Equivalent Condition of the Measure Shadowing Property on Metric Spaces

Abstract

The concept referred to as the measure shadowing property for a dynamical system on compact metric space has recently been introduced, acting as an extension of the classical shadowing property by using the property of the Borel measures on the phase space. In this paper, we extend the concept of the measure shadowing property of continuous flows from compact metric spaces to the general metric spaces and demonstrate the equivalence relation between the measure shadowing property and the shadowing property for flows on metric spaces via the shadowable points.

1. Introduction

The theory of the shadowing property (also known as the pseudo-orbit tracing property) plays an important role in the qualitative theory of dynamical systems. Initially conceptualized by Bowen [1] in 1975, it has since been studied by numerous researchers, in works such as [2,3], among others. Broadly speaking, this theory states that any sequence of sections of orbits (referred to as a pseudo-orbit) with sufficiently small jumps can be approximated by a real orbit of some point on the phase space. Since the orbit of the nearby systems can be regarded as the pseudo-orbit of the original system, there is an intrinsic relationship between the shadowing property and the stability of the dynamics. The shadowing property has become a hot research topic as a result of its connection to the notion of hyperbolicity, which was first introduced by S. Smale [4]. Hyperbolic dynamical systems represent a significant milestone in the dynamical systems theory. In previous decades, hyperbolicity has proven to be an important topic and a huge source of systems with rich dynamical behavior. Hyperbolic systems are known to possess the shadowing property.

There are several ways to define the shadowing property of a system on compact metric spaces. Thomas [5] proposed extending the shadowing property to flows and proved that every continuous expansive flow without fixed points which possesses the shadowing property is topologically stable. In Morales’ study [6], the definition of shadowing for homeomorphisms was generalized by splitting the shadowing property into pointwise shadowings, giving rise to the concept of shadowable points, and the relations were studied among the set of all shadowable points and the shadowing property. The notions of the shadowing property for Borel measures of homeomorphisms were introduced by Lee and Morales in [7], and they proved that every expansive measure that possesses the shadowing property is topologically stable. This represented a measurable version of the classical Walters’ stability theorem on compact metric spaces in [8]. Ahn, Kim and Lee [9] introduced the measure shadowing property of flows on compact metric spaces and discussed the relationship between the measure shadowing property for homeomorphisms and their corresponding suspension flows. Recently, Lee and Nguyen [10] introduced invariantly measure expanding and proved that flows on compact metric spaces is invariantly measure expanding on its chain recurrent set and has the invariantly measure shadowing property on its chain recurrent set then the flows have the spectral decomposition.

Regarding the dynamics of noncompact systems, a well-known difficulty is that some dynamical properties (i.e., properties that are invariant under conjugacy) on compact metric spaces may not be dynamical properties on noncompact metric spaces. For example, in [11], the authors provided an example to demonstrate that the classical shadowing property for flows on noncompact metric spaces depends on the choice of metrics; moreover, they introduced the notion of the shadowing property of continuous flows on metric spaces and showed that the extended definition is a dynamical property.

In this paper, we generalize the notion of the measure shadowing property of the continuous flows from compact metric spaces to the general metric spaces, which is independent of the choice of metrics. Using the property of the shadowable points, we can obtain an equivalent relation among the measure shadowing property and the shadowing property of the continuous flows on metric spaces.

2. Basic Definitions

In this section, we introduce the concept of the shadowing property for a continuous flow on a metric space from a measure theoretic viewpoint. We begin by recalling the corresponding definition for compact metric spaces.

Let $(X,d)$ be a compact metric space. A $flow$ on X is a continuous map $\varphi :X\times \mathbb{R}⟶X$ satisfying $\varphi (x,0)=x$ and $\varphi \left(\varphi \right(x,s),t)=\varphi (x,s+t)$ for $x\in X$ and $s,t\in \mathbb{R}$. For convenience, we write

$$\varphi (x,s)={\varphi }_s\left(x\right)and{\varphi }_{(a,b)}\left(x\right)=\{{\varphi }_t\left(x\right):t\in (a,b)\}.$$

The set ${\mathcal{O}}_{\varphi }\left(x\right)={\varphi }_{\mathbb{R}}\left(x\right)$ is known as the $orbit$ of $\varphi$ through $x\in X$.

We state that a sequence $\{(x_i,t_i):x_i\in X,t_i\ge 1,-\infty \le a\le i\le b\le \infty \}$ is a $(\delta ,1)$-pseudo orbit of $\varphi$ if for any $a\le i\le b-1$,

$$d({\varphi }_{t_i}\left(x_i\right),x_{i+1})\le \delta .$$

We state that a $(\delta ,1)$-pseudo orbit ${\left\{(x_i,t_i)\right\}}_{i=a}^b$ is $\epsilon$-$shadowed$ by $y\in X$ if

$$d({\varphi }_{\tau (y,t)}\left(y\right),{\varphi }_{t-T_i}\left(x_i\right))\le \epsilon ,$$

for a continuous map $\tau :X\times \mathbb{R}\to \mathbb{R}$ such that $\tau (x,·):\mathbb{R}\to \mathbb{R}$ is a strictly increasing homeomorphism with $\tau (x,0)=0$ and for all $T_i\le t<T_{i+1}$ with $a\le i\le b-1$, where

$$T_i=\left\{\begin{array}{cc}{t}_0+t_1+\cdots +t_{i-1}\hfill & \mathrm{for}i>0,\hfill \\ 0\hfill & \mathrm{for}i=0,\hfill \\ -(t_{-1}+t_{-2}+\cdots +t_i)\hfill & \mathrm{for}i<0.\hfill \end{array}\right.$$

We state that a flow $\varphi$ has the shadowing property if, for any $\epsilon >0$, there is $\delta >0$ satisfying any $(\delta ,1)$-pseudo orbit is $\epsilon$-shadowed by $y\in X$.

Here, $\tau$ is referred to as a time reparametrization and $\tau$ satisfies $\tau (x,s+t)=\tau ({\varphi }_{\tau (x,t)}\left(x\right),s)+\tau (x,t)$.

Before we recall the definition of the shadowing property for Borel measures of a continuous flow on a compact metric space X in [9], let us introduce some notations. For any subset $B\subset X$, we write $\mu \left(B\right)=0$ if $\mu \left(A\right)=0$ for every Borel set $A\subset B$. Given $x\in X$, we say that a sequence $\{(x_i,t_i):-\infty \le a\le i\le b\le \infty \}$ of $X\times \mathbb{R}$ is $through$ x if $x_0=x$, and we say that a sequence $\{(x_i,t_i):-\infty \le a\le i\le b\le \infty \}$ is $through$ B if $x_0\in B$.

Let $\mathcal{M}\left(X\right)$ be the set of all Borel probability measures $\mu$ on X. Given a Borel measure $\mu \in \mathcal{M}\left(X\right)$, a flow $\varphi$ has the $\mu$-shadowing property if, for any $\epsilon >0$, we have $\delta >0$, a Borelian B with $\mu (X\setminus B)=0$, and a continuous map $\tau :X\times \mathbb{R}\to \mathbb{R}$ with the following properties: $\tau (x,0)=0,\tau (x,·):\mathbb{R}\to \mathbb{R}$ is a strictly increasing homeomorphism for all $x\in X$ and any $(\delta ,1)$-pseudo orbit through a point in B can be $\epsilon$-shadowed by a point $y\in X$. We state that $\varphi$ has the measure shadowing property if $\varphi$ has the $\mu$-shadowing property for all $\mu \in \mathcal{M}\left(X\right)$.

Next, we recall the definition of the shadowing property for a continuous flow on a metric space X in [11]. We begin by introducing the setting in metric space.

Consider a closed subset A of X, we use ${\mathcal{C}}_A\left(X\right)$ to denote the collection of all continuous functions $\epsilon :X\to [0,\infty )$ such that $\epsilon \left(x\right)=0$ if and only if $x\in A$. We use the notion of 0-ball centered at $x\in X$, $B(x,0)$, for one point set $\left\{x\right\}$. Denote $Fix\left(\varphi \right)=\{x\in X:{\varphi }_t\left(x\right)=x,\forall t\in \mathbb{R}\}$ for the collection of all fixed points of $\varphi$. For simplicity, we write ${\mathcal{C}}_{\varphi }\left(X\right)$ and $\mathcal{C}\left(X\right)$ if $A=Fix\left(\varphi \right)$ and A is empty, respectively. For $\epsilon ,\delta \in {\mathcal{C}}_A\left(X\right)$, we denote $\epsilon \le \delta$ (resp. $\epsilon <\delta$) whenever $\epsilon \left(x\right)\le \delta \left(x\right)$ (resp. $\epsilon \left(x\right)<\delta \left(x\right)$) for all $x\in X$.

For a given $\delta \in {\mathcal{C}}_{\varphi }\left(X\right)$, we state that a sequence $\{(x_i,t_i):x_i\in X,t_i\ge 1,-\infty \le a\le i\le b\le \infty \}$ is a $(\delta ,1)$-pseudo orbit of $\varphi$ if for any $a\le i\le b-1$,

$$d({\varphi }_{t_i}\left(x_i\right),x_{i+1})\le \delta \left({\varphi }_{t_i}\left(x_i\right)\right).$$

We state that a $(\delta ,1)$-pseudo orbit $\{(x_i,t_i):x_i\in X,t_i\ge 1,-\infty \le a\le i\le b\le \infty \}$ is $\epsilon$-shadowed ($\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right)$) by $y\in X$ if

$$d({\varphi }_{\tau (y,t)}\left(y\right),{\varphi }_{t-T_i}\left(x_i\right))\le \epsilon \left({\varphi }_{\tau (y,t)}\left(y\right)\right),$$

for a continuous map $\tau :X\times \mathbb{R}\to \mathbb{R}$ such that $\tau (x,·):\mathbb{R}\to \mathbb{R}$ is a strictly increasing homeomorphism with $\tau (x,0)=0$ and for all $T_i\le t<T_{i+1}$, where $T_i$ is given as before. Finally, we state that a flow $\varphi$ on a metric space X has the shadowing property if for any $\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right)$, there is $\delta \in {\mathcal{C}}_{\varphi }\left(X\right)$, such that any $(\delta ,1)$-pseudo orbit of $\varphi$ can be $\epsilon$-shadowed by a point in X. Note that this definition does not depend on the choice of metric d.

After that, we extend the notions of the shadowing property for Borel measures of a continuous flow from compact metric spaces to the metric spaces.

Definition 1.

Given a Borel measure $\mu \in \mathcal{M}\left(X\right)$, a flow ϕ has the μ-shadowing property if for any $\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right)$ there are $\delta \in {\mathcal{C}}_{\varphi }\left(X\right)$, a Borelian B with $\mu (X\setminus B)=0$, and a continuous map $\tau :X\times \mathbb{R}\to \mathbb{R}$ with the following properties: $\tau (x,0)=0,\tau (x,·):\mathbb{R}\to \mathbb{R}$ is a strictly increasing homeomorphism for all $x\in X$ and any $(\delta ,1)$-pseudo orbit through a point in B can be ε-shadowed by a point $y\in X$. We state that ϕ has the measure shadowing property if ϕ has the μ-shadowing property for all $\mu \in \mathcal{M}\left(X\right)$.

Note that the above definition does not depend on the choice of metric d. Recall that two metrics d and $d^{\prime }$ are said to be equivalent on a toplogical space $(X,\mathcal{T})$ if both d and $d^{\prime }$ induce the topology $\mathcal{T}$ of X. See the following lemma for further details.

Lemma 1.

The measure shadowing property is independent of the choice of equivalent metrics.

Proof. 

Let $\varphi$ be a flow on metric spaces $(X,d)$ while $d^{\prime }$ is the equivalent metric of d on X. Suppose that for any $\mu \in \mathcal{M}\left(X\right)$, $\varphi$ possesses the $\mu$-shadowing property with respect to d and $\varphi$ has the $\mu$-shadowing property with respect to $d^{\prime }$. For any $\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right)$, according to Lemma 2, let ${\epsilon }^{\prime }\in {\mathcal{C}}_{\varphi }\left(X\right)$, such that $d(x,y)\le {\epsilon }^{\prime }\left(x\right)$ implies $d^{\prime }(x,y)\le \epsilon \left(x\right)$. Take ${\delta }^{\prime }\in {\mathcal{C}}_{\varphi }\left(X\right)$ corresponding to ${\epsilon }^{\prime }$ based on the $\mu$-shadowing property of $\varphi$ with respect to d. Let $\delta \in {\mathcal{C}}_{\varphi }\left(X\right)$ be such that $d^{\prime }(x,y)\le \delta \left(x\right)$ implies $d(x,y)\le {\delta }^{\prime }\left(x\right)$. Let ${\left\{(x_i,t_i)\right\}}_{i\in \mathbb{Z}}$ be a $(\delta ,1)$-pseudo orbit of $\varphi$ with respect to $d^{\prime }$. Based on the choice of $\delta$, we find that

$$\begin{array}{c}\hfill d({\varphi }_{t_i}(\left(x_i\right),x_{i+1})\le {\delta }^{\prime }\left({\varphi }_{t_i}\left(x_i\right)\right)\end{array}$$

for all $i\in \mathbb{Z}$, and so ${\left\{(x_i,t_i)\right\}}_{i\in \mathbb{Z}}$ is a $({\delta }^{\prime },1)$-pseudo orbit of $\varphi$ with respect to d. According to the $\mu$-shadowing property of $\varphi$ with respect to d, there is a Borelian B with $\mu (X\setminus B)=0$, and a continuous map $\tau :X\times \mathbb{R}\to \mathbb{R}$ with the following properties: $\tau (x,0)=0,\tau (x,·):\mathbb{R}\to \mathbb{R}$ is a strictly increasing homeomorphism, such that

$$d({\varphi }_{\tau (x,t)}\left(x\right),{\varphi }_{t-T_i}\left(x_i\right))\le {\epsilon }^{\prime }\left({\varphi }_{\tau (x,t)}\left(x\right)\right)$$

for all $T_i\le t<T_{i+1}$ with $i\in \mathbb{Z}$, where $T_i$ is defined as stated previously. Then,

$$\begin{array}{c}\hfill d^{\prime }({\varphi }_{\tau (x,t)}\left(x\right),{\varphi }_{t-T_i}\left(x_i\right))\le \epsilon \left({\varphi }_{\tau (x,t)}\left(x\right)\right)\end{array}$$

for all $T_i\le t<T_{i+1}$ and $i\in \mathbb{Z}$. This demonstrates that $\varphi$ possesses the $\mu$-shadowing property with respect to $d^{\prime }$. Since $\mu$ is arbitrary, we find that $\varphi$ possesses the measure shadowing property with respect to $d^{\prime }$. □

In [5], Thomas proved that, for every $T>0$, a flow has the shadowing property (i.e., with respect to time 1) if and only if it has the shadowing property with respect to time T, and if and only if for every $\epsilon >0$ we can find $\delta >0$, such that every $(\delta ,T,2T)$-pseudo orbit can be $\epsilon$-shadowed in a compact metric space. Indeed, for the metric space case, we obtained the parallel result, and the definition of the measure shadowing property with respect to time $T>0$ is given as follows.

We state that a sequence $\{(x_i,t_i):x_i\in X,t_i\ge T,-\infty \le a\le i\le b\le \infty \}$ is a $(\delta ,T)$-pseudo orbit of $\varphi$ if for any $a\le i\le b-1$,

$$d({\varphi }_{t_i}\left(x_i\right),x_{i+1})\le \delta \left({\varphi }_{t_i}\left(x_i\right)\right).$$

Meanwhile, a sequence of pairs ${\left\{(x_i,t_i)\right\}}_{i\in \mathbb{Z}}$ is a $(\delta ,T_1,T_2)$-pseudo orbit of $\varphi$ if it is a $(\delta ,T_1)$-pseudo orbit of $\varphi$ and satisfies $t_i\le T_2$, for all $i\in \mathbb{Z}$.

A $(\delta ,T)$-pseudo orbit $\{(x_i,t_i):x_i\in X,t_i\ge T,-\infty \le a\le i\le b\le \infty \}$ is referred to as $\epsilon$-shadowed ($\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right)$) by $y\in X$ if

$$d({\varphi }_{\tau (y,t)}\left(y\right),{\varphi }_{t-T_i}\left(x_i\right))\le \epsilon \left({\varphi }_{\tau (y,t)}\left(y\right)\right),$$

for a continuous map $\tau :X\times \mathbb{R}\to \mathbb{R}$, such that $\tau (x,·):\mathbb{R}\to \mathbb{R}$ is a strictly increasing homeomorphism with $\tau (x,0)=0$ and for all $T_i\le t<T_{i+1}$, where $T_i$ is as stated previously. We also state that a flow $\varphi$ on a metric space X has the shadowing property with respect to time $T>0$ if, for any $\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right)$, $\delta \in {\mathcal{C}}_{\varphi }\left(X\right)$, such that any $(\delta ,T)$-pseudo orbit of $\varphi$ can be $\epsilon$-shadowed by a point in X.

Next, we will extend the notion of the shadowing property to Borel measures for a continuous flow on a metric space X.

Definition 2.

Given a Borel measure $\mu \in \mathcal{M}\left(X\right)$, a flow ϕ has the μ-shadowing property with respect to time T if for any $\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right)$ there are $\delta \in {\mathcal{C}}_{\varphi }\left(X\right)$, a Borelian B with $\mu (X\setminus B)=0$, and a continuous map $\tau :X\times \mathbb{R}\to \mathbb{R}$ with the following properties: $\tau (x,0)=0,\tau (x,·):\mathbb{R}\to \mathbb{R}$ is a strictly increasing homeomorphism for all $x\in X$ and any $(\delta ,T)$-pseudo orbit through a point in B can be ε-shadowed by a point $y\in X$. We state that ϕ has the measure shadowing property with respect to time T if ϕ has the μ-shadowing property with respect to time T for all $\mu \in \mathcal{M}\left(X\right)$.

3. Main Theorems

Proposition 1.

Let $T>0$ and ϕ be a flow on a metric space X. Then, the following statements are equivalent.

1. 

For all $\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right)$, we found that $\delta \in {\mathcal{C}}_{\varphi }\left(X\right)$, such that every $(\delta ,T,2T)$-pseudo-orbit is ε-shadowed by an orbit of ϕ;

2. 

ϕ possesses the shadowing property with respect to time T;

3. 

ϕ possesses the shadowing property.

To prove the above proposition, we need to recall the following lemmas in [11].

Lemma 2.

Let $(X,d_X)$ and $(Y,d_Y)$ represent two metric spaces, while f represents a continuous map from X to Y. Then, for any closed subset A of Y and $\epsilon \in {\mathcal{C}}_A\left(Y\right)$, we have $\delta \in {\mathcal{C}}_{f^{-1}\left(A\right)}\left(X\right)$, such that if $d(x,y)\le \delta \left(x\right)(x,y\in X)$, then $d\left(f\right(x),f(y\left)\right)\le \epsilon \left(f\right(x\left)\right)$.

Lemma 3.

For any $\alpha \in {\mathcal{C}}_{\varphi }\left(X\right)$, we have $\gamma \in {\mathcal{C}}_{\varphi }\left(X\right)$, such that $\gamma \left(x\right)\le \inf \left\{\alpha \right(y):y\in B(x,\gamma \left(x\right)\left)\right\}$ for all $x\in X$. Moreover, if $d(x,y)\le \max \left\{\gamma \right(x),\gamma (y\left)\right\}(x,y\in X)$, then $d(x,y)\le \alpha \left(x\right)$.

For simplicity, we write $\gamma \ll \alpha (\gamma ,\alpha \in {\mathcal{C}}_{\varphi }\left(X\right))$ if $\gamma \left(x\right)\le inf\left\{\alpha \right(y):y\in B(x,\gamma \left(x\right)\left)\right\}$ for any $x\in X$.

Proof of Proposition 1. 

We assume that $T>1$. For the other case (when $T<1$), a similar argument can be used.

Suppose that for all $\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right)$, we have $\delta \in {\mathcal{C}}_{\varphi }\left(X\right)$, such that every $(\delta ,T,2T)$ pseudo-orbit is $\epsilon$-shadowed by an orbit of $\varphi$. Let $\epsilon \ll {\epsilon }^{\prime }$. We begin by proving that $\varphi$ possesses the shadowing property with respect to time T.

Let ${\left\{(x_i,t_i)\right\}}_{i\in \mathbb{Z}}$ be any $(\delta ,T)$-pseudo-orbit of $\varphi$. For each $n\in \mathbb{Z}$, we have $m_n\in \mathbb{N}$, such that $t_n=m_{n}T+r_n$ with $T\le r_n<2T$. Let ${\left(s_n^m\right)}_{n\in \mathbb{Z}}$ the sequence of sums associated with $m={\left(m_n\right)}_{n\in \mathbb{Z}}$. Then, we denote $A_n=s_n^m+n$ for all $n\in \mathbb{Z}$ and define the sequence ${\left(y_i\right)}_{i\in \mathbb{Z}}$ on X, such that $y_i={\varphi }_{T\left(i-A_n\right)}\left(x_n\right)$ if $A_n\le i<A_{n+1}$. In addition, we define a sequence $\lambda ={\left({\lambda }_i\right)}_{i\in \mathbb{Z}}$ of real numbers as follows. For each $i\in \mathbb{Z}$, we set

$${\lambda }_i=\left\{\begin{array}{cc}T& \mathrm{if}{A}_n\le i<A_{n+1}-1,\hfill \\ r_n& \mathrm{if}i=A_{n+1}-1.\hfill \end{array}\right.$$

Given $i\in \mathbb{Z}$, we note that $T\le {\lambda }_i<2T$ and let $n\in \mathbb{Z}$ be such that $A_n\le i<A_{n+1}$. We have two such cases.

Case 1: if $i<A_{n+1}-1$, then

$$d\left({\varphi }_{{\lambda }_i}\left(y_i\right),y_{i+1}\right)=d\left({\varphi }_T\left({\varphi }_{T\left(i-A_n\right)}\left(x_n\right)\right),{\varphi }_{T\left(i+1-A_n\right)}\left(x_n\right)\right)=0.$$

Case 2: if $i=A_{n+1}-1$, bearing in mind that $A_{n+1}-A_n=s_{n+1}^m-s_n^m+1=m_n+1$ we obtain

$$\begin{array}{cc}\hfill d\left({\varphi }_{{\lambda }_i}\left(y_i\right),y_{i+1}\right)& =d\left({\varphi }_{r_n}\left({\varphi }_{T\left(A_{n+1}-1-A_n\right)}\left(x_n\right)\right),x_{n+1}\right)\hfill \\ \hfill & =d\left({\varphi }_{r_n}\left({\varphi }_{T{m}_n}\left(x_n\right)\right),x_{n+1}\right)\hfill \\ \hfill & =d\left({\varphi }_{t_n}\left(x_n\right),x_{n+1}\right)\hfill \\ \hfill & \le \delta \left({\varphi }_{t_n}\left(x_n\right)\right)=\delta \left({\varphi }_{{\lambda }_i}\left(y_i\right)\right).\hfill \end{array}$$

That is, ${\left\{(y_i,{\lambda }_i)\right\}}_{i\in \mathbb{Z}}$ is a $(\delta ,T,2T)$-pseudo-orbit of $\varphi$.

Then, we have $z\in X$ and a continuous map $\tau :X\times \mathbb{R}\to \mathbb{R}$ with $\tau (x,0)=0$, $\tau (x,·):\mathbb{R}\to \mathbb{R}$ is a strictly increasing homeomorphism, such that $d\left({\varphi }_{\tau (z,r)}\left(z\right),{\varphi }_{r-s_n^{\lambda }}\left(y_n\right)\right)\le \epsilon \left({\varphi }_{\tau (z,r)}\left(z\right)\right)$ where $s_n^{\lambda }\le r<s_{n+1}^{\lambda }$ and $\left(s_i^{\lambda }\right)$ is the sequence of sums associated with $\lambda ={\left({\lambda }_i\right)}_{i\in \mathbb{Z}}$. Let $w\in \mathbb{R}$ and $n\in \mathbb{Z}$, such that $s_n^t\le w<s_{n+1}^t$, where $\left(s_n^t\right)$ is associated with $t={\left(t_i\right)}_{i\in \mathbb{Z}}$. Since $s_n^t=s_{A_n}^{\lambda }$, then $s_{A_n}^{\lambda }\le w<s_{A_{n+1}}^{\lambda }=s_{A_n+m_n+1}^{\lambda }$. Hence, we have $0\le j\le m_n$, such that $s_{A_n+j}^{\lambda }\le w<s_{A_n+j+1}^{\lambda }$ and then

$$\begin{array}{cc}\hfill \epsilon \left({\varphi }_{\tau (z,w)}\left(z\right)\right)& \ge d\left({\varphi }_{\tau (z,w)}\left(z\right),{\varphi }_{w-s_{A_n+j}^{\lambda }}\left(y_{A_n+j}\right)\right)\hfill \\ \hfill & =d\left({\varphi }_{\tau (z,w)}\left(z\right),{\varphi }_{w-s_n^t}\left({\varphi }_{s_n^t-s_{A_n+j}^{\lambda }}\left(y_{A_n+j}\right)\right)\right)\hfill \\ \hfill & =d\left({\varphi }_{\tau (z,w)}\left(z\right),{\varphi }_{w-s_n^t}\left({\varphi }_{s_n^t-s_{A_n+j}^{\lambda }}\left({\varphi }_{Tj}\left(x_n\right)\right)\right)\right)\hfill \\ \hfill & =d\left({\varphi }_{\tau (z,w)}\left(z\right),{\varphi }_{w-s_n^t}\left(x_n\right)\right).\hfill \end{array}$$

It follows that $\varphi$ possesses the shadowing property with respect to time T.

Now, we shall prove that the flow possesses the shadowing property. We begin by fixing $m\in \mathbb{N}$, such that $m\ge T$. Given $\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right)$, we choose $\delta \in {\mathcal{C}}_{\varphi }\left(X\right)$ satisfying the following conditions:

• Every $(\delta ,T)$-pseudo-orbit is $\frac{\epsilon }{2}$-shadowed.
• For each $0\le t\le 2m$, we have $d\left({\varphi }_t\left(x\right),{\varphi }_t\left(y\right)\right)<\frac{\epsilon }{2}\left({\varphi }_t\left(x\right)\right)$, whenever $d(x,y)<\delta \left(x\right)$.

We let $m{\delta }^{\prime }\ll \delta$ and take $0<\beta <{\delta }^{\prime }$, such that $d(x,y)<\beta \left(x\right)$ implies that $d\left({\varphi }_t\left(x\right),{\varphi }_t\left(y\right)\right)<{\delta }^{\prime }\left({\varphi }_t\left(x\right)\right)$ for $0\le t\le 2m$. Then, we let ${\left\{(x_n,t_n)\right\}}_{n\in \mathbb{Z}}$ be a $(\beta ,1)$-pseudo-orbit for $\varphi$ with $1\le t_n\le 2$ for $n\in \mathbb{Z}$. Next, we consider the sequence of pairs ${\left(x_{im},{\lambda }_i\right)}_{i\in \mathbb{Z}}$ where ${\lambda }_i={\sum }_{j=0}^{m-1}{t}_{j+im}$ for every $i\in \mathbb{Z}$. We denote ${\lambda }_i\left(k\right)={\sum }_{j=k}^{m-1}{t}_{j+im}$ with $0\le k<m$. Then,

$$\begin{array}{cc}\hfill d\left({\varphi }_{{\lambda }_i}\left(x_{im}\right),x_{(i+1)m}\right)& \le {\displaystyle \sum _{r=1}^m}d\left({\varphi }_{{\lambda }_i\left(r\right)}\left({\varphi }_{t_{im+r-1}}\left(x_{im+r-1}\right)\right),{\varphi }_{{\lambda }_i\left(r\right)}\left(x_{im+r}\right)\right)\hfill \\ \hfill & \le {\displaystyle \sum _{r=1}^m}{\delta }^{\prime }\left({\varphi }_{{\lambda }_i\left(r\right)}\left({\varphi }_{t_{im+r-1}}\left(x_{im+r-1}\right)\right)\right)\hfill \\ \hfill & \le m\max \{{\delta }^{\prime }\left({\varphi }_{{\lambda }_i\left(1\right)}\left({\varphi }_{t_{im}}\left(x_{im}\right)\right)\right),\cdots ,{\delta }^{\prime }\left({\varphi }_{{\lambda }_i\left(m\right)}\left({\varphi }_{t_{im+m-1}}\left(x_{im+m-1}\right)\right)\right)\}\hfill \\ \hfill & \le \delta ({\varphi }_{{\lambda }_i}\left(x_{im}\right).\hfill \end{array}$$

Since $T\le {\lambda }_i\le 2m$, ${\left(x_{im},{\lambda }_i\right)}_{i\in \mathbb{Z}}$ is a $(\delta ,T)$-pseudo-orbit for $\varphi$. Hence, we have $z\in X$ and a continuous map $\tau :X\times \mathbb{R}\to \mathbb{R}$ with $\tau (x,0)=0$, $\tau (x,·):\mathbb{R}\to \mathbb{R}$ is a strictly increasing homeomorphism, such that $d\left({\varphi }_{\tau (z,t)}\left(z\right),{\varphi }_{t-s_n^{\lambda }}\left(x_{nm}\right)\right)\le \frac{\epsilon }{2}\left({\varphi }_{\tau (z,t)}\left(z\right)\right)$ where $s_n^{\lambda }\le t<s_{n+1}^{\lambda }$. Now, for $0\le k<m$, we denote $s_k^t\left(r\right)={\sum }_{j=r}^{k-1}{t}_j$, and as such, we find that

$$\begin{array}{cc}\hfill d\left({\varphi }_{s_k^t}\left(x_0\right),x_k\right)& \le {\displaystyle \sum _{r=1}^k}d\left({\varphi }_{s_k^t\left(r\right)}\left({\varphi }_{t_{r-1}}\left(x_{r-1}\right)\right),{\varphi }_{s_k^t\left(r\right)}\left(x_r\right)\right)\hfill \\ \hfill & <{\displaystyle \sum _{r=1}^k}{\delta }^{\prime }\left({\varphi }_{s_k^t\left(r\right)}\left({\varphi }_{t_{r-1}}\left(x_{r-1}\right)\right)\right)\hfill \\ \hfill & <k\max \{{\delta }^{\prime }\left({\varphi }_{s_k^t\left(1\right)}\left({\varphi }_{t_0}\left(x_0\right)\right)\right),\cdots ,{\delta }^{\prime }\left({\varphi }_{s_k^t\left(k\right)}\left({\varphi }_{t_{k-1}}\left(x_{k-1}\right)\right)\right)\}\hfill \\ \hfill & <\delta \left({\varphi }_{s_k^t}\left(x_0\right)\right).\hfill \end{array}$$

Then, for $s_k^t\le t<s_{k+1}^t$, we find that

$$\begin{array}{cc}\hfill d\left({\varphi }_{\tau (z,t)}\left(z\right),{\varphi }_{t-s_k^t}\left(x_k\right)\right)& \le d\left({\varphi }_{t-s_k^t}\left(x_k\right),{\varphi }_{t-s_k^t}\left({\varphi }_{s_k^t}\left(x_0\right)\right)\right)+d\left({\varphi }_t\left(x_0\right),{\varphi }_{\tau (z,t)}\left(z\right)\right)\hfill \\ \hfill & \le \frac{\epsilon }{2}\left({\varphi }_{t-s_k^t}\left({\varphi }_{s_k^t}\left(x_0\right)\right)\right)+\frac{\epsilon }{2}\left({\varphi }_{\tau (z,t)}\left(z\right)\right)\hfill \\ \hfill & \le {\epsilon }^{\prime }\left({\varphi }_{\tau (z,t)}\left(z\right)\right).\hfill \end{array}$$

For $m\le k<2m$, we continue in the same manner, and thus demonstrate that the flow possesses the shadowing property. □

We state that a sequence ${\left\{(x_i,t_i)\right\}}_{i\in \mathbb{Z}}$ of $X\times \mathbb{R}$ is obtained via a subset of $K\subset X$ if $x_0\in K$. We also state that a flow $\varphi$ possesses the measure shadowing property through K, if given $\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right)$, there exists $\delta \in {\mathcal{C}}_{\varphi }\left(X\right)$, such that every $(\delta ,1)$-pseudo orbit ${\left\{(x_i,t_i)\right\}}_{i\in \mathbb{Z}}$ of $\varphi$ passing through K can be $\epsilon$-shadowed. For any $\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right),T>0$, we state that a point $x\in X$ is $\epsilon$-shadowable with respect to time T if we have $\delta \in {\mathcal{C}}_{\varphi }\left(X\right)$, such that every $(\delta ,T)$-pseudo orbit through x is $\epsilon$-shadowed. We write $S{h}_{\epsilon ,T}\left(\varphi \right)$ to denote the collection of all $\epsilon$-shadowable points of $\varphi$ with respect to time $T>0$. For simplicity, we write $S{h}_{\epsilon ,1}\left(\varphi \right)$ by $S{h}_{\epsilon }\left(\varphi \right)$. A point x is said to be shadowable for a flow $\varphi$ if it belongs to $Sh\left(\varphi \right)={\bigcap }_{\epsilon >0}S{h}_{\epsilon }\left(\varphi \right)$. Referring to Proposition 1, we can obtain $S{h}_{\epsilon ,T}\left(\varphi \right)\subset S{h}_{2{\epsilon }^{\prime },S}\left(\varphi \right)$ for any $T,S>0$.

A measure $\mu$ is called invariant if $t\in \mathbb{R}$ satisfies $\mu \left(A\right)=\mu \left({\varphi }_t\left(A\right)\right)$. Let $\mathcal{M}(X,\varphi )$ be the set of all Borel probability-invariant measures $\mu$ on X. In [10], the authors found that for a flow of $\varphi$ on a compact metric space X, $\varphi$ has the shadowing property if and only if $\mu \left(Sh\right(\varphi \left)\right)=1$ for all $\mu \in \mathcal{M}(X,\varphi )$. We observe that this result is false if X was noncompact, as shown in Example 1 below.

Example 1.

Let $X=\{\frac{1}{n}:n\in \mathbb{N}\}\cup \{1-\frac{1}{n}:n\ge 2\}$, and f be a homeomorphism on X, satisfying that $f\left(a_i\right)=a_{i+1}$, $i\in \mathbb{N}$ and $f\left(1\right)=1$. Here, $a_i=1/(i-2)$, for $i\le 0$; $a_i=1-1/(i+1)$, for $i>0$. Consider that ϕ is the suspension flow of f; if this is the case, we have $\mu \left(Sh\right(\varphi \left)\right))=1$, but ϕ does not have the shadowing property on X.

Proof. 

In contradiction, we suppose $\varphi$ will have the shadowing property. For $\epsilon =\frac{1}{4}$, we take $\delta >0$ corresponding $\epsilon$ by the shadowing property. Then, we choose n of sufficient size such that $d(a_n,a_{n+1})<\delta$. Since $d({\varphi }_1\left(x_i\right),x_{i+1})=d(a_{n+1},a_n)<\delta$, then we can obtain ${\left\{(x_i,t_i)\right\}}_{i\in \mathbb{Z}}={\left\{(a_n,1)\right\}}_{n\in \mathbb{Z}}$ is a ($\delta$, 1)-pseudo orbit of $\varphi$. Since $\varphi$ has the shadowing property, we find that there are $x\in X$ and continuous map $\tau :X\times \mathbb{R}\to \mathbb{R}$ with $\tau (x,0)=0$, $\tau (x,·):\mathbb{R}\to \mathbb{R}$ is a strictly increasing homeomorphism such that for all $t\in \mathbb{R}$

$$d({\varphi }_{\tau (x,t)}\left(x\right),{\varphi }_{t-T_i}\left(x_i\right))\le \epsilon .$$

But as t goes to infinity, any $x\in \overline{X}$, ${\varphi }_{\tau (x,t)}\left(x\right)$ will tend to 1, and ${\varphi }_{t-T_i}\left(x_i\right)$ will stay nearby $a_n$. This contradiction shows that $\varphi$ does not have the shadowing property.

Next, we prove that $1\in Sh\left(f\right)$. For any $\epsilon >0$, take $\delta >0$, such that $\delta <\epsilon$. And take N, such that all $n\ge N$ will satisfy $d(a_n,1)<\frac{\delta }{2}$. Let ${\left\{x_i\right\}}_{i\in \mathbb{Z}}$ be a $\delta$-pseudo orbit of f, such that $x_0=1$. Since $d(f\left(x_0\right),x_1)<\delta$ implies $d(1,x_1)<\delta$. Then, we can obtain $x_1\in \{a_n:n\ge N\}\cup \left\{1\right\}$, and in the same way, we can obtain that $x_i\in \{a_n:n\ge N\}\cup \left\{1\right\}$ for all $i\ge 0$. For $i<0$, there are only two possibilities. One is that $x_1\in \{a_n:n\ge N\}\cup \left\{1\right\}$, for all $i<0$, and thus, the shadowing point is $x=1$; another one is $x_i=a_{N-1}$ for some $i<0$, and then the shadowing point is $x=f^{-1}\left(a_{N-1}\right)$. So we obtain $1\in Sh\left(f\right)$. By the Theorem 2.7 in [12], we can obtain $(1,0)\in Sh\left(f\right)\times [0,1]/\sim =Sh\left(\varphi \right)$. Since $Sh\left(\varphi \right)$ is invariant, we have ${\varphi }_{\mathbb{R}}(1,0)\subset Sh\left(\varphi \right)$. Let $\mu \in \mathcal{M}(X,\varphi )=\{\mathcal{L}:\mathit{the}\mathit{Lebesgue}\mathit{measure}\mathit{supported}\mathit{on}{\varphi }_{\mathbb{R}}(1,0)\}$, since $\mathcal{L}\left({\varphi }_{\mathbb{R}}(1,0)\right)=1$, then $\mathcal{L}\left(Sh\left(\varphi \right)\right)\ge \mathcal{L}\left({\varphi }_{\mathbb{R}}(1,0)\right)=1$. So, for any $\mu \in \mathcal{M}(X,\varphi )$, we have $\mu \left(Sh\right(\varphi \left)\right)=1$. □

The following theorem shows that $\varphi$ possesses the shadowing property if and only if $\mu \left(Sh\right(\varphi \left)\right)=1$ for all $\mu \in \mathcal{M}\left(X\right)$ on metric spaces.

Theorem 1.

Let ϕ be a flow on a metric space X. Then, the following are equivalent:

1. 

ϕ has the shadowing property;

2. 

ϕ has the measure shadowing property;

3. 

$\mu \left(Sh\right(\varphi \left)\right)=1$ for all $\mu \in \mathcal{M}\left(X\right)$.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)\Rightarrow \left(3\right)$, as shown by the definitions.

Now, we shall show $\left(3\right)\Rightarrow \left(1\right)$. Suppose $\mu \left(Sh\right(\varphi \left)\right)=1$ for any $\mu \in \mathcal{M}\left(X\right)$. We begin by proving the claim.

Claim: For any ${\epsilon }^{\prime }\in {\mathcal{C}}_{\varphi }\left(X\right)$, we take $\epsilon \ll {\epsilon }^{\prime }$; if $x\in S{h}_{\epsilon }\left(\varphi \right)$ for some $\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right)$, then for any $T>0$, we find that ${\delta }_T\in {\mathcal{C}}_{\varphi }\left(X\right)$, such that every $({\delta }_T,T)$-pseudo orbit ${\left\{(y_n,t_n)\right\}}_{n\in \mathbb{Z}}$ with $d(y_0,x)<{\delta }_T\left(x\right)$ can be $3{\epsilon }^{\prime }$-shadowed.

For all ${\epsilon }^{\prime }\in {\mathcal{C}}_{\varphi }\left(X\right)$, we take $\epsilon \ll {\epsilon }^{\prime }$, and let $x\in S{h}_{\epsilon }\left(\varphi \right)$. Then, according to Proposition 1, we find that $x\in S{h}_{2{\epsilon }^{\prime },T}\left(\varphi \right)$, and so we have $\delta \in {\mathcal{C}}_{\varphi }\left(X\right)$, such that any $(\delta ,T)$-pseudo orbit ${\left\{(x_n,t_n)\right\}}_{n\in \mathbb{Z}}$ through x with $T\le t_n\le 2T(n\in \mathbb{Z})$ can be $2{\epsilon }^{\prime }$-shadowed. We let ${\delta }_T\in {\mathcal{C}}_{\varphi }\left(X\right)$ with $2{\delta }_T\ll \delta$ be such that if $d(y,z)<{\delta }_T\left(y\right)$, then $d({\varphi }_t\left(y\right),{\varphi }_t\left(z\right))<\min \{\frac{\delta \left({\varphi }_t\left(y\right)\right)}{2},\epsilon \left({\varphi }_t\left(y\right)\right)\}$ for all $t\in [0,2T]$. Then, we let ${\left\{(y_n,t_n)\right\}}_{n\in \mathbb{Z}}$ be a $({\delta }_T,T)$-pseudo orbit with $d(y_0,x)<{\delta }_T\left(x\right)$. We define a sequence ${\left\{(z_n,s_n)\right\}}_{n\in \mathbb{Z}}$ as

$$(z_n,s_n)=(x,t_0)\mathrm{for}n=0,\mathrm{and}(z_n,s_n)=(y_n,t_n)\mathrm{for}n\ne 0.$$

Since

$$\begin{array}{cc}\hfill d({\varphi }_{t_{-1}}\left(y_{-1}\right),x)& \le d({\varphi }_{t_{-1}}\left(y_{-1}\right),y_0)+d(y_0,x)\hfill \\ \hfill & <{\delta }_T\left({\varphi }_{t_{-1}}\left(y_{-1}\right)\right)+{\delta }_T\left(x\right)\hfill \\ \hfill & <2\max \{{\delta }_T\left({\varphi }_{t_{-1}}\left(y_{-1}\right)\right),{\delta }_T\left(x\right)\}\hfill \\ \hfill & \le \delta \left({\varphi }_{t_{-1}}\left(y_{-1}\right)\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & d({\varphi }_{t_0}\left(x\right),y_1)\le d({\varphi }_{t_0}\left(x\right),{\varphi }_{t_0}\left(y_0\right))+d({\varphi }_{t_0}\left(y_0\right),y_1)\hfill \\ & <\frac{\delta \left({\varphi }_{t_0}\left(x\right)\right)}{2}+{\delta }_T\left({\varphi }_{t_0}\left(y_0\right)\right)\le \delta \left({\varphi }_{t_0}\left(x\right)\right),\hfill \end{array}$$

we find that ${\left\{(z_n,s_n)\right\}}_{n\in \mathbb{Z}}$ is a $(\delta ,T)$-pseudo orbit of $\varphi$ through x. Since $x\in S{h}_{2{\epsilon }^{\prime },T}\left(\varphi \right)$, we have $z\in X$ and a continuous map $\tau :X\times \mathbb{R}\to \mathbb{R}$ with $\tau (x,0)=0$, $\tau (x,·):\mathbb{R}\to \mathbb{R}$ is a strictly increasing homeomorphism, such that

$$d({\varphi }_{\tau (z,t)}\left(z\right),{\varphi }_{t-T_n}\left(z_n\right))\le 2{\epsilon }^{\prime }\left({\varphi }_{\tau (z,t)}\left(z\right)\right),\forall t\in [T_n,T_{n+1})\mathrm{with}n\in \mathbb{Z},$$

where $T_n$ is given as stated previously. For $T_0\le t<T_1$, we have

$$\begin{array}{cc}\hfill d({\varphi }_{\tau (z,t)}\left(z\right),{\varphi }_t\left(y_0\right))& \le d({\varphi }_{\tau (z,t)}\left(z\right),{\varphi }_t\left(x\right))+d({\varphi }_t\left(x\right),{\varphi }_t\left(y_0\right))\hfill \\ \hfill & <2{\epsilon }^{\prime }\left({\varphi }_{\tau (z,t)}\left(z\right)\right)+\epsilon \left({\varphi }_t\left(x\right)\right)\hfill \\ \hfill & \le 3{\epsilon }^{\prime }\left({\varphi }_{\tau (z,t)}\left(z\right)\right)).\hfill \end{array}$$

Additionally, when $T_i\le t<T_{i+1}$ with $i\ne 0$, we find that

$$d({\varphi }_{\tau (z,t)}\left(z\right),{\varphi }_{t-T_n}\left(y_n\right))=d({\varphi }_{\tau (z,t)}\left(z\right),{\varphi }_{t-T_n}\left(z_n\right))\le 2{\epsilon }^{\prime }\left({\varphi }_{\tau (z,t)}\left(z\right)\right)<3{\epsilon }^{\prime }\left({\varphi }_{\tau (z,t)}\left(z\right)\right).$$

Thus, it is clear that ${\left\{(y_n,t_n)\right\}}_{n\in \mathbb{Z}}$ is $3{\epsilon }^{\prime }$-shadowed by z, which proves that our claim is correct.

Next, we let ${\beta }^{\prime }\in {\mathcal{C}}_{\varphi }\left(X\right)$ and take $\epsilon \ll {\epsilon }^{\prime }\ll \beta \ll {\beta }^{\prime }$, and thus, we let $x\in X$. Since the $\omega$-limit set $\omega \left(x\right)$ is a nonempty closed invariant set, we have $\mu \in \mathcal{M}\left(X\right)$, such that $\mu \left(\omega \right(x\left)\right)=1$. Since $\mu \left(Sh\right(\varphi \left)\right)=1$, we have $\omega \left(x\right)\cap Sh\left(\varphi \right)\ne \varnothing$. Then, we let $y\in \omega \left(x\right)\cap Sh\left(\varphi \right)$. For any ${\epsilon }^{\prime }\in {\mathcal{C}}_{\varphi }\left(X\right)$, based on the above claim, we take $\epsilon \ll {\epsilon }^{\prime }$, if $x\in S{h}_{\epsilon }\left(\varphi \right)$ for some $\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right)$, and as a result, ${\delta }_1\in {\mathcal{C}}_{\varphi }\left(X\right)$, such that any (${\delta }_1$, 1)-pseudo orbit ${\left\{(y_n,t_n)\right\}}_{n\in \mathbb{Z}}$ with $d(y,y_0)<{\delta }_1\left(x\right)$ can be ${\epsilon }^{\prime }$-shadowed. Since $y\in \omega \left(x\right)$, we have $T>0$, such that $d({\varphi }_T\left(x\right),y)<{\delta }_1\left({\varphi }_T\left(x\right)\right)$. Then, we obtain ${\varphi }_T\left(x\right)\in S{h}_{{\epsilon }^{\prime }}\left(\varphi \right)$.

Next, we show that $x\in S{h}_{8{\beta }^{\prime }}\left(\varphi \right)$. Based on the above claim, for any $\beta \in {\mathcal{C}}_{\varphi }\left(X\right)$ and take ${\epsilon }^{\prime }\ll \beta$, if ${\varphi }_T\left(x\right)\in S{h}_{{\epsilon }^{\prime }}\left(\varphi \right)$ for some ${\epsilon }^{\prime }\in {\mathcal{C}}_{\varphi }\left(X\right)$, then we have ${\delta }_T\in {\mathcal{C}}_{\varphi }\left(X\right)$, such that any $({\delta }_T,T)$-pseudo orbit ${\left\{(y_n,t_n)\right\}}_{n\in \mathbb{Z}}$ with $d(y_0,{\varphi }_T\left(x\right))<{\delta }_T\left({\varphi }_T\left(x\right)\right)$ can be $3\beta$-shadowed. Let ${\delta }^{\prime }<{\delta }_T$ be such that if $d(x,y)<{\delta }^{\prime }\left(x\right)$, then $d({\varphi }_t\left(x\right),{\varphi }_t\left(y\right))<{\delta }_T\left({\varphi }_t\left(x\right)\right)$ for $-T\le t\le T$. Let ${\left\{(x_n,t_n)\right\}}_{n\in \mathbb{Z}}$ be a $({\delta }^{\prime },T)$-pseudo orbit of $\varphi$ through x with $T\le t_n\le 2T$ for all $n\in \mathbb{Z}$. We define a sequence ${\left\{(y_n,s_n)\right\}}_{n\in \mathbb{Z}}$ as follows:

$$(y_n,s_n)=\left\{\begin{array}{ccc}\hfill (x_{n+1},t_{n+1})& \hfill & \hfill \mathrm{if}n<-1,\\ \hfill (x_0,T)& \hfill & \hfill \mathrm{if}n=-1,\\ \hfill ({\varphi }_{T-t_0}\left(x_1\right),t_1+t_0-T)& \hfill & \hfill \mathrm{if}n=0,\\ \hfill (x_{n+1},t_{n+1})& \hfill & \hfill \mathrm{if}n>0.\end{array}\right.$$

Since $d({\varphi }_{t_0}\left(x_0\right),x_1)<{\delta }^{\prime }\left({\varphi }_{t_0}\left(x_0\right)\right)$, we find that

$$d({\varphi }_{T-t_0}\left({\varphi }_{t_0}\left(x_0\right)\right),{\varphi }_{T-t_0}\left(x_1\right))=d({\varphi }_T\left(x_0\right),{\varphi }_{T-t_0}\left(x_1\right))<{\delta }_T\left({\varphi }_T\left(x_0\right)\right).$$

Then, we can see that ${\left\{(y_n,s_n)\right\}}_{n\in \mathbb{Z}}$ is a $({\delta }_T,T)$-pseudo orbit of $\varphi$ through ${\varphi }_{T-t_0}\left(x_1\right)$. Moreover, we have

$$d(y_0,{\varphi }_T\left(x\right))=d({\varphi }_{T-t_0}\left(x_1\right),{\varphi }_{T-t_0}\left({\varphi }_{t_0}\left(x_0\right)\right))=d({\varphi }_{T-t_0}\left(x_1\right),{\varphi }_T\left(x_0\right))<{\delta }_T\left({\varphi }_T\left(x_0\right)\right).$$

As a result, we have $z\in X$ and a continuous map $\tau :X\times \mathbb{R}\to \mathbb{R}$ with $\tau (x,0)=0$, $\tau (x,·):\mathbb{R}\to \mathbb{R}$ is a strictly increasing homeomorphism, such that

$$d({\varphi }_{\alpha (z,t)}\left(z\right),{\varphi }_{t-T_n}\left(y_n\right))\le 3\beta \left({\varphi }_{\alpha (z,t)}\left(z\right)\right),\forall t\in [T_n,T_{n+1})\mathrm{with}n\in \mathbb{Z},$$

where $T_n$ is given as before. Since

$$\begin{array}{cc}\hfill & d({\varphi }_{\alpha (z,t)}\left(z\right),{\varphi }_t\left({\varphi }_T\left(x\right)\right))\hfill \\ \hfill & \le d({\varphi }_{\alpha (z,t)}\left(z\right),{\varphi }_t\left({\varphi }_{-t_0+T}\left(x_1\right)\right))+d({\varphi }_t\left({\varphi }_{-t_0+T}\left(x_1\right)\right),{\varphi }_t\left({\varphi }_T\left(x\right)\right))\hfill \\ \hfill & <3\beta \left({\varphi }_{\alpha (z,t)}\left(z\right)\right)+\epsilon \left({\varphi }_t\left({\varphi }_T\left(x\right)\right)\right)\hfill \\ \hfill & <4\beta \left({\varphi }_{\alpha (z,t)}\left(z\right)\right))\hfill \end{array}$$

for all $t\in [0,t_0-T]$, we find that ${\left\{(x_n,t_n)\right\}}_{n\in \mathbb{Z}}$ is $4\beta$-shadowed by z. According to Proposition 1, we also have $x\in S{h}_{8{\beta }^{\prime }}\left(\varphi \right)$. Since ${\beta }^{\prime }$ is arbitrary, we obtain $x\in Sh\left(\varphi \right)$. This shows that $\varphi$ possesses the shadowing property on X. □

Based on the following theorem, we prove that $\varphi$ has the shadowing property with respect to time T if and only if $\varphi$ has the measure shadowing property with respect to time T. The proof of the next theorem is similar to Theorem 1. The main difference is the method used to tackle the parameter T, for the completeness of the context. The proof is as follows.

Theorem 2.

Let ϕ be a flow on a metric space X. Then, the following are equivalent:

1. 

ϕ possesses the shadowing property with respect to time T;

2. 

ϕ possesses the measure shadowing property with respect to time T;

3. 

$\mu \left(S{h}_T\left(\varphi \right)\right)=1$ for all $\mu \in \mathcal{M}\left(X\right)$.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)\Rightarrow \left(3\right)$, as is clear from the definitions.

Now, we shall show $\left(3\right)\Rightarrow \left(1\right)$. Suppose $\mu \left(S{h}_T\left(\varphi \right)\right)=1$ for any $\mu \in \mathcal{M}\left(X\right)$. We begin by proving the following claim.

Claim: For any ${\epsilon }^{\prime }\in {\mathcal{C}}_{\varphi }\left(X\right)$ and take $\epsilon \ll {\epsilon }^{\prime }$, if $x\in S{h}_{\epsilon ,T}\left(\varphi \right)$ for some $\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right)$, then for any $S>0$, we find that ${\delta }_S\in {\mathcal{C}}_{\varphi }\left(X\right)$, such that every $({\delta }_S,S)$-pseudo orbit ${\left\{(y_n,t_n)\right\}}_{n\in \mathbb{Z}}$ with $d(y_0,x)<{\delta }_S\left(x\right)$ can be $3{\epsilon }^{\prime }$-shadowed.

For all ${\epsilon }^{\prime }\in {\mathcal{C}}_{\varphi }\left(X\right)$, take $\epsilon \ll {\epsilon }^{\prime }$, and let $x\in S{h}_{\epsilon ,T}\left(\varphi \right)$. Then, according to Proposition 1, we have $x\in S{h}_{2{\epsilon }^{\prime },S}\left(\varphi \right)$, and so we have $(\delta ,S)$-pseudo orbit ${\left\{(x_n,t_n)\right\}}_{n\in \mathbb{Z}}$ through x with $S\le t_n\le 2S(n\in \mathbb{Z})$ can be $2{\epsilon }^{\prime }$-shadowed. Let ${\delta }_S\in {\mathcal{C}}_{\varphi }\left(X\right)$ with $2{\delta }_S\ll \delta$ be such that if $d(y,z)<{\delta }_S\left(y\right)$, then $d({\varphi }_t\left(y\right),{\varphi }_t\left(z\right))<\min \{\frac{\delta \left({\varphi }_t\left(y\right)\right)}{2},\epsilon \left({\varphi }_t\left(y\right)\right)\}$ for all $t\in [0,2S]$. Let ${\left\{(y_n,t_n)\right\}}_{n\in \mathbb{Z}}$ be a $({\delta }_S,S)$-pseudo orbit with $d(y_0,x)<{\delta }_S\left(x\right)$. We define a sequence ${\left\{(z_n,s_n)\right\}}_{n\in \mathbb{Z}}$ by

$$(z_n,s_n)=(x,t_0)\mathrm{for}n=0,\mathrm{and}(z_n,s_n)=(y_n,t_n)\mathrm{for}n\ne 0.$$

Since

$$\begin{array}{cc}\hfill d({\varphi }_{t_{-1}}\left(y_{-1}\right),x)& \le d({\varphi }_{t_{-1}}\left(y_{-1}\right),y_0)+d(y_0,x)\hfill \\ \hfill & <{\delta }_S\left({\varphi }_{t_{-1}}\left(y_{-1}\right)\right)+{\delta }_S\left(x\right)\hfill \\ \hfill & <2\max \{{\delta }_S\left({\varphi }_{t_{-1}}\left(y_{-1}\right)\right),{\delta }_S\left(x\right)\}\hfill \\ \hfill & \le \delta \left({\varphi }_{t_{-1}}\left(y_{-1}\right)\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill d({\varphi }_{t_0}\left(x\right),y_1)& \le d({\varphi }_{t_0}\left(x\right),{\varphi }_{t_0}\left(y_0\right))+d({\varphi }_{t_0}\left(y_0\right),y_1)\hfill \\ \hfill & <\frac{\delta \left({\varphi }_{t_0}\left(x\right)\right)}{2}+{\delta }_S\left({\varphi }_{t_0}\left(y_0\right)\right)\hfill \\ \hfill & \le \delta \left({\varphi }_{t_0}\left(x\right)\right),\hfill \end{array}$$

we see that ${\left\{(z_n,s_n)\right\}}_{n\in \mathbb{Z}}$ is a $(\delta ,S)$-pseudo orbit of $\varphi$ through x. Since $x\in S{h}_{2{\epsilon }^{\prime },S}\left(\varphi \right)$, we have $z\in X$ and a continuous map $\tau :X\times \mathbb{R}\to \mathbb{R}$ with $\tau (x,0)=0$, $\tau (x,·):\mathbb{R}\to \mathbb{R}$ is a strictly increasing homeomorphism, such that

$$d({\varphi }_{\tau (z,t)}\left(z\right),{\varphi }_{t-T_n}\left(z_n\right))\le 2{\epsilon }^{\prime }\left({\varphi }_{\tau (z,t)}\left(z\right)\right),\forall t\in [T_n,T_{n+1})\mathrm{with}n\in \mathbb{Z},$$

where $T_n$ is given as stated previously. For $T_0\le t<T_1$, we find that

$$\begin{array}{cc}\hfill d({\varphi }_{\tau (z,t)}\left(z\right),{\varphi }_t\left(y_0\right))& \le d({\varphi }_{\tau (z,t)}\left(z\right),{\varphi }_t\left(x\right))+d({\varphi }_t\left(x\right),{\varphi }_t\left(y_0\right))\hfill \\ \hfill & <2{\epsilon }^{\prime }\left({\varphi }_{\tau (z,t)}\left(z\right)\right)+\epsilon \left({\varphi }_t\left(x\right)\right)\hfill \\ \hfill & \le 3{\epsilon }^{\prime }\left({\varphi }_{\tau (z,t)}\left(z\right)\right).\hfill \end{array}$$

Additionally, when $T_i\le t<T_{i+1}$ with $i\ne 0$, we can obtain that

$$\begin{array}{cc}\hfill d({\varphi }_{\tau (z,t)}\left(z\right),{\varphi }_{t-T_n}\left(y_n\right))& =d({\varphi }_{\tau (z,t)}\left(z\right),{\varphi }_{t-T_n}\left(z_n\right))\hfill \\ \hfill & \le 2{\epsilon }^{\prime }\left({\varphi }_{\tau (z,t)}\left(z\right)\right)\hfill \\ \hfill & <3{\epsilon }^{\prime }\left({\varphi }_{\tau (z,t)}\left(z\right)\right).\hfill \end{array}$$

Thus, it is clear that ${\left\{(y_n,t_n)\right\}}_{n\in \mathbb{Z}}$ is $3{\epsilon }^{\prime }$-shadowed by z, which proves our claim.

Let ${\beta }^{\prime }\in {\mathcal{C}}_{\varphi }\left(X\right)$ and take $\epsilon \ll {\epsilon }^{\prime }\ll \beta \ll {\beta }^{\prime }$. Let $x\in X$. Since the $\omega$-limit set $\omega \left(x\right)$ is a nonempty closed invariant set, we have $\mu \in \mathcal{M}\left(X\right)$, such that $\mu \left(\omega \right(x\left)\right)=1$. Since $\mu \left(S{h}_T\left(\varphi \right)\right)=1$, we have $\omega \left(x\right)\cap S{h}_T\left(\varphi \right)\ne \varnothing$. Let $y\in \omega \left(x\right)\cap S{h}_T\left(\varphi \right)$. For any ${\epsilon }^{\prime }\in {\mathcal{C}}_{\varphi }\left(X\right)$, based on the above claim, we take $\epsilon \ll {\epsilon }^{\prime }$, if $y\in S{h}_{\epsilon ,T}\left(\varphi \right)$ for some $\epsilon \in {\mathcal{C}}_{\varphi }\left(X\right)$, which gives us ${\delta }_T\in {\mathcal{C}}_{\varphi }\left(X\right)$, such that any (${\delta }_T$, T)-pseudo orbit ${\left\{(y_n,t_n)\right\}}_{n\in \mathbb{Z}}$ with $d(y,y_0)<{\delta }_T\left(x\right)$ can be ${\epsilon }^{\prime }$-shadowed. Since $y\in \omega \left(x\right)$, we have $S>0$, such that $d({\varphi }_S\left(x\right),y)<{\delta }_T\left({\varphi }_S\left(x\right)\right)$. Then, we have ${\varphi }_S\left(x\right)\in S{h}_{{\epsilon }^{\prime },T}\left(\varphi \right)$.

Next, we show that $x\in S{h}_{8{\beta }^{\prime },T}\left(\varphi \right)$. Based on the above claim, for any $\beta \in {\mathcal{C}}_{\varphi }\left(X\right)$, we take ${\epsilon }^{\prime }\ll \beta$, if ${\varphi }_S\left(x\right)\in S{h}_{{\epsilon }^{\prime },T}\left(\varphi \right)$ for some ${\epsilon }^{\prime }\in {\mathcal{C}}_{\varphi }\left(X\right)$, then we have ${\delta }_S\in {\mathcal{C}}_{\varphi }\left(X\right)$, such that any $({\delta }_S,S)$-pseudo orbit ${\left\{(y_n,t_n)\right\}}_{n\in \mathbb{Z}}$ with $d(y_0,{\varphi }_S\left(x\right))<{\delta }_S\left({\varphi }_S\left(x\right)\right)$ can be $3\beta$-shadowed. Let ${\delta }^{\prime }<{\delta }_S$ be such that if $d(x,y)<{\delta }^{\prime }\left(x\right)$, then $d({\varphi }_t\left(x\right),{\varphi }_t\left(y\right))<{\delta }_S\left({\varphi }_t\left(x\right)\right)$ for $-S\le t\le S$. Let ${\left\{(x_n,t_n)\right\}}_{n\in \mathbb{Z}}$ be a $({\delta }^{\prime },S)$-pseudo orbit of $\varphi$ through x with $S\le t_n\le 2S$ for all $n\in \mathbb{Z}$. We define a sequence ${\left\{(y_n,s_n)\right\}}_{n\in \mathbb{Z}}$ as

$$(y_n,s_n)=\left\{\begin{array}{ccc}\hfill (x_{n+1},t_{n+1})& \hfill & \hfill \mathrm{if}n<-1,\\ \hfill (x_0,S)& \hfill & \hfill \mathrm{if}n=-1,\\ \hfill ({\varphi }_{S-t_0}\left(x_1\right),t_1+t_0-S)& \hfill & \hfill \mathrm{if}n=0,\\ \hfill (x_{n+1},t_{n+1})& \hfill & \hfill \mathrm{if}n>0.\end{array}\right.$$

Since $d({\varphi }_{t_0}\left(x_0\right),x_1)<{\delta }^{\prime }\left({\varphi }_{t_0}\left(x_0\right)\right)$, we have

$$d({\varphi }_{S-t_0}\left({\varphi }_{t_0}\left(x_0\right)\right),{\varphi }_{S-t_0}\left(x_1\right))=d({\varphi }_S\left(x_0\right),{\varphi }_{S-t_0}\left(x_1\right))<{\delta }_S\left({\varphi }_S\left(x_0\right)\right).$$

Then, we can see that ${\left\{(y_n,s_n)\right\}}_{n\in \mathbb{Z}}$ is a $({\delta }_S,S)$-pseudo orbit of $\varphi$ through ${\varphi }_{S-t_0}\left(x_1\right)$. Moreover, we have

$$\begin{array}{cc}\hfill d(y_0,{\varphi }_S\left(x\right))& =d({\varphi }_{S-t_0}\left(x_1\right),{\varphi }_{S-t_0}\left({\varphi }_{t_0}\left(x_0\right)\right))\hfill \\ \hfill & =d({\varphi }_{S-t_0}\left(x_1\right),{\varphi }_S\left(x_0\right))\hfill \\ \hfill & <{\delta }_S\left({\varphi }_S\left(x_0\right)\right).\hfill \end{array}$$

Then, we have $z\in X$ and a continuous map $\alpha :X\times \mathbb{R}\to \mathbb{R}$ with $\alpha (x,0)=0$, $\alpha (x,·):\mathbb{R}\to \mathbb{R}$ is a strictly increasing homeomorphism, such that

$$d({\varphi }_{\alpha (z,t)}\left(z\right),{\varphi }_{t-T_n}\left(y_n\right))\le 3\beta \left({\varphi }_{\alpha (z,t)}\left(z\right)\right),\forall t\in [T_n,T_{n+1})\mathrm{with}n\in \mathbb{Z},$$

where $T_n$ is given as before. Since

$$\begin{array}{cc}\hfill & d({\varphi }_{\alpha (z,t)}\left(z\right),{\varphi }_t\left({\varphi }_S\left(x\right)\right))\hfill \\ \hfill & \le d({\varphi }_{\alpha (z,t)}\left(z\right),{\varphi }_t\left({\varphi }_{-t_0+S}\left(x_1\right)\right))+d({\varphi }_t\left({\varphi }_{-t_0+S}\left(x_1\right)\right),{\varphi }_t\left({\varphi }_S\left(x\right)\right))\hfill \\ \hfill & <3\beta \left({\varphi }_{\alpha (z,t)}\left(z\right)\right)+\epsilon \left({\varphi }_t\left({\varphi }_S\left(x\right)\right)\right)\hfill \\ \hfill & <4\beta \left({\varphi }_{\alpha (z,t)}\left(z\right)\right)\hfill \end{array}$$

for all $t\in [0,t_0-S]$, we find that ${\left\{(x_n,t_n)\right\}}_{n\in \mathbb{Z}}$ is $4\beta$-shadowed by z. According to Proposition 1, we have $x\in S{h}_{8{\beta }^{\prime },T}\left(\varphi \right)$. Since ${\beta }^{\prime }$ is arbitrary, we find that $x\in S{h}_T\left(\varphi \right)$. This demonstrates that $\varphi$ possesses the shadowing property on X with respect to time T. □

According to Proposition 1, Theorems 1 and 2, we can obtain the following corollary, which provides an equivalent definition of the measure shadowing property.

Corollary 1.

Let $T>0$ and ϕ be a flow on a metric space X. ϕ has the measure shadowing property if and only if ϕ has the measure shadowing property with respect to time T.

4. Conclusions

In this paper, we extend the concept of the measure shadowing property of continuous flows from compact metric spaces to the general metric spaces. Moreover, by using the property of the shadowable points, we demonstrate the equivalence relation between the measure shadowing property and the shadowing property for flows on metric spaces. In the following research, we will further consider the topological stability and spectral decomposition of the measure shadowing property of continuous flows on metric spaces.