{w_0}(x),x_1\right) \le d\left( f_{w_0}(x),f_{w_0}(x_0)\right) +d\left( f_{w_0}(x_0),x_1\right)<\frac{\delta _1}{2}+\delta <\delta _1. \end{aligned}$$

so $\{y_i\}_{i=0}^{\infty }$ is a $\delta _1$-pseudo-orbit for G with $y_0=x$, then there exists $y\in X$, such that $d\left( f_{\omega }^n(y),y_n\right) <\frac{\varepsilon }{2}$ for every $n\ge 0$. Then, we have

$$\begin{aligned} d(y,x_0)\le d(y,x)+d(x,x_0)<\frac{\varepsilon }{2}+\delta <\varepsilon , \end{aligned}$$

and

$$\begin{aligned} d\left( f_{\omega }^n(y),x_n\right) =d\left( f_{\omega }^n(y),y_n\right)<\frac{\varepsilon }{2}<\varepsilon \end{aligned}$$

for all $n\ge 1$. Hence, y is a $\varepsilon$-shadowing point of $\{x_i\}_{i=0}^{\infty }$. $\square$

Lemma 3.2

If $f_0,\cdots ,f_{m-1}$are continuous self-maps of a compact metric space and G is the free semigroup generated by $\{f_0,\cdots ,f_{m-1}\}$, then G has the POTP through a compact subset K if and only if every point in K is a shadowable point of G.

Proof

By the previous remark we only have to prove the sufficiency. Now we use the finite cover theorem to prove the conclusion. Take $\varepsilon >0$, $x\in K$. There exists $\delta _x>0$ by Lemma 3.1. Consider the open cover $\{B(x,\delta _x)~|x\in K\}$, obviously it is the open cover of K. Since K is compact, there exists a finite open cover $\{B(x_1,\delta _1),\cdots ,B(x_n,\delta _n)\}$ such that $K\subset \bigcup \limits _{j=1}^nB(x_j,\delta _j)$. Let $\delta =\underset{1\le j \le n}{\min }\{\delta _j\}$. Take $\delta$-pseudo-orbit $\{x_i\}_{i=0}^{\infty }$ for G with $x_0\in K$, then there exists $1\le k\le n$ such that $x_0\in B(x_k,\delta _k)$. Obviously $\{x_i\}_{i=0}^{\infty }$ is also a $\delta _k$-pseudo-orbit. Then $\{x_i\}_{i=0}^{\infty }$ can be $\varepsilon$-shadowed by Lemma 3.1. So G has the POTP through K. $\square$

Lemma 3.3

If $f_0,\cdots ,f_{m-1}$are continuous self-maps of a compact metric space and G is the free semigroup generated by $\{f_0,\cdots ,f_{m-1}\}$, then $CR(G)\bigcap Sh(G)\subset \Omega (G)$.

Proof

Take $x\in CR(G)\bigcap Sh(G)$. Since $x\in Sh(G)$, for every $\varepsilon >0$, there exists $\delta >0$ such that every $\delta$-pseudo-orbit $\{z_i\}_{i=0}^{\infty }$ for G with $z_0=x$ can be $\varepsilon$-shadowed. Since $x\in CR(G)$ and $\delta >0$, there is a $(w,\delta )$-chain from x to itself for some $w\in F_m^+$, assume the $(w,\delta )$-chain is $\{x_0=x,x_1,\cdots ,x_n=x\}$ with $d(f_{w_i}(x_i),x_{i+1})<\delta$ for $0\le i\le n-1$ and $w=(w_0,w_1,\cdots ,w_{n-1})$. Define the sequence $\xi _{kn+r}=x_r$, where $0\le r\le n-1$ and $k\in {\mathbb {N}}$. Let $v=(w_0,w_1,\cdots ,w_{n-1})^{\infty }$. Then we obtain $d(f_{v_i}(\xi _i),\xi _{i+1})<\delta$. So $\{\xi _i\}_{i=0}^{\infty }$ is a $\delta$-pseudo-orbit for G with $\xi _0=x_0=x$, then there exists $y\in X$ such that $d(f_v^j(y),\xi _j)<\varepsilon$ for every $j\in {\mathbb {N}}$. Then we have

$$\begin{aligned} d(y,\xi _0)=d(y,x)<\varepsilon , \end{aligned}$$

and

$$\begin{aligned} d(f_v^n(y),\xi _n)=d(f_v^n(y),x)<\varepsilon \end{aligned}$$

that is $(f_v^n)^{-1}B(x,\varepsilon )\bigcap B(x,\varepsilon )\ne \emptyset$, so $x\in \Omega (G)$. We obtain $CR(G)\bigcap Sh(G)\subset \Omega (G)$ by the arbitrariness of x. $\square$

Remark 3.4

From the proof process of Lemma 3.3, we can know that the point y satisfies $d(f_v^{kn}(y),x)<\varepsilon$ for every $k\in {\mathbb {N}}$, so the orbit of point y will periodically enter the neighbourhood of x.

Lemma 3.5

Let $f_0,\cdots ,f_{m-1}$ be continuous self-maps of a compact metric space and G the free semigroup generated by $\{f_0,\cdots ,f_{m-1}\}$, then Sh(G) is a Borel set.

Proof

Given $\varepsilon >0$, let $Sh_{\varepsilon }(G)$ be the set of points $x\in X$ satisfying for $\varepsilon >0$, there exists $\delta _{\varepsilon }>0$ such that every $\delta _{\varepsilon }$-pseudo-orbit $\{z_i\}_{i=0}^{\infty }$ with $z_0=x$ can be $\varepsilon$-shadowed. Also, given $\varepsilon >0$ and $\delta >0$, let $S_{\delta ,\varepsilon }(G)$ be the set of points $x\in X$ satisfying for every pseudo-orbit $\{y_i\}_{i=0}^{\infty }$ with $y_0=x$ and $d(f_{w_i}(y_i),y_{i+1})<\delta$ for all $i\ge 0$ where $\omega =(w_0,w_1,\cdots )\in \Sigma _m^+$, there exists $y\in X$ such that $d(f_{\omega }^i(y),y_i)\le \varepsilon$ for all $i\ge 0$.

Clearly from the definition of Sh(G) we easily have

$$\begin{aligned} Sh(G)=\bigcap _{k\in \mathbb {N^+}}Sh_{\frac{1}{k}}(G), \end{aligned}$$

(4)

and for every $k\in \mathbb {N^+}$

$$\begin{aligned} Sh_{\frac{1}{k}}(G)=\bigcup _{m\in \mathbb {N^+}}S_{\frac{1}{m},\frac{1}{k}}(G). \end{aligned}$$

(5)

For any given $k,m\in \mathbb {N^+}$, we prove that $S_{\frac{1}{m},\frac{1}{k}}(G)$ is a closed subset of X for all $m\in \mathbb {N^+}$. Assume $\{z_n\}_{n\in {\mathbb {N}}}\subset S_{\frac{1}{m},\frac{1}{k}}(G)$ and $\lim \limits _{n\rightarrow \infty }z_n=z$ for some $z\in X$. Next we show $z\in S_{\frac{1}{m},\frac{1}{k}}(G)$. Given $\frac{1}{m}$-pseudo-orbit $\{x_i\}_{i=0}^{\infty }$ for G with $x_0=z$ and $d(f_{w_i}(x_i),x_{i+1})<\frac{1}{m}$ where $\omega =(w_0,w_1,\cdots )\in \Sigma _m^+$. For every $n\in {\mathbb {N}}$, consider sequence $\{x_i^{(n)}\}_{i=0}^{\infty }$ satisfying $x_0^{(n)}=z_n$ and $x_i^{(n)}=x_i$ for $i>0$. Since $d(f_{w_0}(x_0),x_1)<\frac{1}{m}$, there exists $t>0$ such that $d(f_{w_0}(x_0),x_1)<t<\frac{1}{m}$. We can choose $\delta >0$ such that $d(a,b)<\delta$ implies $d(f_j(a),f_j(b))<\frac{1}{m}-t$ for every $j\in \{0,1,\cdots ,m-1\}$ and every a, $b\in X$. Since $\lim \limits _{n\rightarrow \infty }z_n=z$, we have $d(z_n,z)<\delta$ for sufficiently large $n\in {\mathbb {N}}$, so $d(f_{w_0}(z_n),f_{w_0}(z))<\frac{1}{m}-t$. Then we have

$$\begin{aligned} \begin{aligned} d(f_{w_0}(x_0^{(n)}),x_1^{(n)})=d(f_{w_0}(z_n),x_1)&\le d(f_{w_0}(z_n),f_{w_0}(z))+d(f_{w_0}(z),x_1)\\&<\frac{1}{m}-t+t=\frac{1}{m}, \end{aligned} \end{aligned}$$

and for $i>0$

$$\begin{aligned} \begin{aligned} d(f_{w_i}(x_i^{(n)}),x_{i+1}^{(n)})=d(f_{w_i}(x_i),x_{i+1})<\frac{1}{m}, \end{aligned} \end{aligned}$$

when n is large enough. For such $n\in {\mathbb {N}}$, the pseudo-orbit $\{x_i^{(n)}\}_{i=0}^{\infty }$ is a $\frac{1}{m}$-pseudo-orbit for G with $x_0^{(n)}=z_n\in S_{\frac{1}{m},\frac{1}{k}}(G)$ and $d(f_{w_i}(x_i^{(n)}),x_{i+1}^{(n)})<\frac{1}{m}$, so there exists $y_n\in X$ such that $d(f_{\omega }^i(y_n),x_i^{(n)})<\frac{1}{k}$ for every $i\ge 0$. Take s subsequence $\{y_{n_j}\}_{j\in {\mathbb {N}}}$ and assume $\lim \limits _{j\rightarrow \infty }y_{n_j}=y$ for some $y\in X$. Then we have

$$\begin{aligned} d(y,x_0)=d(y,z)=\lim \limits _{j\rightarrow \infty }d(z_{n_j},y_{n_j})=\lim \limits _{j\rightarrow \infty } d(x_0^{(n_j)},y_{n_j})\le \frac{1}{k}, \end{aligned}$$

and for every $i>0$

$$\begin{aligned} d(f_{\omega }^i(y),x_i)=\lim \limits _{j\rightarrow \infty }d(f_{\omega }^i(y_{n_j}),x_i^{(n_j)})\le \frac{1}{k}. \end{aligned}$$

Hence $\{x_i\}_{i=0}^{\infty }$ can be $\frac{1}{k}$-shadowed, that is $z\in S_{\frac{1}{m},\frac{1}{k}}(G)$. From this we know that $S_{\frac{1}{m},\frac{1}{k}}(G)$ is a closed subset of X, and therefore we obtain $Sh_{\frac{1}{k}}(G)$ is a Borel set in X for every $k\in \mathbb {N^+}$ by (5). Thus Sh(G) is a Borel set in X by (4). $\square$

Lemma 3.6

If $f_0,\cdots ,f_{m-1}$are continuous self-maps of a compact metric space X and G is the free semigroup generated by $\{f_0,\cdots ,f_{m-1}\}$, F is the skew-product transformation corresponding to $\{f_0,\cdots ,f_{m-1}\}$ which satisfies $Sh(F)=(\Sigma _m^+\times X)^{deg}$, then $Sh(G)=X^{deg}$.

Proof

Firstly we prove $Sh(G)\subset X^{deg}$. Take $x\in Sh(G)$ and $\varepsilon >0$. There exists $k\in {\mathbb {N}}$ such that $\frac{1}{2^k}<\varepsilon$ and $\delta '>0$ such that every $\delta '$-pseudo-orbit $\{z_i\}_{i=0}^{\infty }$ for G with $z_0=x$ can be $\varepsilon$-shadowed. Let $\delta =\min \{\frac{1}{2^{k+1}},\delta '\}$, $\omega \in \Sigma _m^+$. Now take a $\delta$-pseudo-orbit $\{\xi _i\}_{i=0}^{\infty }$ of F with $\xi _{i}=(\omega ^{(i)},x_i)$ and $\xi _0=(\omega ,x)$, where $\omega ^{(i)}=(w_0^{(i)},w_1^{(i)},\cdots )$. Therefore

$$\begin{aligned} D(F(\xi _{i}),\xi _{i+1})=\max \{d_1(\sigma \omega ^{(i)},\omega ^{(i+1)}),d(f_{w_0^{(i)}}(x_i), x_{i+1})\}<\delta ~for~all~i\ge 0. \end{aligned}$$

So we have

$$\begin{aligned} d_1(\sigma \omega ^{(i)},\omega ^{(i+1)})<\delta \le \frac{1}{2^{k+1}}. \end{aligned}$$

That is $w^{(i)}_{j+1}=w^{(i+1)}_j$ for every $0\le j\le k+1$, $i\ge 0$. Consider $v=(w_0^{(0)},w_0^{(1)},w_0^{(2)},\cdots )$. Since $d(f_{w_0^{(i)}}(x_i),x_{i+1})<\delta <\delta '$ by $D(F(\xi _{i}),\xi _{i+1})<\delta$, and $x_0=x\in Sh(G)$, there exists $y\in X$, such that $d(f_v^n(y),x_n)<\varepsilon$ for all $n\ge 0$. In addition for this v, one can see that $d_1(\sigma ^n(v),\omega ^{(n)})\le \frac{1}{2^{k+3}}<\varepsilon$ for all $n\ge 0$. Therefore we have

$$\begin{aligned} D(F^n(v,y),\xi _{n})<\varepsilon \quad for~all~n\ge 0. \end{aligned}$$

Hence $(\omega ,x)\in Sh(F)=(\Sigma _m^+\times X)^{deg}$. Next, we will prove that $x\in X^{deg}$.

Suppose by contradiction that $x\notin X^{deg}$. Then the connected component E of X containing x(which is compact) has positive diameter diam(E)>0. Obviously $\{\omega \}\times F$ is also a connected component of $\Sigma _m^+\times X$ containing $(\omega ,x)$. This contradicts with $(\omega ,x)\in (\Sigma _m^+\times X)^{deg}$. Hence $Sh(G)\subset X^{deg}$.

   On the other hand, for any $x\in X^{deg}$, we claim that for any $\omega \in \Sigma _m^+$ we have $(\omega ,x)\in (\Sigma _m^+\times X)^{deg}$. If not, there exists $v\in \Sigma _m^+$, such that $(v,x)\notin (\Sigma _m^+\times X)^{deg}$. Then we obtain a connected component E of $\Sigma _m^+\times X$ containing (v, x) and

$E\setminus (v,x)\ne \emptyset$. We divide into two cases:

$\textbf{Case}~1$: If $E\mid _X=\{x\}$, we have $E=A\times \{x\}$ where $A\subset \Sigma _m^+$. Since $\Sigma _m^+$ is totally disconnected, there exists open sets $U_1,U_2\subset \Sigma _m^+$ such that $(U_1\cap A)\cap (U_2\cap A)=\emptyset$ and $A\subset U_1\cup U_2$, then we have $[(U_1\times \{x\})\cap E]\cap [(U_2\times \{x\})\cap E]=\emptyset$ and $E\subset (U_1\times \{x\})\cup (U_2\times \{x\})$. According to the definition of connectivity we know that E is not connected. It’s wrong.

$\textbf{Case}~2$: If $E\mid _X\setminus \{x\}\ne \emptyset$, $E\mid _X$ is not connected by $x\in X^{deg}$. There exists open sets $V_1,V_2\subset X$ such that $(V_1\cap E\mid _X)\cap (V_2\cap E\mid _X)=\emptyset$ and $E\mid _X\subset V_1\cup V_2$, then we have $[(\Sigma _m^+\times V_1)\cap E]\cap [(\Sigma _m^+\times V_2)\cap E]=\emptyset$ and $E\subset \Sigma _m^+\times V_1\cup \Sigma _m^+\times V_2$. So E is not connected. This contradicts the fact that E is connected.

   In all, we obtain $E=\{(v,x)\}$, therefore $(v,x)\in (\Sigma _m^+\times X)^{deg}$. It is conflicts with $(v,x)\notin (\Sigma _m^+\times X)^{deg}$, so we have $(\omega ,x)\in (\Sigma _m^+\times X)^{deg}=Sh(F)$ for every $\omega \in \Sigma _m^+$.

   Next, we show $x\in Sh(G)$. Take $\omega \in \Sigma _m^+$ and $0<\varepsilon <\frac{1}{2}$. we have $(\omega ,x)\in Sh(F)$ according to the above proof. For $0<\varepsilon <\frac{1}{2}$, there exists $\delta _{\omega }>0$ by Lemma 2.9, such that

$$\begin{aligned} \Sigma _m^+\subset \bigcup _{\omega \in \Sigma _m^+}B(\omega ,\delta _{\omega }). \end{aligned}$$

Actually $\Sigma _m^+$ is compact with respect to metric $d_1$ by Tychonoff theorem. So there exists finite open covers such that

$$\begin{aligned} \Sigma _m^+\subset \bigcup _{i=1}^nB(\omega ^{(i)},\delta _i), \end{aligned}$$

where $\omega ^{(i)}\in \Sigma _m^+$ and $(\omega ^{(i)},x)\in Sh(F)$. Let $\delta =\underset{1\le i \le n}{\min }\{\delta _i\}$, given $\delta$-pseudo-orbit $\{x_i\}_{i=0}^{\infty }$ for G with $x_0=x$ and $d(f_{v_i}(x_i),x_{i+1})<\delta$ for all $i\ge 0$ where $v=(v_0,v_1,\cdots )\in \Sigma _m^+$. Consider sequence $\{\xi _i\}_{i=0}^{\infty }$ with $\xi _i=(\sigma ^iv,x_i)$, $\xi _0=(v,x)$. Obiously $\{\xi _i\}_{i=0}^{\infty }$ is a $\delta$-pseudo-orbit of F. Since $v\in \Sigma _m^+$, there exists $1\le k\le n$, such that $v\in B(\omega ^{(k)},\delta _k)$. Also we have

$$\begin{aligned} D(\xi _0,(\omega ^{(k)},x))=\max \{d(x,x),d_1(v,\omega ^{(k)})\}<\delta _k. \end{aligned}$$

For above $\varepsilon$ and Lemma 2.9, we know $\{\xi _i\}_{i= 0}^{\infty }$ can be $\varepsilon$-shadowed. There exists $(u,y)\in \Sigma _m^+\times X$ such that

$$\begin{aligned} D(F^i(u,y),\xi _{i})<\varepsilon , \end{aligned}$$

that is $d_1(\sigma ^iu,\sigma ^iv)<\varepsilon$ and $d(f_u^i(y),x_i)<\varepsilon$ for every $i\ge 0$. Since $d_1(\sigma ^iu,\sigma ^iv)<\varepsilon <\frac{1}{2}$ we obtain $u=v$. Hence $d(f_u^i(y),x_i)=d(f_v^i(y),x_i)<\varepsilon$ for all $i\ge 0$, that is $x\in Sh(G)$. Therefore $X^{deg}\subset Sh(G)$. $\square$

   The following lemma is an improvement of Theorem 2.13 and with some modifications on the proof process of Theorem 2.13.

Lemma 3.7

Let $f:X\rightarrow X$ be a continuous map of a compact metric space X. If there exists $\delta >0$ satisfying $B(\gamma (Sh(f)),\delta )\subset R(f)$, then $Sh(f) \subset X^{deg}$.

Proof

Take $z\in Sh(f)$ and suppose by contradiction that $z\notin X^{deg}$. Then the connected component $\gamma (\{z\})$, denoted by E, is compact and diam(E)>0. Take $0<\varepsilon <\min \{\frac{1}{11}$diam(E), $\delta \}$. Obviously $z\in B(\gamma (Sh(f)),\delta )$, so $z\in R(f)\subset \Omega (f)$. There exists $k\in {\mathbb {N}}^+$ and $y\in X$ such that $f^{nk}(y)\in B(z,\varepsilon )$ for every $n\ge 0$ by Lemma 2.11. Define $g=f^k$. Then

$$\begin{aligned} g^n(y)\in B(z,\varepsilon )~for~all~n\ge 0. \end{aligned}$$

(6)

On the other hand $z\in Sh(g)$ by Lemma 2.12. Then for above $\varepsilon$,there exists $\delta '>0$ by Lemma 3.1. We can assume $\delta '<\varepsilon$. Since E is compact and connected, we can choose a sequence $y=p_1,p_2,\cdots ,p_N\in E$ such that $d(p_i,p_{i+1})\le \frac{\delta '}{2}$ for $1\le i\le N-1$ and

$$\begin{aligned} E\subset \bigcup \limits _{i=1}^NB(p_i,\delta '). \end{aligned}$$

(7)

Also $R(g)=R(f^k)=R(f)$ by Lemma 2.14, and $p_i\in E\subset B(\gamma (Sh(f)),\delta )\subset R(f)=R(g)$. From this we can find positive integers $c(i)~(1\le i\le N)$ such that $d(p_i,g^{c(i)}(p_i))\le \frac{\delta '}{2}$ for all $1\le i\le N$. Since $z\in E\subset \bigcup \limits _{i=1}^NB(p_i,\delta ')$, there exists $p_{i_z}~(1\le i_z\le N)$ such that $z\in B(p_{i_z},\delta ')$. We define the sequence $\{\xi _i\}_{i=0}^{\infty }$ as follow

$$\begin{aligned} \begin{aligned} \xi _i&=g^i(p_{i_z}){} & {} \text{ if } 0\le i\le c(i_z)-1, \\ \xi _{c(i_z)+i}&=g^i(p_{i_z+1}){} & {} \text{ if } 0 \le i \le c(i_z+1)-1, \\&\vdots{} & {} \\ \xi _{c(i_z)+\cdots +c(N-2)+i}&=g^i(p_{N-1}){} & {} \text{ if } 0 \le i \le c(N-1)-1, \\ \xi _{c(i_z)+\cdots +c(N-1)+i}&=g^i\left( p_N\right){} & {} \text{ if } 0 \le i \le c(N)-1, \\ \xi _{c(i_z)+\cdots +c(N)+i}&=g^i\left( p_{N-1}\right){} & {} \text{ if } 0 \le i \le c(N-1)-1, \\&\vdots{} & {} \\ \xi _{c(1)+\cdots +c(i_z-1)+2\{c(i_z)+\cdots +c(N-1)\}+c(N)+i}&=g^i\left( p_1\right){} & {} \text{ if } i \ge 0. \end{aligned} \end{aligned}$$

Obviously $\{\xi _i\}_{i=0}^{\infty }$ is a $\delta '$-pseudo-orbit of g with $\xi _0=p_{i_z}\in B(z,\delta ')$. Then there exists $x\in X$ such that $d(g^n(x),\xi _n)<\varepsilon$ for every $n\ge 0$ by Lemma 2.10. From the definition of $\{\xi _i\}_{i= 0}^{\infty }$ we conclude that there exists integers $n_1,\cdots ,n_N$ satisfying $d(g^{n_i}(x),p_i)<\varepsilon$ for every $1\le i\le N$. Take $c=c(1)+\cdots +c(i_z-1)+2\{c(i_z)+\cdots +c(N-1)\}+c(N)$, we obtain

$$\begin{aligned} d(g^{i+c}(x),g^i(y))<\varepsilon \quad (for~i\ge 0). \end{aligned}$$

This combined with (6) yields

$$\begin{aligned} g^i(x)\in B(z,2\varepsilon )~whenever~i\ge c. \end{aligned}$$

Since $d(g^{n_i}(x),p_i)<\varepsilon \le \delta$ and $p_i\in E$ for every $1\le i\le N$, we have $g^{n_i}(x)\in B(\gamma (sh(f)),\delta )\subset R(g)$. Thus for $g^{n_i}(x)$, there exists $k_i\ge c$ such that $d(g^{k_i}(x),g^{n_i}(x))<\varepsilon$. Now take $w\in E$. It follows from (7) that $d(w,p_i)<\delta '$ for some $1\le i\le N$. Then $d(g^{n_i}(x),w)\le d(g^{n_i}(x),p_i)+d(p_i,w)<\varepsilon +\delta '<2\varepsilon$. Now we have two cases:

$\textbf{Case}~1$: If $0\le n_i<c$, then

$$\begin{aligned} \begin{aligned} d(w,z)&\le d(w,g^{n_i}(x))+d(g^{n_i}(x),g^{k_i}(x))+d(g^{k_i}(x),g^{k_i-c}(p_1))+d(g^{k_i-c}(p_1),z)\\&<2\varepsilon +\varepsilon +\varepsilon +\varepsilon \\&=5\varepsilon . \end{aligned} \end{aligned}$$

$\textbf{Case}~2$: If $n_i\ge c$, then

$$\begin{aligned} d(w,z)\le & {} d(w,g^{n_i}(x))+d(g^{n_i}(x),g^{n_i-c}(p_1))+d(g^{n_i-c}(p_1),z)\\< & {} 72\varepsilon +\varepsilon +\varepsilon \\= & {} 4\varepsilon . \end{aligned}$$

We conclude that $E\subset B(z,5\varepsilon )$ by the arbitrariness of w and adove cases, so diam(E)$<10\varepsilon$. This contradicts the choice of $\varepsilon$ therefore $z\in X^{deg}$. As $z\in Sh(f)$ is arbitrary, we obtain $Sh(f)\subset X^{deg}$. $\square$

Remark 3.8

Since F is not pointwise recurrent, we need to change the condition to Lemma 3.7, which only focuses on the local properties of the shadowable points, and we can prove that if f is a pointwise recurrent continuous map, then there exists $\delta >0$ such that $B(\gamma (Sh(f)),\delta )\subset R(f)$. But the inverse may not necessarily hold true. For example, let $a,b,c,p\in {\mathbb {R}}$ satisfying $a\ne p$ and $a,p\notin [b,c]$. Consider compact metric space $([b,c]\bigcup \{a\}\bigcup \{p\},d)$, where d is Euclidean metric. Define map:

$$\begin{aligned} f(x)= {\left\{ \begin{array}{ll} p&{}x=p\\ b&{}x=a\\ x&{}x\in [b,c] \end{array}\right. } \end{aligned}$$

Then we have $Sh(f)=\{p\}$ by Theorem 2.13, so it holds for $B(\gamma (Sh(f)),\delta )\subset R(f)$, but f is not recurrent by $f^{n}a=b$ for every $n\ge 1$.

Proof of Theorem 1.1

Let $f_0,f_1,\cdots ,f_{m-1}$ be continuous self-maps on compact metric space X and G the free semigroup generated by $\{f_0,f_1,\cdots ,f_{m-1}\}$. We have that Sh(G) is a Borel set by Lemma 3.5. Take $K=X$ in Lemma 3.2, then we obtain that G has the POTP if and only if $Sh(G)=X$. Finally, since $\Omega (G)\subset CR(G)$ we have that if $CR(G)\subset Sh(G)$, then $CR(G)=\Omega (G)$ by Lemma 3.3. $\square$

Proof of Theorem 1.2

Let $f_0,f_1,\cdots ,f_{m-1}$ be continuous self-maps on compact metric space X, G the free semigroup generated by $\{f_0,f_1,\cdots ,f_{m-1}\}$ and F the skew-product transformation corresponding to $\{f_0,\cdots ,f_{m-1}\}$. If there exists $\delta >0$ such that $B(\gamma (Sh(F)),\delta )\subset R(F)$, we obtain $Sh(F)\subset (\Sigma _m^+\times X)^{deg}$ by Lemma 3.7. From the proof process of Lemma 3.6, it can be seen that $Sh(G)\subset X^{deg}$. $\square$

The following example will state the rationality of the condition in Theorem 1.2.

Example 3.9

Let $X={\mathbb {S}}^1$ is one-dimensional torus, $f_0,f_1$ are all irrational rotation with irrational independence. Assume $f_0(x)=x+\alpha _0~(mod~1)$ and $f_1(x)=x+\alpha _1~(mod~1)$ with $\alpha _0+\alpha _1\in {\mathbb {R}}\setminus {\mathbb {Q}}$. The periodic point $\omega =(0,1)^{\infty }$ is a minimal point in the dynamical system $(\Sigma _2^+,\sigma )$. For minimal sub-system $(\overline{Orb(\omega ,\sigma )},\sigma )$, define skew-product transformation $F: \overline{Orb(\omega ,\sigma )} \times {\mathbb {S}}^1 \rightarrow \overline{Orb(\omega ,\sigma )} \times {\mathbb {S}}^1$ corresponding to $\{f_0,f_1\}$. There exists $\delta >0$ such that $B(\gamma (Sh(F)),\delta )\subset R(F)$ according to the following proof.

Clearly $\overline{Orb(\omega ,\sigma )}=\{\omega ,\sigma (\omega )\}$, then we prove that $R(F)=\overline{Orb(\omega ,\sigma )}\times {\mathbb {S}}^1$. Take $x\in {\mathbb {S}}^1$. For $(\omega ,x)\in \overline{Orb(\omega ,\sigma )}\times {\mathbb {S}}^1$, define $g_0(x)=f_1\circ f_0(x)=x+\alpha _0+\alpha _1~(mod~1)$. Since $\alpha _0+\alpha _1\in {\mathbb {R}}\setminus {\mathbb {Q}}$, we obtain $g_0$ is minimal, that is $R(g_0)={\mathbb {S}}^1$. For $x\in R(g_0)$, there exists an increasing sequence $\{n_k\}_{k\ge 0}$ of positive integers such that $\lim \limits _{k\rightarrow \infty }g_0^{n_k}(x)=x$, then we have

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }F^{2n_k}(\omega ,x)=\lim \limits _{k\rightarrow \infty }(\sigma ^{2n_k} (\omega ),f_{\omega }^{2n_k}(x))=\lim \limits _{k\rightarrow \infty }(\omega ,g_0^{n_k}(x))=(\omega ,x), \end{aligned}$$

that is $(\omega ,x)\in R(F)$. For $(\sigma (\omega ),x)$, define $g_1(x)=f_0\circ f_1(x)=x+\alpha _1+\alpha _1~(mod~1)$. Similarly we have $R(g_1)={\mathbb {S}}^1$. For $x\in R(g_1)$, there exists an increasing sequence $\{n_i\}_{i\ge 0}$ of positive integers such that $\lim \limits _{i\rightarrow \infty }g_1^{n_i}(x)=x$, then we have

$$\begin{aligned} \lim \limits _{i\rightarrow \infty }F^{2n_i}(\sigma (\omega ),x)=\lim \limits _{i\rightarrow \infty } (\sigma ^{2n_i+1}(\omega ),f_{\sigma (\omega )}^{2n_i}(x))=\lim \limits _{i\rightarrow \infty } (\sigma (\omega ),g_1^{n_i}(x))=(\sigma (\omega ),x), \end{aligned}$$

that is $(\sigma (\omega ),x)\in R(F)$. In all we have $R(F)=\overline{Orb(\omega ,\sigma )}\times {\mathbb {S}}^1$, so there exists $\delta >0$ such that $B(\gamma (Sh(F)),\delta )\subset R(F)$.