Shadowing Property of Hyperspace for Free Semigroup Actions

Abstract

In this paper, we introduce a notion of shadowing property for a free semigroup action on a compact metric space, which is different of the notion of the shadowing property introduced by Bahabadi, called chain shadowing property. We study the relation between the shadowing property of a free semigroup action on a compact metric space X and the shadowing property of the induced free semigroup action on the hyperspace 2^X. Specially, we not only theoretically prove that $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the (chain) shadowing property if and only if $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$ has the (chain) shadowing property, but also give examples to illustrate it. Finally, we compare the two notions of shadowing for free semigroup actions and obtain an interesting result that if $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the shadowing property, then it has the chain shadowing property, but not vice versa.

1 Introduction

Let (X, T, d) be a topological dynamical system, where X is a compact metric space with metric d and $T:X\rightarrow X$ is a continuous map. The map T induces in a natural way a map $2^{T}: 2^{X}\rightarrow 2^{X}$ defined by 2^T(A) = T(A) on the space 2^X, called hyperspace, of all nonempty closed subsets equipped with the Vietoris topology which coincides with the topology induced by the Hausdorff metric d_H. Then 2^T is also a continuous map and the hyperspace 2^X is also a compact metric space with the Hausdorff metric d_H that means (2^X,2^T,d_H) is also a topological dynamical system. The study of relationship about topological properties between T and 2^T was initiated by Bauer and Sigmund [2]. After that, several results dealing with this new relationship continue to appear. For instance, Banks in [4] showed that the transitivity of (2^X,2^T,d_H) is coincident to its weakly mixing property. In [8] authors showed that if T is a chain transitive map then 2^T is also chain transitive. And the relationship between the entropy of the map T and the entropy of the induced map 2^T are studied in [9]. Kwietniak and Oprocha in [15] studied the relationship of entropy between the hyperspace and the base space and showed that under some nonrecurrence assumption the induced map 2^T is always topologically chaotic. Li, Oprocha and Ye in [16] systematically studied the question when hyperspace is recurrent. Recently, Ji, Chen and Zhou in [14] introduced the entropy order concept for (2^X,2^T,d_H) and proved that it is coincides with the topological entropy of (X, T, d). Blank in [5] proposed the ε-trajectory and introduced the average shadowing property while studying chaotic dynamical systems. Motivated by the work of Bauer on hyperspace, Gómez Rueda, Illanes and Mendez in [10] study some dynamical properties (like periodicity, recoverability and shadowing property) in hyperspace. Specially, Femández and Good in [7] proposed that following theorem.

Theorem 1.1

[7, Theorem 3.8] Let X be a compact metric space and let $T:X\rightarrow X$ be a continuous map, then 2^T has the shadowing property if and only if T has the shadowing property.

Recently, many research works have been devoted to the applications of free semigroup actions. For instance, Carvalho, Rodrigues and Varandas in [6] studied finitely generated free semigroup actions on a compact metric space and obtained quantitative information on Poincare recurrence. Maria, Rodrigues and Paulo in [17] introduced a notion of measure theoretical entropy for free semigroup actions and established a variational principle. In [22], Zhu and Ma introduced the notion of topological r-entropy for free semigroup actions on a compact metric space and provided some properties of it. Therefore, we want to generalize the above theorem to free semigroup actions. In [3], Bahabadi and Zamani first introduced the notions of chain shadowing property and average shadowing property for free semigroup actions (IFSs). Osipov and Tikhomirov in [21] proposed the shadowing property of finite generated group actions on compact metric spaces and studied the related properties. In the present paper, we push this direction further, considering the relationship of shadowing property for a free semigroup action between base space dynamical system and hyperspace dynamical system.

The rest of the paper is organized as follows. In Section 2, some basic concepts and notes are introduced. In Section 3, we note the definition of shadowing property for a free semigroup action from Hui and Ma [11] as the chain shadowing property and obtained $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$ has the chain shadowing property if and only if $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the chain shadowing property. In Section 4, the definition of shadowing property for a free semigroup action is introduced. We also have $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$ has the shadowing property if and only if $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the shadowing property. Moreover, we give examples to illustrate that the two shadowing properties for free semigroup actions are inequivalent.

2 Preliminaries

First of all, we introduce some basic definitions of hyperspace.

2.1 Hyperspace

Let X be a compact metric space with metric d. Denote by 2^X the hyperspace on X,

$$ 2^{X}=\{A\subseteq X:A\ne\emptyset, \bar{A}=A\}, $$

that is, the collection of all nonempty closed subsets of X. Then 2^X is also a compact metric space, equipped with the Hausdorff metric d_H, defined as

$$ d_{H}(A,C)=\inf\{\varepsilon >0:A\subseteq B(C,\varepsilon) \quad\text{and}\quad C\subseteq B(A,\varepsilon)\} $$

where B(A, ε) = {x ∈ X : d(x, A) < ε} for a set A, that is, B(A, ε) is an ε-neighborhood of A in X.

The following family

$$ \{\langle U_{1},\cdots,U_{n}\rangle :U_{1},\cdots,U_{n}\ {\text {are nonempty open subsets of}}\ X, n\in\mathbb{N}\}, $$

forms a basis for the topology of 2^X called the Vietoris topology, where

$$ \langle S_{1},\cdots,S_{n}\rangle \doteq\left\{A\in 2^{X} : A\subset \bigcup\limits_{i=1}^{n} S_{i}\ {\text{and}}\ A\cap S_{i}\neq\emptyset \ {\text{for each}}\ i=1,\cdots,n\right\} $$

is defined for arbitrary nonempty subsets $S_{1},\cdots ,S_{n}\subseteq X$. It is not hard to see that the Hausdorff topology (the topology induced by the Hausdorff metric d_H) and the Vietoris topology for 2^X coincide [12 Theorem 3.1].

Let F_n(X) = {A ∈ 2^X : ♯A ≤ n}, where ♯A denotes the cardinality of the set A.

Denote by $F(X)=\bigcup \limits _{n=1}^{\infty } F_{n}(X)$ the collection of all finite subsets of X. Obviously, F(X) is a dense subset of 2^X.

For more details on hyperspaces please see [19] and [20].

2.2 Free Semigroup Action

The free semigroup, written $F_{m}^{+}$, is the set of all finite words of symbols 0,1,⋯ ,m − 1. In $F_{m}^{+}$, we consider the semigroup operation of concatenation defined as usual: if $w=w_{n}w_{n-1}{\cdots } w_{1}\in F_{m}^{+}$ and $w^{\prime }=w_{k}^{\prime } w_{k-1}^{\prime } {\cdots } w_{1}^{\prime }\in F_{m}^{+}$, then $ww^{\prime }=w_{n}w_{n-1}{\cdots } w_{1}w_{k}^{\prime } w_{k-1}^{\prime } {\cdots } w_{1}^{\prime }\in F_{m}^{+}$. For any $w\in F_{m}^{+}, |w|$ denotes the length of w, that is, if $w=w_{n}w_{n-1}{\cdots } w_{1}\in F_{m}^{+}$, then |w| = n and the identity $e\in F_{m}^{+}$ which length is 0, called empty word. We write $w\leq w^{\prime }$ if there exists a word $w^{\prime \prime }$ such that $w^{\prime }=w^{\prime \prime } w$.

Remark 2.1

For any $w\in F_{m}^{+}, |w|\neq 0,$ we have e < w.

Let X be a compact metric space and let $f_{0},f_{1},\dots ,f_{m-1}$ be continuous maps on X. We denote by $(F_{m}^{+},\mathcal {F})\curvearrowright X$ the free semigroup action generated by $\mathcal {F}$=$\{f_{0},f_{1},\dots ,f_{m-1}\}$. One way to interpret this statement is to consider a map $\varphi : F_{m}^{+}\times X \longrightarrow X$ such that, for any $w=w_{n}w_{n-1}{\cdots } w_{1}\in F_{m}^{+}$,

$$ \begin{aligned} \varphi:F_{m}^{+}\times X& \longrightarrow X\\ (w,x)& \longmapsto f_{w}(x), \end{aligned} $$

where $f_{w}(x)=f_{w_{n}}\circ f_{w_{n-1}}\circ {\cdots } f_{w_{1}}(x)$. Specially, (e, x)↦f_e(x) ≡ x. Then for any map $f_{i}\in \mathcal {F}$, the map $2^{f_{i}}$,

$$ \begin{aligned} 2^{f_{i}}:2^{X}&\longrightarrow 2^{X}\\ A& \longmapsto f_{i}(A), \end{aligned} $$

is a continuous map on 2^X. Thus the map φ naturally induces a map 2^φ,

$$ \begin{aligned} 2^{\varphi}:F_{m}^{+}\times 2^{X}& \longrightarrow 2^{X}\\ (w,A)& \longmapsto 2^{f_{w}}(A). \end{aligned} $$

Therefore, $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$ is also a free semigroup action.

Next, we set up some notations about the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$.

For sets $A\subseteq X$ and $K\subseteq F_{m}^{+}$ and a word $w\in F_{m}^{+}$, we write

$$ wA=\{f_{w}(x): x\in A\}, \ \ \ \ Kx=\{f_{w}(x): w\in K\},\ \ \ \ KA=\{f_{w}(x): w\in K \ \text {and} \ x\in A\}. $$

The $F_{m}^{+}$-orbit of a point x ∈ X is the set $F_{m}^{+}x$. We say a set $A\subseteq X$ is $F_{m}^{+}$-invariant if $F_{m}^{+}A\subseteq A$.

Throughout this paper, $(F_{m}^{+},\mathcal {F})\curvearrowright X$ is a free semigroup action on a compact metric space with the metric denoted by d.

3 The Chain Shadowing Property

In this section, we study the relation of the chain shadowing property between base space dynamical system and hyperspace dynamical system. The definition of chain shadowing property under the free semigroup action refers to [11]. In order to distinguish the definition of shadowing property in the next section, we call shadowing defined in [11] as chain shadowing.

Denote the set of all one-side infinite sequences of symbols 0,1,⋯ ,m − 1 by ${\Sigma }_{m}^{+}$, i.e.,

$$ {\Sigma}_{m}^{+}=\{\omega=w_{1}w_{2}\cdots|w_{i}\in\{0,1,\cdots,m-1\}\}. $$

Let $\omega =w_{1}w_{2}{\cdots } \in {\Sigma }_{m}^{+}$ and $a\leq b \in \mathbb {Z}^{+}$. We write ω|_{[a, b]} = w_aw_a+ 1⋯w_b− 1w_b and $\overline {\omega |_{[a,b]}}=w_{b}{\cdots } w_{a}$.

For $\omega =w_{1}w_{2}{\cdots } \in {\Sigma }_{m}^{+}$, the ω-orbit of x ∈ X under the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ is the sequence $\{f_{\omega }^{n}(x)\}_{n=0}^{\infty }$, where

$$ f_{\omega}^{n}(x)=f_{w_{n}}\circ f_{w_{n-1}}\circ{\cdots} f_{w_{1}}(x) \quad {\text{and}} \quad f_{\omega}^{0}(x)=f_{e}(x)\equiv x. $$

It is obvious that

$$ f_{\omega}^{n}(x)=f_{\overline{\omega|_{[1,n]}}}(x)=f_{w}(x), $$

where $w=w_{n}w_{n-1}{\cdots } w_{1}=\overline {\omega |_{[1,n]}}$. Specially, we denote e = ω|_[1,0].

For $k\in \mathbb {Z}^{+}$, let

$$ D_{k}=\{\omega_{k}=w_{1}w_{2}{\cdots} w_{k}: 0\leq w_{i}\leq m-1,1\leq i\leq k\}. $$

For ω_k = w₁w₂⋯w_k ∈ D_k, the ω_k-orbit of x ∈ X under the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ is the finite sequence $\{f_{\omega _{k}}^{n}(x)\}_{n=0}^{k}$.

Definition 3.1

For δ > 0, a sequence $\tau =\{x_{i}\}_{i=0}^{\infty }\subseteq X$ is called a δ-pseudo orbit for the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ if there exists $\omega =w_{1}w_{2}\cdots \in {\Sigma }_{m}^{+}$, such that

$$ d(f_{w_{i+1}}(x_{i}),x_{i+1})<\delta $$

(3.1)

for all $i\in \mathbb {N}$, where ω is called a compatible element on τ.

Denote by $S(\tau )=\left \{\omega \in {\Sigma }_{m}^{+}:\omega {\text {is a compatible element on} \tau }\right \}$.

It is obvious that τ is called a δ-pseudo orbit, if S(τ)≠∅.

Definition 3.2

A δ-pseudo orbit $\tau =\{x_{i}\}_{i=0}^{\infty }$ is ε-chain shadowed by some point of X, if for any ω ∈ S(τ), there exists z_ω ∈ X, such that

$$ d\left( {f_{w}^{i}}(z_{\omega}),x_{i} \right)<\varepsilon \quad {\text{for all}} \quad i\in\mathbb{N}. $$

Definition 3.3

The free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the chain shadowing property if for any ε > 0, there exists δ > 0, such that every δ-pseudo orbit can be ε-chain shadowed by some point of X.

Definition 3.4

For δ > 0, a finite sequence $\tau _{r}=\{x_{i}\}_{i=0}^{r}$ is called a finite δ-pseudo orbit for the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ provided that there exists ω_r = w₁w₂⋯w_r ∈ D_r, such that

$$ d(f_{w_{i+1}}(x_{i}),x_{i+1})<\delta $$

(3.2)

for all 0 ≤ i ≤ r − 1.

Similarly, denote by S(τ_r) = {ω_r ∈ D_r : ω_ris a compatible element onτ_r}.

Definition 3.5

A finite δ-pseudo orbit $\tau _{r}=\{x_{i}\}_{i=0}^{r}$ is ε-chain shadowed by some point of X, if for any ω_r ∈ S(τ_r) there exists $z_{\omega _{r}}\in X$, such that

$$ d\left( f_{\omega_{r}}^{i}(z_{\omega_{r}}),x_{i}\right)<\varepsilon \quad {\text{for all}} \quad 0\leq i\leq r. $$

Definition 3.6

The free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the finite chain shadowing property if for any ε > 0, there exists δ > 0, such that for any $r\in \mathbb {N}$ the finite δ-pseudo orbit τ_r can be ε-chain shadowed by some point of X.

First we give a classic theorem about compact space for the proof of our first result.

Definition 3.7

A collection $\mathcal {C}$ of subsets of X is said to have the finite intersection property if for every finite subcollection {C₁,…,C_n} of $\mathcal {C}$, the intersection C₁ ∩… ∩ C_n is nonempty.

Lemma 3.8

[18, Theorem 26.9] The topological space X is compact space if and only if for every collection $\mathcal {C}$ of closed sets in X having the finite intersection property, then the intersection $\bigcap \limits _{C\in \mathcal {C}}C$ is nonempty.

Next, we give some results about the chain shadowing property for a free semigroup action.

Theorem 3.9

The free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the chain shadowing property if and only if it has the finite chain shadowing property.

Proof

Suppose the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the chain shadowing property. Let ε > 0 and δ > 0 given by the chain shadowing property for $(F_{m}^{+},\mathcal {F})\curvearrowright X$. For any $r\in \mathbb {N}$, let the finite sequence $\tau _{r}=\{x_{i}\}_{i=0}^{r}$ be a finite δ-pseudo orbit, that means S(τ_r)≠∅. For any ω_r = w₁w₂⋯w_r ∈ S(τ_r), we can extend τ_r into an infinite sequence $\tau =\{x_{i}\}_{i=0}^{\infty }$ by the following way:

$$ f_{0}(x_{i})=x_{i+1} \quad {\text{for all}} \quad i\geq r. $$

Then $\tau =\{x_{i}\}_{i=0}^{\infty }$ is a δ-pseudo orbit. Since $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the chain shadowing property, for the ω = w₁w₂⋯w_r00⋯ ∈ S(τ), there exists z_ω ∈ X, such that

$$ d\left( f_{\omega}^{i}(z_{\omega}),x_{i}\right)<\varepsilon \quad {\text{for all}} \quad i\in\mathbb{N}. $$

And ω|_[1,r] = ω_r, then

$$ d\left( f_{\omega}^{i}(z_{\omega}\right),x_{i})<\varepsilon \quad {\text{for all}} \quad 0\leq i\leq r. $$

Therefore, the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the finite chain shadowing property.

In the converse direction, since $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the finite chain shadowing property. For any ε > 0 there exists δ > 0 such that every finite δ-pseudo orbit is a $\frac {\varepsilon }{2}$-chain shadowed by some point in X. Assume that an infinite sequence $\tau =\{x_{i}\}_{i=0}^{\infty }$ is δ-pseudo orbit. For any ω ∈ S(τ) and any $k\in \mathbb {N}$, note ω_k = ω|_[1,k]. Then ω_k is a compatible element on τ_k, where $\tau _{k}= \{x_{i}\}_{i=0}^{k}$. By the finite chain shadowing property, we know there exists $z_{\omega _{k}}$, such that

$$ d\left( f_{\omega_{k}}^{i}(z_{\omega_{k}}),x_{i}\right)<\frac{\varepsilon}{2} \quad {\text{for all}} \quad 0\leq i\leq k. $$

It is obvious that $f_{\omega _{k}}^{i}(z_{\omega _{k}})\in B\left (x_{i},\frac {\varepsilon }{2}\right )$, then

$$ z_{\omega_{k}} \in \left( f_{\omega_{k}}^{i}\right)^{-1}\left( B\left( x_{i},\frac{\varepsilon}{2}\right)\right) , $$

that means

$$ z_{\omega_{k}}\in \overline{\left( f_{\omega_{k}}^{i}\right)^{-1}\left( B\left( x_{i},\frac{\varepsilon}{2}\right)\right)} \quad {\text{for all}} \quad 0\leq i\leq k. $$

And ω_k = ω|_[1,k], we have

$$ \bigcap\limits_{i=0}^{k}\overline{\left( f_{\omega}^{i}\right)^{-1}\left( B\left( x_{i},\frac{\varepsilon}{2}\right)\right)}\neq\emptyset. $$

By Lemma 3.8 and the arbitrary of k, we have

$$ \bigcap\limits_{i\in \mathbb{N}}\overline{\left( f_{\omega}^{i}\right)^{-1}\left( B\left( x_{i},\frac{\varepsilon}{2}\right)\right)}\neq\emptyset. $$

Hence, there exists z ∈ X,

$$ z\in \bigcap\limits_{i\in \mathbb{N}}\overline{\left( f_{\omega}^{i}\right)^{-1}\left( B\left( x_{i},\frac{\varepsilon}{2}\right)\right)}, $$

then

$$ z\in \left( f_{\omega}^{i}\right)^{-1}\left( B\left( x_{i},\frac{\varepsilon}{2}\right)\right), \quad {\text { i.e.,}} \quad f_{\omega}^{i}(z)\in B(x_{i},\varepsilon) \quad {\text{for all}} \quad i\in \mathbb{N}. $$

Thus for any δ-pseudo orbit $\tau =\{x_{i}\}_{i=0}^{\infty }$ in X and any ω ∈ S(τ). There exists z ∈ X, such that

$$ d(f_{\omega}^{i}(z),x_{i})<\varepsilon $$

for all $i\in \mathbb {N}$, which implies $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the chain shadowing property. □

Theorem 3.10

Let $(F_{m}^{+},\mathcal {F})\curvearrowright X$ be the free semigroup action and Y be a dense invariant subset of X, Then the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the finite chain shadowing property if and only if the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright Y$ has the finite chain shadowing property.

Proof

Assume $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the finite chain shadowing property. Let ε > 0 and choose δ > 0 such that every δ-pseudo orbit in X is $\frac {\varepsilon }{2}$-chain shadowed. Let $\tau _{r}=\{y_{i}\}_{i=0}^{r}$ be a finite δ-pseudo orbit in Y. Then τ_r is a finite δ-pseudo orbit in X. Since $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the finite chain shadowing property, then for any ω_r = w₁w₂⋯w_r ∈ S(τ_r) there exists $z_{\omega _{r}}\in X$, such that

$$ d\left( f_{\omega_{r}}^{i}(z_{\omega_{r}}),y_{i}\right)<\frac{\varepsilon}{2} $$

for all 0 ≤ i ≤ r, which implies

$$ f_{\omega_{r}}^{i}(z_{\omega_{r}})\in B\left( y_{i},\frac{\varepsilon}{2}\right). $$

Then

$$ z_{\omega_{r}}\in \left( f_{\omega_{r}}^{i}\right)^{-1}\left( B\left( y_{i},\frac{\varepsilon}{2}\right)\right) $$

for all 0 ≤ i ≤ r, that means,

$$ \bigcap\limits_{0\leq i\leq r}\left( f_{\omega_{r}}^{i}\right)^{-1}\left( B\left( y_{i},\frac{\varepsilon}{2}\right)\right)\neq \emptyset. $$

For $(F_{m}^{+},\mathcal {F})\curvearrowright X$ is continuous action, thus

$$ \bigcap\limits_{0\leq i\leq r}\left( f_{\omega_{r}}^{i}\right)^{-1}\left( B\left( y_{i},\frac{\varepsilon}{2}\right)\right)\cap Y\neq \emptyset. $$

That means there exists z ∈ Y and

$$ z\in \bigcap\limits_{0\leq i\leq r}\left( f_{\omega_{r}}^{i}\right)^{-1}\left( B\left( y_{i},\frac{\varepsilon}{2}\right)\right) \quad {\text{i.e.,}} \quad f_{\omega_{r}}^{i}(z)\in B(y_{i},\varepsilon). $$

Thus, for any finite δ-pseudo orbit τ_r in Y, and any ω_r ∈ S(τ_r) there exists z ∈ Y, such that

$$ d\left( f_{\omega_{r}}^{i}(z),y_{i}\right)<\varepsilon $$

for all 0 ≤ i ≤ r, which means $(F_{m}^{+},\mathcal {F})\curvearrowright Y$ has the finite chain shadowing property.

In the converse direction, we assume $(F_{m}^{+},\mathcal {F})\curvearrowright Y$ has the finite chain shadowing property. Let ε > 0 and $\tau ^{\prime }_{r}=\{x_{i}\}_{i=0}^{r}$ be a finite $\frac {\delta }{3}$-pseudo orbit in X, where δ > 0 is given by the finite chain shadowing property in $(F_{m}^{+},\mathcal {F})\curvearrowright Y$ for $\frac {\varepsilon }{2}$. Since $f_{i}\in \mathcal {F}$ is a continuous map and X is compact, then $f_{i}\in \mathcal {F}$ is a uniformly continuous map for any i ∈{0,1⋯m − 1}. That means for $\frac {\delta }{3}>0$ there exists η > 0 with $\eta <\frac {\delta }{3}$ and $\eta <\frac {\varepsilon }{2}$ for any y ∈ B(x, η), we have

$$ d(f_{i}(x),f_{i}(y))<\frac{\delta}{3} \quad {\text{for all }} \quad 0\leq i \leq m-1. $$

For any $\omega _{r}=w_{1}w_{2}{\cdots } w_{r} \in S(\tau ^{\prime }_{r})$, we can construct a finite sequence $\tau _{r}^{\ast }=\{y_{i}\}_{i=0}^{r}$ in Y such that ω_r is compatible on $\tau _{r}^{\ast }$.

For any $x_{i} \in \tau ^{\prime }_{r}$, we take y_i ∈ B(x_i,η) ∩ Y, then

$$ d(f_{w_{i+1}}(x_{i}),f_{w_{i+1}}(y_{i}))<\frac{\delta}{3}. $$

Therefore

$$ \begin{aligned} d(f_{w_{i+1}}(y_{i}),y_{i+1})&<d(f_{w_{i+1}}(y_{i}),f_{w_{i+1}}(x_{i}))+d(f_{w_{i+1}}(x_{i}),x_{i+1}) +d(x_{i+1},y_{i+1})\\ &<\frac{\delta}{3}+\frac{\delta}{3}+\frac{\delta}{3}=\delta. \end{aligned} $$

Hence $\tau _{r}^{\ast }$ is δ-pseudo orbit in Y and $\omega _{r} \in S(\tau _{r}^{\ast })$. Since $(F_{m}^{+},\mathcal {F})\curvearrowright Y$ has the finite chain shadowing property, there exists $y_{\omega _{r}}\in Y$, such that

$$ d\left( f_{\omega_{r}}^{i}(y_{\omega_{r}}),y_{i}\right)<\frac{\varepsilon}{2} \quad {\text{for all }} \quad 0\leq i \leq r. $$

Then

$$ \begin{aligned} \left( f_{\omega_{r}}^{i}(y_{\omega_{r}}),x_{i}\right)&<d\left( f_{\omega_{r}}^{i}(y_{\omega_{r}}),y_{i}\right)+d(y_{i},x_{i})\\ &<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon, \end{aligned} $$

for any 0 ≤ i ≤ r. Therefore, the $\frac {\delta }{3}$-pseudo orbit $\tau =\{x_{i}\}_{i=0}^{r}$ can be ε-chain shadowed by some point in X.

Hence $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the finite chain shadowing property. □

Theorem 3.11

If the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$ has the chain shadowing property, then $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the chain shadowing property.

Proof

Suppose $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$ has the chain shadowing property. Let ε > 0 and choose δ > 0 be given by the chain shadowing property for $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$. Let a sequence $\tau =\{x_{i}\}_{i=0}^{\infty }$ be a δ-pseudo orbit for $(F_{m}^{+},\mathcal {F})\curvearrowright X$, that means, S(τ)≠∅. For any ω = w₁w₂⋯ ∈ S(τ), we have

$$ d(f_{w_{i+1}}(x_{i}),x_{i+1})<\delta, $$

for all $i\in \mathbb {N}$, that means,

$$ d_{H}(f_{w_{i+1}}(\{x_{i}\}),\{x_{i+1}\})<\delta \quad {\text{for all }} \quad i\in\mathbb{N}. $$

(3.3)

Then $\tau ^{\ast }=\{\{x_{i}\}\}_{i=0}^{\infty }$ is a δ-pseudo orbit for $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$. Since $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$ has the chain shadowing property, and ω ∈ S(τ^∗) there exists A_ω ∈ 2^X, we have

$$ d_{H}\left( f_{\omega}^{i}(A_{\omega}),\{x_{i}\} \right)<\varepsilon \quad {\text{for all }} \quad i\in\mathbb{N}. $$

Then

$$ f_{\omega}^{i}(A_{\omega})\in B(\{x_{i}\},\varepsilon) \quad {\text{for all }} \quad i\in\mathbb{N}. $$

That is,

$$ d\left( f_{\omega}^{i}(z),x_{i}\right)<\varepsilon \quad {\text{for all }} \quad z\in A_{\omega}, i\in\mathbb{N}. $$

Thus $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the chain shadowing property. □

Theorem 3.12

If the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the finite chain shadowing property, then $(F_{m}^{+},\mathcal {F})\curvearrowright F(X)$ has the finite chain shadowing property.

Proof

Let ε > 0 and choose δ > 0 be given by the finite chain shadowing property for $(F_{m}^{+},\mathcal {F})\curvearrowright X$. Let $\tau =\{A_{i}\}_{i=0}^{r}$ be a finite δ-pseudo orbit in F(X). Suppose $A_{i}={\left \{{x_{i}^{1}},{x_{i}^{2}},\cdots ,x_{i}^{n_{i}}\right \}}$ and ♯A_i = n_i, for each 1 ≤ i ≤ r.

Claim. There exist a family of δ-pseudo orbits in X, denoted by {τ_j : j ≤ n}, where $\tau _{j}={\left ({x_{0}^{j}},{x_{1}^{j}},\cdots ,{x_{r}^{j}}\right )}$, for some $n\in \mathbb {N}$, such that,

$$ A_{r}=\left\{{x_{r}^{j}}:j\leq n\right\} \quad {\text{for each}} \quad 0\leq i\leq r. $$

We first construct a δ-pseudo orbit in X. Since $\tau =\{A_{i}\}_{i=0}^{r}$ is a finite δ-pseudo orbit, for any ω_r = (w₁w₂⋯w_r) ∈ D_r, we have

$$ d_{H}(f_{w_{i}}(A_{i-1}),A_{i})<\delta \quad {\text{for any}} \quad 1\leq i\leq r. $$

For any ${x^{j}_{r}}\in A_{r}$, we can chose $x_{r-1}^{j}\in A_{r-1}$, such that

$$ d\left( f_{w_{r}}\left( x_{r-1}^{j}\right),{x_{r}^{j}}\right)<\delta. $$

Again, there exists $x_{r-2}^{j}\in A_{r-2}$, such that

$$ d\left( f_{w_{r-1}}\left( x_{r-2}^{j}\right),x_{r-1}^{j}\right)<\delta. $$

By induction, we have a family of δ-pseudo orbits $\tau _{j}={\left ({x_{0}^{j}},{x_{1}^{j}},\cdots ,{x_{r}^{j}}\right )}$ for each 1 ≤ j ≤ n_r such that $A_{r}=\left \{{x_{r}^{j}}:j\leq n_{r}\right \}$ and $\left \{{x_{i}^{j}}:j\leq n_{r}\right \}\subseteq A_{i}$ for each i ≤ r − 1.

Let $k=max\left \{i<r:\left \{{x_{i}^{j}}:j\leq n_{r}\right \}\neq A_{i}\right \}$ (if no such k exists, then we are done), and write $A_{k}-\left \{{x_{k}^{j}}:j\leq n_{r}\right \}=\left \{{x_{k}^{j}}:n_{r}<j\leq n_{k}^{\prime }\right \}$. Continuing in above way, there exists $x_{k-1}^{j}\in A_{k-1}$, such that

$$ d\left( f_{w_{k}}\left( x_{k-1}^{j}\right),{x_{k}^{j}}\right)<\delta \quad {\text{for each}} \quad n_{r}<j\leq n_{k}^{\prime}. $$

Thus, we can construct a δ-pseudo orbit $\tau _{j}^{\prime }={\left ({x_{0}^{j}}, {x_{1}^{j}},\cdots ,{x_{k}^{j}}\right )}$ where ${x_{i}^{j}}\in A_{i}$, i ≤ k. Now, since $f_{w_{k+1}}\left ({x_{k}^{j}}\right )\in f_{w_{k+1}}(A_{k})$ and $d_{H}(f_{w_{k+1}}(A_{k}),A_{k+1})<\delta$, then there exists $x_{k+1}^{j}\in A_{k+1}$ such that

$$ d\left( f_{w_{k+1}}\left( {x_{k}^{j}}\right),x_{k+1}^{j}\right)<\delta. $$

Similarly, for each $n_{r}<j\leq n_{k}^{\prime }$, and k ≤ i < r, there exists $x_{i+1}^{j}\in A_{i+1}$ such that

$$ d\left( f_{w_{i+1}}\left( {x_{i}^{j}}\right),x_{i+1}^{j}\right)<\delta. $$

Thus, we can extend $\tau _{j}^{\prime }$ to a δ-pseudo orbit $\tau _{j}={\left ({x_{0}^{j}},{x_{1}^{j}},\cdots ,{x_{r}^{j}}\right )}$ for each $n_{r}<j\leq n_{k}^{\prime }$.

Repeating this process, we can construct the collection $\left \{\tau _{j}=\left \{{x_{i}^{j}}\right \}_{i=0}^{r}:j\leq n\right \}$ of δ-pseudo orbits in X, satisfying $\bigcup \limits _{j} \left \{{x_{i}^{j}}\right \}= A_{i}$, for any 0 ≤ i ≤ r. We complete the claim.

Since $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the finite chain shadowing property, for each $\tau _{j}=\left \{{x_{i}^{j}}\right \}_{i=0}^{r}$ and ω_r = (w₁w₂⋯ ) ∈ S(τ_j), there exists a point z_j ∈ X such that

$$ d\left( f_{\omega_{r}}^{i}(z_{j}),{x_{i}^{j}}\right)<\varepsilon \quad {\text{for any}} \quad 0\leq i\leq r. $$

Let B = {z₁,z₂,⋯ ,z_j : j ≤ n}∈ F(X). By construction we have

$$ d_{H}\left( f_{\omega_{r}}^{i}(B),A_{i}\right)<\varepsilon \quad {\text{for any}} \quad 0\leq i\leq r. $$

Consequently, $(F_{m}^{+},\mathcal {F})\curvearrowright F(X)$ has the finite chain shadowing property. □

Theorem 3.13

The free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the chain shadowing property if and only if $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$ has the chain shadowing property.

Proof

By Theorem 3.11, we have if the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$ has the chain shadowing property, then $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the chain shadowing property. Conversely, if the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the chain shadowing property, then $(F_{m}^{+},\mathcal {F})\curvearrowright F(X)$ has the finite chain shadowing property by Theorem 3.12, and F(x) is an invariant dense subset of 2^X, so by Theorem 3.10, $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$ has the chain shadowing property. □

Next, we give an example of a free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ with the chain shadowing property.

Example 3.14

Define the two continuous maps f₀,f₁ on ${\Sigma }(2)=\{0,1\}^{\mathbb {N}}$, as follows:

$$ f_{0}(x)=(x_{2}x_{3}x_{4}\cdots);\quad f_{1}(x)=(x_{3}x_{4}x_{5}\cdots), $$

where x = (x₀x₁x₂⋯ ) ∈Σ(2), the metric

$$ d(x,y)=1/2^{\delta(x,y)} \quad {\text{where}} \quad \delta(x,y)=\min\{s:s\in \mathbb{N} \quad {\text{and}} \quad x_{s}\neq y_{s}\}. $$

The free semigroup action $(F_{2}^{+},\mathcal {F})\curvearrowright {\Sigma }(2)$ generated by $\mathcal {F}=\{f_{0},f_{1}\}$.

Next we show $(F_{2}^{+},\mathcal {F})\curvearrowright {\Sigma }(2)$ has the chain shadowing property.

For any ε > 0, choose $\delta =\frac {1}{2^{N}}<\varepsilon$, where N is a positive integer. Let $\tau =\{x^{i}\}_{i=0}^{\infty }$ be a δ-pseudo orbit for $(F_{2}^{+},\mathcal {F})\curvearrowright {\Sigma }(2)$, and ω = w₁w₂⋯ ∈ S(τ). We take $z_{\omega }=(x^{0}|_{[0,1+w_{1}]}x^{1}|_{[0,1+w_{2}]}x^{2}|_{[0,1+w_{3}]}\cdots )$. We show that z_ω satisfies

$$ d\left( f^{i}_{\omega}(z_{\omega}),x^{i}\right)<\varepsilon \quad {\text{for all } } \quad i\in \mathbb{N}. $$

By the definition of compatible element ω on τ, we have

$$ d(f_{w_{i+1}}(x_{i}),x_{i+1})<\delta \quad \ i\in \mathbb{N}, $$

that means

$$ x^{i}|_{[2+w_{i+1},N+2+w_{i+1}]}=x^{i+1}|_{[0,N]}. $$

(3.4)

For any $i\in \mathbb {N}$, we have

$$ f^{i}_{\omega}(z_{\omega})=\left( x^{i}_{[0,1+w_{i+1}]}x^{i+1}_{[0,1+w_{i+2}]}\cdots\right). $$

Without loss of generality, we suppose there exists $r_{i} \in \mathbb {N}$ such that

$$ N=2+w_{i+1}+(1+w_{i+2})+{\cdots} +(1+w_{i+r_{i}}). $$

(3.5)

Then

$$ x^{i}|_{[0,N]}=x^{i}|_{[0,1+w_{i+1}]}x^{i}|_{[2+w_{i+1},N]}. $$

(3.6)

By (3.4) and (3.5), we have

$$ x^{i}|_{[2+w_{i+1},N]}=x^{i+1}|_{\left[0,(1+w_{i+2})+{\cdots} +(1+w_{i+r_{i}}) \right]}. $$

Again

$$ x^{i+1}|_{\left[0,(1+w_{i+2})+{\cdots} +(1+w_{i+r_{i}}) \right]}=x^{i+1}|_{[0,1+w_{i+2}]}x^{i+1}|_{\left[2+w_{i+2},(1+w_{i+2})+{\cdots} +(1+w_{i+r_{i}})\right]}. $$

By (3.4), we have

$$ x^{i+1}|_{\left[2+w_{i+2},(1+w_{i+2})+{\cdots} +\left( 1+w_{i+r_{i}}\right)\right]}=x^{i+2}|_{\left[0,(1+w_{i+3})+{\cdots} +\left( 1+w_{i+r_{i}}\right) \right]}. $$

Again and again, until

$$ x^{i+r_{i} -2}|_{\left[0,\left( 1+w_{i+r_{i} -1}\right)+\left( 1+w_{i+r_{i}}\right) \right]}=x^{i+r_{i} -2}|_{\left[0,1+w_{i+r_{i} -1}\right]}x^{i+r_{i} -2}|_{\left[2+w_{i+r_{i} -1},(1+w_{i+r_{i} -1})+(1+w_{i+r_{i}}) \right]}. $$

By (3.4), we have

$$ x^{i+r_{i} -2}|_{\left[2+w_{i+r_{i} -1},(1+w_{i+r_{i} -1})+(1+w_{i+r_{i}}) \right]}=x^{i+r_{i} -1}|_{\left[0,1+w_{i+r_{i}} \right]}. $$

Hence

$$ x^{i}|_{[0,N]}=x^{i}|_{[0,1+w_{i+1}]}x^{i+1}|_{[0,1+w_{i+2}]}{\cdots} x^{i+r_{i} -1}|_{[0,1+w_{i+r_{i}} ]}. $$

Therefore, for any $i\in \mathbb {N}$,

$$ d\left( f^{i}_{\omega}\left( z_{\omega}\right),x^{i}\right)\leq \delta <\varepsilon, $$

which implies $(F_{2}^{+},\mathcal {F})\curvearrowright {\Sigma }(2)$ has the chain shadowing property.

4 The Shadowing Property

In this section, we introduce a notation of shadowing property for a free semigroup action, which is consistent with the definition of shadowing property of finitely generated group action in [21].

Definition 4.1

For δ > 0, a sequence $\{x_{w}\}_{w\in F_{m}^{+}}\subseteq X$ is called a δ-pseudo orbit for the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ provided that

$$ d(f_{j}(x_{w}),x_{jw})<\delta \quad {\text {for any}} \quad f_{j}\in \mathcal{F}, \ w\in F_{m}^{+}. $$

Definition 4.2

For δ > 0, a finite sequence $\{x_{w}\}_{\shortmid w\shortmid \leq n }\subseteq X$ where $w\in F_{m}^{+}$ is called a finite δ-pseudo orbit for the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ provided that

$$ d(f_{j}(x_{w}),x_{jw})<\delta \quad {\text {for any}} \quad f_{j}\in \mathcal{F}, \ w\in F_{m}^{+} \quad {\text {with}} \quad \shortmid w\shortmid \leq n-1. $$

Definition 4.3

The free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the shadowing property if for any ε > 0, there exists δ > 0, such that for any δ-pseudo orbit $\{x_{w}\}_{w\in F_{m}^{+}}$ there exists a point z_e ∈ X such that

$$ d(f_{w}(z_{e}),x_{w})<\varepsilon \quad {\text {for any}} \quad w\in F_{m}^{+}. $$

Definition 4.4

The free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the finite shadowing property if for any ε > 0, there exists δ > 0, such that for any finite δ-pseudo orbit $\{x_{w}\}_{\shortmid w\shortmid \leq n}$ there exists a point z_e ∈ X such that

$$ d(f_{w}(z_{e}),x_{w})<\varepsilon \quad {\text {for any}} \quad \shortmid w\shortmid \leq n. $$

Theorem 4.5

The free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the shadowing property if and only if it has the finite shadowing property.

Proof

Suppose the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the shadowing property. Let ε > 0 and δ > 0 be given by the shadowing property for $(F_{m}^{+},\mathcal {F})\curvearrowright X$. For any $k \in \mathbb {N}$ let $\tau = \{x_{w}\}_{\shortmid w\shortmid \leq k}$ be a finite δ-pseudo orbit. We extend it into a sequence $\{x_{w}\}_{w\in F_{m}^{+}}$ by the following way

$$ f_{w^{\prime}}(x_{w})=x_{w^{\prime}w} \quad {\text {for any}} \quad \shortmid w\shortmid = k \quad {\text {and}} \quad w^{\prime}\in F_{m}^{+}. $$

Then $\{x_{w}\}_{w\in F_{m}^{+}}$ is a δ-pseudo orbit for $(F_{m}^{+},\mathcal {F})\curvearrowright X$, and $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the shadowing property, that means, there exists z_e ∈ X, satisfying

$$ d(f_{w}(z_{e}),x_{w})<\varepsilon \quad {\text {for every}} \quad w\in F_{m}^{+}. $$

Thus,

$$ d(f_{w}(z_{e}),x_{w})<\varepsilon \quad {\text {for any}} \quad \shortmid w\shortmid \leq k. $$

In the converse direction, suppose $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the finite shadowing. For any ε > 0, we take δ > 0 depend on $\frac {\varepsilon }{2}$. For any δ-pseudo orbit $\{x_{w}\}_{w\in F_{m}^{+}}$ and any $n\in \mathbb {N}$, we take the subsequence $\{x_{w}\}_{\shortmid w\shortmid \leq n}\subseteq \{x_{w}\}_{w\in F_{m}^{+}}$. By the definition of finite shadowing property, we know there exists ${z_{e}^{n}}\in X$, such that

$$ d\left( f_{w}\left( {z_{e}^{n}}\right),x_{w}\right)<\frac{\varepsilon}{2} \quad {\text {for any}} \quad \shortmid w\shortmid \leq n. $$

It is obvious that $f_{w}({z_{e}^{n}})\in B(x_{w},\varepsilon )$, then

$$ {z_{e}^{n}}\in f_{w}^{-1}\left( B\left( x_{w},\frac{\varepsilon}{2}\right)\right) \quad {\text {for any}} \quad \shortmid w\shortmid \leq n. $$

That means

$$ {z_{e}^{n}}\in \overline{f_{w}^{-1}\left( B\left( x_{w},\frac{\varepsilon}{2}\right)\right)}. $$

Therefore, for any finite positive integer $n\in \mathbb {N}$

$$ \bigcap\limits_{j=0}^{n}\overline{f_{w}^{-1}\left( B\left( x_{w},\frac{\varepsilon}{2}\right)\right)}\neq\emptyset,\quad \shortmid w\shortmid \leq n $$

By Lemma 3.8, we have

$$ \bigcap\limits_{w\in F_{m}^{+}}\overline{f_{\omega}^{-1}\left( B\left( x_{w},\frac{\varepsilon}{2}\right)\right)}\neq\emptyset. $$

That is, there exists z_e ∈ X, such that,

$$ z_{e}\in \bigcap\limits_{w\in F_{m}^{+}}\overline{f_{\omega}^{-1}\left( B\left( x_{w},\frac{\varepsilon}{2}\right)\right)} \quad {\text{i.e.,}} \quad f_{\omega}(z_{e})\in B(x_{w},\varepsilon). $$

Consequently, for any δ-pseudo orbit $\{x_{w}\}_{w\in F_{m}^{+}}$, we can find some z_e such that

$$ d(f_{w}(z_{e}),x_{w})<\varepsilon \quad {\text {for any}} \quad w\in F_{m}^{+}. $$

That means $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the shadowing property. □

Next to prove a general result about shadowing.

Theorem 4.6

Let $(F_{m}^{+},\mathcal {F})\curvearrowright X$ be the free semigroup action, and Y is a dense invariant subset of X. Then the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the finite shadowing property if and only if the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright Y$ has the finite shadowing property.

Proof

Assume $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the shadowing property. Let ε > 0 and choose δ > 0 such that for every finite δ-pseudo orbit in X is $\frac {\varepsilon }{2}$-shadowed. Let $\{x_{w}\}_{\shortmid w\shortmid \leq n}$ be a finite δ-pseudo orbit in Y. Then $\{x_{w}\}_{\shortmid w\shortmid \leq n}$ is also a finite δ-pseudo orbit in X. Since $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the finite shadowing property, there exists $z_{e}^{\prime }\in X$ satisfying

$$ d(f_{w}(z_{e}^{\prime}),x_{w})<\frac{\varepsilon}{2} \quad {\text {for any}} \quad \shortmid w\shortmid \leq n. $$

It is obvious that $f_{w}(z_{e}^{\prime })\in B\left (x_{w},\frac {\varepsilon }{2}\right )$, then

$$ z_{e}^{\prime}\in f_{w}^{-1}\left( B\left( x_{w},\frac{\varepsilon}{2}\right)\right) \quad {\text {for any}} \quad \shortmid w\shortmid \leq n. $$

That means

$$ \bigcap\limits_{\shortmid w\shortmid \leq n}f_{w}^{-1}\left( B\left( x_{w},\frac{\varepsilon}{2}\right)\right)\neq\emptyset. $$

For $F_{m}^{+}$ is finite generated free semigroup then the members with $\shortmid w\shortmid \leq n$ is finite. Thus

$$ \bigcap\limits_{\shortmid w\shortmid \leq n}f_{w}^{-1}\left( B\left( x_{w},\frac{\varepsilon}{2}\right)\right) $$

is nonempty open subset of X. That means

$$ \bigcap\limits_{\shortmid w\shortmid \leq n}f_{w}^{-1}\left( B\left( x_{w},\frac{\varepsilon}{2}\right)\right) \bigcap Y \neq\emptyset. $$

Therefore, there exists z_e ∈ Y, such that,

$$ f_{w}(z_{e})\in B(x_{w},\varepsilon)\quad {\text {for any}} \quad \shortmid w\shortmid \leq n. $$

By combing the process above, we know for any δ-pseudo orbit $\{x_{w}\}_{\shortmid w\shortmid \leq n}$ in Y, we can find some z_e ∈ Y such that

$$ d(f_{w}(z_{e}),x_{w})<\varepsilon \quad {\text {for any}} \quad \shortmid w\shortmid \leq n. $$

That means $(F_{m}^{+},\mathcal {F})\curvearrowright Y$ has the finite shadowing property.

In the converse direction, suppose $(F_{m}^{+},\mathcal {F})\curvearrowright Y$ has the finite shadowing. Let ε > 0 and δ > 0 given by $\frac {\varepsilon }{2}$-shadowing property for $(F_{m}^{+},\mathcal {F})\curvearrowright Y$ and $\{x_{w}\}_{\shortmid w\shortmid \leq n}$ is $\frac {\delta }{3}$-pseudo orbit in X. Since $(F_{m}^{+},\mathcal {F})\curvearrowright X$ is a continuous action and X is compact, then $(F_{m}^{+},\mathcal {F})\curvearrowright X$ is a uniformly continuous action that means for any $\frac {\delta }{3}>0$ there exists a η > 0 with $\eta <\min \limits \{\frac {\delta }{3},\frac {\varepsilon }{2}\}$, for each y_w ∈ B(x_w,η) ∩ Y we have

$$ d(f_{j}(y_{w}),f_{j}(x_{w}))<\frac{\delta}{3} \quad {\text {for any}} \quad f_{j}\in \mathcal{F}. $$

Hence $\{y_{w}\}_{\shortmid w\shortmid \leq n}$ is a δ-pseudo orbit in Y, for

$$ \begin{aligned} d(f_{j}(y_{w}),y_{jw})&<d(f_{j}(y_{w}),f_{j}(x_{w}))+d(f_{j}(x_{w}),x_{jw}) +d(x_{jw},y_{jw})\\ &<\frac{\delta}{3}+\frac{\delta}{3}+\frac{\delta}{3}=\delta. \end{aligned} $$

Since $(F_{m}^{+},\mathcal {F})\curvearrowright Y$ has the shadowing property, there is a point z_e ∈ Y such that

$$ d(f_{w}(z_{e}),y_{w})<\frac{\varepsilon}{2} \quad {\text {for any}} \quad \shortmid w\shortmid \leq n. $$

Then

$$ \begin{aligned} d(f_{w}(z_{e}),x_{w})&<d(f_{w}(z_{e}),y_{w})+d(y_{w},x_{w})\\ &<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon. \end{aligned} $$

Therefore, for any $\frac {\delta }{3}$-pseudo orbit $\{x_{w}\}_{\shortmid w\shortmid \leq n}$ in X, there is a point $z_{e}\in Y\subseteq X$ satisfying

$$ d(f_{w}(z_{e}),x_{w})<\varepsilon \quad {\text {for any}} \quad \shortmid w\shortmid \leq n. $$

Hence the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the finite shadowing property. □

Theorem 4.7

If the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the finite shadowing property, then the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright F(X)$ has the finite shadowing property.

Proof

For any ε > 0 and choose δ > 0 be given by the shadowing property for $(F_{m}^{+},\mathcal {F})\curvearrowright X$. For any finite δ-pseudo orbit $\{A_{w}\}_{\shortmid w\shortmid \leq n}$ in F(X), we have

$$ d_{H}(f_{j}(A_{w}),A_{jw})<\delta \quad {\text {for all}} \quad f_{j}\in \mathcal{F}, \ w\in F_{m}^{+} \quad {\text {with}} \quad \shortmid w\shortmid \leq n-1. $$

That means, for any x_w ∈ A_w there exists x_jw ∈ A_jw such that,

$$ d(f_{j}(x_{w}),x_{jw})<\delta \quad {\text {for all}} \quad f_{j}\in \mathcal{F}, \ w\in F_{m}^{+} \quad {\text {with}} \quad \shortmid w\shortmid \leq n-1. $$

(4.1)

Similarly, for any y_jw ∈ A_jw there exists y_w ∈ A_w such that,

$$ d(f_{j}(y_{w}),y_{jw})<\delta \quad {\text {for all}} \quad f_{j}\in \mathcal{F}, \ w\in F_{m}^{+} \quad {\text {with}} \quad \shortmid w\shortmid \leq n-1. $$

(4.2)

We claim that for any given $\shortmid w^{\prime }\shortmid \leq n$ and any $x^{\prime }\in A_{w^{\prime }}$, we can construct a finite δ-pseudo orbit $\{x_{w}\}_{\shortmid w\shortmid \leq n}$ in X with x_w ∈ A_w and $x_{w^{\prime }}=x^{\prime }$, by the following way.

Case 1. HCode $\shortmid w^{\prime }\shortmid =0$

That means $x^{\prime }\in A_{e}$, by (4.1) we can find x_i ∈ A_i such that

$$ d(f_{i}(x^{\prime}),x_{i})<\delta \quad {\text {for all}} \quad f_{i}\in \mathcal{F}. $$

Again, by (4.1) we can find x_ji ∈ A_ji such that

$$ d(f_{j}(x_{i}),x_{ji})<\delta \quad {\text {for all}} \quad f_{j}\in \mathcal{F}. $$

Hence by induction, for any $w\in F_{m}^{+}$ with $\shortmid w\shortmid \leq n-1$ where x_w ∈ A_w, we can find x_iw ∈ A_iw such that

$$ d(f_{i}(x_{w}),x_{iw})<\delta \quad {\text {for all}} \quad f_{i}\in \mathcal{F}. $$

Case 2. HCode $\shortmid w^{\prime }\shortmid >0$.

We can assume $w^{\prime }=w^{\prime }_{r}w^{\prime }_{r-1}{\cdots } w^{\prime }_{1}$ where 0 < r ≤ n. Then for any $x^{\prime }\in A_{w^{\prime }}$ by (4.2), we can find $x_{w^{r-1}}\in A_{w^{r-1}}$, where $w^{r-1}=w^{\prime }_{r-1}w^{\prime }_{r-2}{\cdots } w^{\prime }_{1}$, satisfying

$$ d(f_{w^{\prime}_{r}}(x_{w^{r-1}}),x_{w^{\prime}_{r}w^{r-1}})=d(f_{w^{\prime}_{r}}(x_{w^{r-1}}),x_{w^{\prime}})<\delta. $$

For $x_{w^{r-1}},$ by (4.2) again we can find $x_{w^{r-2}}\in A_{w^{r-2}}$, where $w^{r-2}=w^{\prime }_{r-2}w^{\prime }_{r-3}{\cdots } w^{\prime }_{1}$, such that

$$ d(f_{w^{\prime}_{r-1}}(x_{w^{r-2}}),x_{w^{\prime}_{r-1}w^{r-2}})=d(f_{w^{\prime}_{r-1}}(x_{w^{r-2}}),x_{w^{r-1}})<\delta. $$

Now applying (4.2) again and again until there exists x_e ∈ A_e such that

$$ d(f_{w^{1}}(x_{e}),x_{w^{1}})<\delta. $$

For x_e, we repeat the process of Case 1. In particular, for the process of taking $x_{w^{i}}$, we take the values obtained in the above process, for every 0 ≤ i ≤ r. Thus the above method establish our claim.

Then, for each x_w ∈ A_w, we can find a finite δ-pseudo orbit and A_w ∈ F(X), that means the number of all x_w ∈ A_w with $\shortmid w\shortmid \leq n$ is finite. Without loss of generality, we suppose there exists $m\in \mathbb {N}$ such that each sequence $\left \{{x^{j}_{w}}\right \}_{\shortmid w\shortmid \leq n }$ is finite δ-pseudo orbit satisfying $A_{w}=\left \{{x_{w}^{j}}:j=1,2,\cdots ,m\right \}$ for every $\shortmid w\shortmid \leq n$. For each finite δ-pseudo orbit $\left \{{x^{j}_{w}}\right \}_{\shortmid w\shortmid \leq n }$, by the shadowing property of the action $(F_{m}^{+},\mathcal {F})\curvearrowright X$, we can find ${z^{j}_{e}}\in X$, such that

$$ d\left( f_{w}\left( {z^{j}_{e}}\right),{x^{j}_{w}}\right)<\varepsilon \quad {\text {for any}} \quad \shortmid w\shortmid \leq n . $$

Let $Z=\left \{{z^{j}_{e}}:j=1,2,\cdots ,m\right \}$, we say Z is ε-shadowing $\{A_{w}\}_{\shortmid w\shortmid \leq n}$.

For any x_w ∈ A_w, there exists positive integer 1 ≤ j ≤ m, such that $x_{w}={x_{w}^{j}}$ and ${z_{e}^{j}}$ satisfying

$$ d\left( f_{w}\left( {z^{j}_{e}}\right),{x^{j}_{w}}\right)<\varepsilon. $$

That means

$$ A_{w}\subseteq B(f_{w}(Z),\varepsilon). $$

Similarly,

$$ f_{w}(Z)\subseteq B(A_{w},\varepsilon). $$

Hence, we completed the whole proof. □

Theorem 4.8

The free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the shadowing property if and only if the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$ has the shadowing property.

Proof

Suppose $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$ has the shadowing property. Let ε > 0, and choose δ be given by shadowing for $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$. For any δ-pseudo orbit $\{x_{w}\}_{w\in F_{m}^{+}}$ in X, we can turn it into $\{\{x_{w}\}\}_{w\in F_{m}^{+}}$, which is a δ-pseudo orbit in 2^X. Since $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$ has the shadowing property, then there exists a point Z ∈ 2^X, such that

$$ d_{H}(f_{w}(Z),\{x_{w}\})<\varepsilon \quad {\text {for any}} \quad w\in F_{m}^{+}. $$

Then

$$ f_{w}(Z)\subseteq B(x_{w},\varepsilon) \quad {\text {for any}} \quad w\in F_{m}^{+}. $$

That is, every point z ∈ Z is ε-shadowing $\{x_{w}\}_{w\in F_{m}^{+}}$.

Conversely, if $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the shadowing property, then $(F_{m}^{+},\mathcal {F})\curvearrowright F(X)$ has the finite shadowing by Theorem 4.7. And F(X) is an invariant dense subset of 2^X, by Theorem 4.5 and Theorem 4.6, thus $(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$ also has the shadowing property. □

Naturally, we want to know the direct relationship between these two shadowing property definitions.

Corollary 4.9

If the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the shadowing property, then it has the chain shadowing property.

The following example shows that the converse to Corollary 4.9 is not true.

Example 4.10

By Example 3.14, we know the free semigroup action $(F_{2}^{+},\mathcal {F})\curvearrowright {\Sigma }(2)$ has the chain shadowing property. We take $\varepsilon _{0}= \frac {1}{4}$ for any δ > 0. Without loss of generality, we suppose $\delta =\frac {1}{2^{n}}$, where n is a positive integer and n ≥ 2. Let $\tau ^{\prime }=\{{x^{1}_{i}}\}^{\infty }_{i=0}$ where ${x_{0}^{1}}=(00{\cdots } 00\cdots )\in {\Sigma }(2)$.

$$ {x_{i}^{1}}=(\underbrace{00 {\cdots} 0}\limits_{n+3-2i}\underbrace{ 1{\cdots} 1 }\limits_{2i}00\cdots)\in {\Sigma}_(2), \ \ 1\leq i\leq \left\lfloor\frac{n+3}{2}\right\rfloor $$

and

$$ {x^{1}_{i}}=x^{1}_{i+1}=(\underbrace{11 {\cdots} 1}\limits_{n+3}00 \cdots) \quad \text{ for all} \quad i\geq \left\lceil \frac{n+3}{2}\right\rceil. $$

(4.3)

Therefore $\tau ^{\prime }$ is a δ-pseudo orbit for $\omega ^{\prime }=00{\cdots } \in S(\tau ^{\prime })$.

Let $\tau ^{\prime \prime }=\left \{{x^{2}_{i}}\right \}^{\infty }_{i=0}$ where ${x_{0}^{2}}=(00{\cdots } 00\cdots )\in {\Sigma }(2)$

$$ {x_{i}^{2}}=(\underbrace{00 {\cdots} 0}\limits_{n+4-3i}\underbrace{ 100100{\cdots} 100 }\limits_{3i}00\cdots)\in {\Sigma}_{2}, \ \ 1\leq i\leq \left\lfloor\frac{n+4}{3}\right\rfloor $$

and

$$ {x^{2}_{i}}=x^{2}_{i+1}=(\underbrace{s100 {\cdots} 100}\limits_{n+4}00 \cdots) \quad \text{ for all} \quad i\geq \lfloor\frac{n+4}{3}\rfloor. $$

(4.4)

Where s = 0 when n + 4 ≡ 1(mod 3), s = 00 when n + 4 ≡ 2(mod 3), specially, if n + 4 ≡ 0(mod 3) we note s = 100.

Therefore $\tau ^{\prime \prime }$ is a δ-pseudo orbit for $\omega ^{\prime \prime }=11{\cdots } \in S(\tau ^{\prime \prime })$.

Next, we construct a δ-pseudo orbit $\tau =\{x_{w}\}_{w\in F_{2}^{+}}$ for $F_{2}^{+}$ by the following way.

If $w=00\cdots 00\in F_{2}^{+}$, then we take $x_{w}=x^{1}_{|w|}$; if $w=11\cdots 11\in F_{2}^{+}$ then we take $x_{w}=x^{2}_{|w|}$. For others $w\in F_{2}^{+}$ there exists a positive integer m such that w = w_mw_m− 1⋯w₁. We consider the smallest i such that w_i≠w_i+ 1, then note $w^{\prime }=w_{i}w_{i-1}{\cdots } w_{1}$.

If $w^{\prime }=00\cdots 0$, then we take

$$ x_{w}=f_{w_{m}}\circ f_{w_{m-1}}\circ\cdots\circ f_{w_{i+1}}\left( {x_{i}^{1}}\right). $$

If $w^{\prime }=11\cdots 1$ then we take

$$ x_{w}=f_{w_{m}}\circ f_{w_{m-1}}\circ\cdots\circ f_{w_{i+1}}\left( {x_{i}^{2}}\right). $$

Suppose there exists z ∈Σ(2) satisfying z ε-shadows τ, that means

$$ d(f_{w}(z),x_{w})<\varepsilon \quad {\text {for any}} \quad w\in F_{2}^{+}. $$

Therefore, for any $\omega \in {\Sigma }_{m}^{+}$, if there exists a sequence $\overline {\tau }=\{y_{w^{i}}\}^{\infty }_{i=0}\subseteq \tau$ where $w^{i}=\overline {\omega |_{[1,i]}}$ such that $\omega \in S(\overline {\tau })$, we have

$$ d(f_{\omega}^{i}(z),y_{w^{i}})<\varepsilon \quad {\text {for any}}\quad i\in \mathbb{N}. $$

For $\omega ^{\prime }$ and $\omega ^{\prime \prime }$, we have

$$ d\left( f_{\omega^{\prime}}^{n}(z),{x_{n}^{1}}\right)<\varepsilon_{0} \quad {\text {and}} \quad d\left( f_{\omega^{\prime\prime}}^{m}(z),{x_{m}^{2}}\right)<\varepsilon_{0} \quad {\text {for any}}\quad n,m\in \mathbb{N}. $$

(4.5)

By the construction of ${x_{n}^{1}}$, ${x_{m}^{2}}$ and the values of $\varepsilon _{0}=\frac {1}{4}$, we can express (4.5) into

$$ \left( f_{\omega^{\prime}}^{n}(z)\right)|_{[0,2]}=\left( {x_{n}^{1}}\right)|_{[0,2]} \quad {\text {and}} \quad \left( f_{\omega^{\prime\prime}}^{m}(z)\right)|_{[0,2]}=\left( {x_{m}^{2}}\right)|_{[0,2]} \quad {\text {for any}}\quad n,m\in \mathbb{N}. $$

(4.6)

Let $N=\max \limits \left \{\left \lceil \frac {n+3}{2}\right \rceil ,\left \lfloor \frac {n+4}{3}\right \rfloor \right \}$. By (4.3) and (4.6), we have

$$ \left( f_{\omega^{\prime}}^{3N}(z)\right)|_{[0,2]}=\left( x_{3N}^{1}\right)|_{[0,2]}=111 \quad {\text {and}} \quad \left( f_{\omega^{\prime\prime}}^{2N}(z)\right)|_{[0,2]}=\left( x_{2N}^{2}\right)|_{[0,2]}. $$

By the definitions of f₀ and f₁, we know $f_{\omega ^{\prime }}^{3N}(z)=f_{0}^{3N}(z)=f_{1}^{2N}(z)=f_{\omega ^{\prime \prime }}^{2N}(z)$. But, by (4.3) and (4.4) we realize that $\left (x_{3N}^{1}\right )|_{[0,2]}\neq \left (x_{2N}^{2}\right )|_{[0,2]}$. That is a contradiction.

Take a look at the process. There exists $\varepsilon _{0}=\frac {1}{4}$. For any δ > 0, we can construct a δ-pseudo orbit $\tau =\{x_{w}\}_{w\in F_{2}^{+}}$. For any z ∈Σ(2), z can’t ε-shadows τ, that means $(F_{2}^{+},\mathcal {F})\curvearrowright {\Sigma }(2)$ doesn’t have the shadowing property.

Next, we give examples with both the chain shadowing property and the shadowing property.

Example 4.11

Define two continuous maps g₀,g₁ on Σ(2) as follows:

$$ g_{0}(x)=(0x_{0}x_{1}x_{2}\cdots);\quad g_{1}(x)=(1x_{0}x_{1}x_{2}\cdots), $$

where x = (x₀x₁x₂⋯ ) ∈Σ(2).

The free semigroup action $(G_{2}^{+},\mathcal {G})\curvearrowright {\Sigma }(2)$ generated by $\mathcal {G}=\{g_{0},g_{1}\}$. Then the free semigroup action $(G_{2}^{+},\mathcal {G})\curvearrowright {\Sigma }(2)$ has the chain shadowing property [3, Example 1.2].

Next we show $(G_{2}^{+},\mathcal {G})\curvearrowright {\Sigma }(2)$ has the shadowing property. For any $\varepsilon =\frac {1}{2^{n}}>0$, choose $\delta =\frac {1}{2^{n+1}}<\varepsilon$. For any δ-pseudo orbit $\{x_{w}\}_{w\in G^{+}_{2}}$ for $(G_{2}^{+},\mathcal {G})\curvearrowright {\Sigma }(2)$, i.e.,

$$ d(g_{w_{i}}x_{w},x_{w_{i}w})<\delta \quad {\text {for any}} \quad g_{w_{i}}\in \mathcal{G}, \ w\in G^{+}_{2}. $$

(4.7)

By the definitions of g₀,g₁, for any x₁,x₂ ∈Σ(2), if d(x₁,x₂) < δ then

$$ d(g_{w_{i}}(x_{1}),g_{w_{i}}(x_{2}))<\frac{\delta}{2} \quad \text{for all} \quad g_{w_{i}}\in \mathcal{G}_{2}. $$

(4.8)

Consider z = x_e. For any $w\in G_{2}^{+}$, write w = w_mw_m− 1⋯w₁, by (4.7) and (4.8) we have

$$ \begin{aligned} d(g_{w}(z),x_{w})&=d\left( g_{w_{m}}\left( g_{w_{m-1}^{\prime}}(z)\right),x_{w}\right)\\ &<d\left( g_{w_{m}}\left( g_{w_{m-1}^{\prime}}(z)\right),g_{w_{m}}\left( x_{w_{m-1}^{\prime}}\right)\right)+d\left( g_{w_{m}}\left( x_{w_{m-1}^{\prime}}\right),x_{w}\right)\\ &<\frac{d\left( g_{w_{m-1}^{\prime}}(z),x_{w_{m-1}^{\prime}}\right)}{2}+\delta\\ &<\frac{1}{2}\left( d\!\left( g_{w_{m-1}}\!\left( g_{w_{m-2}^{\prime}}(z)\!\right)\!,g_{w_{m-1}}\!\left( x_{w_{m-2}^{\prime}}\!\right)\!\right) + d\!\left( \!g_{w_{m-1}}\!\!\left( x_{w_{m-2}^{\prime}}\right)\!,x_{w_{m-1}^{\prime}}\right)\!\right)+\delta\\ &<\frac{1}{2}\left( \frac{d\left( g_{w_{m-2}^{\prime}}(z),x_{w_{m-2}^{\prime}}\right)}{2}+\delta\right)+\delta\\ &=\frac{d\left( g_{w_{m-2}^{\prime}}(z),x_{w_{m-2}^{\prime}}\right)}{4}+\frac{\delta}{2}+\delta\\ & \quad \quad \quad \quad \quad \quad \vdots\\ &<\frac{d\left( g_{w_{1}^{\prime}}(z),x_{w_{1}^{\prime}}\right)}{2^{m-1}}+\frac{\delta}{2^{m-2}}+\cdots+\frac{\delta}{2}+\delta\\ &<\frac{\delta}{2^{m-1}}+\frac{\delta}{2^{m-2}}+\cdots+\frac{\delta}{2}+\delta\\ &<2\delta<\varepsilon, \end{aligned} $$

where $w_{i}^{\prime }<w$ and $\shortmid w_{i}^{\prime }\shortmid =i$. Thus $\{x_{w}\}_{w\in G^{+}_{2}}$ is ε-shadowed by z.

Consequently, we shown $(G_{2}^{+},\mathcal {G})\curvearrowright {\Sigma }(2)$ has the shadowing property.

In [13], authors constructed an action in S¹ whose minimal set K is a Cantor set and has the shadowing property. And Barzanouni in [1] presented an example to illustrate the shadowing property depends on the metric on non-compact spaces.

Remark 4.12

For any free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ with the discrete metric, we have that the free semigroup action $(F_{m}^{+},\mathcal {F})\curvearrowright X$ has the shadowing property.

Example 4.13

The free semigroup action $(G_{2}^{+},\mathcal {G})\curvearrowright 2^{\Sigma (2)}$, which inducted by Example 4.11, has the chain shadowing property and the shadowing property.

The proof process is similar to the above example, so we will only briefly describe it.

For any A₁,A₂ ∈ 2^Σ(2),δ > 0 if d_H(A₁,A₂) < δ, then

$$ d_{H}(g_{w_{i}}(A_{1}),g_{w_{i}}(A_{2}))<\frac{\delta}{2} \quad \text{for all} \quad g_{w_{i}}\in \mathcal{G}. $$

Similarly, let Z = A_e, that is what we want.