On Ergodic and Inverse Shadowing Properties of Set-Valued Mapping

1. Introduction

The shadowing property, which describes the approximation of pseudo-trajectories by exact trajectories in dynamical systems, has been widely studied in modern global dynamical systems theory (for example,^1,2). This concept plays an important role in understanding the stability of dynamical behavior, since it guarantees that approximate trajectories produced by numerical computations or perturbations can be closely followed by true trajectories of the system.

In contrast to the classical shadowing problem, the inverse shadowing problem investigates whether every exact trajectory of a dynamical system can be approximated by trajectories generated by certain methods that produce pseudo-trajectories (see^3,4). The study of inverse shadowing provides another perspective for analyzing the robustness of dynamical systems and has attracted increasing attention in recent years.

Set-valued dynamical systems have also received considerable interest in the literature. In this direction, Román-Flores,⁵ studied the relationship between transitivity and the associated dynamical system. Following this work, several authors investigated dynamical properties of set-valued discrete systems, including transitivity and mixing properties.^6,7 Furthermore, Shabani and Ahmadi⁸ showed that the ergodic shadowing property is related to chain mixing in non-autonomous discrete-time dynamical systems.

More recently, further developments have appeared in the study of ergodic shadowing. In particular, Koo and Lee,⁹ proved that the uniform ergodic shadowing property holds for sequences of homeomorphisms on non-compact metric spaces. In another work, Al-Sharaa and Russl A.,¹⁰ investigated the shadowing and $w$ -expansive properties for generic non-autonomous discrete dynamical systems and analyzed the relationship between them. More recently, Koo and Lee,¹¹ introduced dynamical systems on non-compact metric spaces and studied their ergodic shadowing property.

Motivated by these developments, in this paper we study the relationship between the inverse shadowing property and the ergodic shadowing property for set-valued mappings, and analyze these properties through the associated inverse limit space. We introduce suitable definitions of inverse shadowing and ergodic shadowing properties for set-valued mappings and investigate how these properties behave for the induced dynamical system on the inverse limit space. Our results extend several known relationships between shadowing properties and inverse limit spaces from single-valued dynamical systems to the framework of set-valued dynamical systems.

The remainder of the paper is organized as follows. In the next section, we present several basic definitions and preliminary concepts that will be used throughout the paper.

2. Preliminaries

Let $X$ be a compact metric space with metric $\mathcal{d}$ . Let ${\left\{{\mathtt{H}}_{\mathrm{\eta }}\right\}}_{\mathrm{\eta }\in \mathbb{N}}$ be a sequence of mappings ${\mathtt{H}}_{\mathrm{\eta }}:X⟶X$ . Then the sequence ${\mathtt{H}}_{1,\infty }=({\mathtt{H}}_1,{\mathtt{H}}_2,\dots )$ defines a non–autonomous discrete system $(X,{\mathtt{H}}_{1,\infty })$ . For any point $\mathcal{x}\in X$ , its trajectory is defined by $\mathrm{orb}(\mathcal{x},{\mathtt{H}}_{1,\infty })=({{\mathtt{H}}_1}^{\mathrm{\eta }}\left(\mathcal{x}\right):\mathrm{\eta }\in \mathbb{N})$ , where ${{\mathtt{H}}_1}^{\mathrm{\eta }}={\mathtt{H}}_{\mathrm{\eta }}\circ \dots \circ {\mathtt{H}}_1$ , and ${{\mathtt{H}}_1}^0$ denotes the identity mapping similarly ${{\mathtt{H}}_{\mathrm{\eta }}}^{\mathrm{k}}={\mathtt{H}}_{\mathrm{\eta }+\mathrm{k}-1}\circ \dots \circ {\mathtt{H}}_{\mathrm{\eta }+1}\circ {\mathtt{H}}_{\mathrm{\eta }}$ .

Let $\mathcal{K}\left(X\right)$ denote the hyperspace of all non-empty compact subsets of $X$ , endowed with the Hausdorff metric ${\mathcal{d}}_{\mathtt{H}}$ , defined by ${\mathcal{d}}_{\mathtt{H}}(\mathrm{{\rm A}},\mathrm{{\rm B}})=max\{{sup}_{\mathcal{x}\in \mathrm{{\rm A}}}{inf}_{\mathcal{y}\in \mathrm{{\rm B}}}\mathcal{d}(\mathcal{x},\mathcal{y}),{sup}_{\mathcal{y}\in \mathrm{{\rm B}}}{inf}_{\mathcal{x}\in \mathrm{{\rm A}}}\mathcal{d}(\mathcal{x},\mathcal{y})\}$ for any $\mathrm{{\rm A}},\mathrm{{\rm B}}\in \mathcal{K}\left(X\right)$ . Then $(\mathcal{K}\left(X\right),{\mathcal{d}}_{\mathtt{H}})$ is a compact metric space.

Let $\mathtt{H}\mathtt{:}X\to \mathcal{K}\left(X\right)$ be a set–valued mapping, that is, for each $\mathcal{x}\in X$ , the image $\mathtt{H}\left(\mathcal{x}\right)$ is a non-empty compact subset of $X$ . The pair $(X,\mathtt{H})$ is called a set–valued dynamical system.

We denote by $X^{\mathrm{{\rm Z}}}$ the set of all bi–infinite sequences in $X$ , that is, $X^{\mathrm{{\rm Z}}}=\{{\left({\mathcal{x}}_{¡}\right)}_{¡\in \mathbb{Z}}:{\mathcal{x}}_{¡}\in X\}$ . This space is endowed with the product topology, under which it is a compact metric space.

A sequence ${\left\{{\mathcal{x}}_{¡}\right\}}_{¡\in \mathbb{Z}}\subset X$ is called a trajectory of $\mathtt{H}$ if ${\mathcal{x}}_{¡+1}\in \mathtt{H}\left({\mathcal{x}}_{¡}\right)$ , for all $¡\in \mathbb{Z}$ . Let $\mathrm{\delta }>0$ . A sequence ${\left\{{\mathcal{x}}_{¡}\right\}}_{¡\in \mathbb{Z}}\subset X$ is called a $\mathrm{\delta }▁$ pseudo trajectory of $\mathtt{H}$ if $\mathcal{d}(\mathtt{H}\left({\mathcal{x}}_{¡}\right),{\mathcal{x}}_{¡+1})<\mathrm{\delta }$ , for all $¡\in \mathbb{Z}$ , where the distance between a set and a point is defined by $\mathcal{d}(\mathtt{H}\left({\mathcal{x}}_{¡}\right),{\mathcal{x}}_{¡+1})=\underset{\mathcal{y}\in \mathtt{H}\left({\mathcal{x}}_{¡}\right)}{inf}\mathcal{d}(\mathcal{y},{\mathcal{x}}_{¡+1})$ .

Let ${\mathrm{\Phi }}_{\mathtt{H}}\left(\mathrm{\delta }\right)$ denote the set of all $\mathrm{\delta }▁$ pseudo trajectories of $\mathtt{H}$ . A mapping $\mathbf{ⱷ}:\mathrm{X}\to X^{\mathrm{{\rm Z}}}$ satisfying ${\mathbf{ⱷ}}_0\left(\mathcal{x}\right)=\mathcal{x}$ , for all $\mathcal{x}\in \mathrm{X}$ is called a $\mathrm{\delta }▁$ method for $\mathtt{H}$ .

We say that $\mathtt{H}$ has inverse shadowing property (ISP) if for every $\mathrm{ϵ}>0$ , there exists $\mathrm{\delta }>0$ such that for every $\mathrm{\delta }▁$ method $\mathbf{ⱷ}$ and every $\mathcal{x}\in \mathrm{X}$ , there exists $\mathcal{y}\in \mathrm{X}$ such that $\mathcal{d}({\mathcal{x}}_{¡},{\mathbf{ⱷ}}_{¡}\left(\mathcal{y}\right))<\mathrm{ϵ}$ , for all $¡\in \mathbb{Z}$ , where ${\left\{{\mathcal{x}}_{¡}\right\}}_{¡\in \mathbb{Z}}$ is a trajectory of $\mathtt{H}$ with ${\mathcal{x}}_0=\mathcal{x}$ .

For a sequence $${\left\{{\mathcal{x}}_{¡}\right\}}_{¡\in \mathbb{Z}}$$ , define

and

The sequence ${\left\{{\mathcal{x}}_{¡}\right\}}_{¡\in \mathbb{Z}}$ is called a $\mathrm{\delta }▁$ ergodic pseudo trajectory if

We say that $\mathtt{H}$ has the ergodic shadowing property (ESP) if every $\mathrm{ϵ}>0$ , there exists a $\mathrm{\delta }>0$ such that every $\mathrm{\delta }▁$ ergodic pseudo trajectory is $\mathrm{ϵ}▁$ ergodically shadowed by some point in $X$ .

The inverse limit space associated with $\mathtt{H}$ is defined by ${\mathrm{X}}_{\mathtt{H}}=\{{\left\{{\mathcal{x}}_{¡}\right\}}_{¡\in \mathbb{Z}}\in X^{\mathrm{{\rm Z}}}:{\mathcal{x}}_{¡+1}\mathtt{\in }\mathtt{H}\left({\mathcal{x}}_{¡}\right)\}$ . The shift mapping ${\mathrm{\sigma }}_{\mathtt{H}}:{\mathrm{X}}_{\mathtt{H}}\to {\mathrm{X}}_{\mathtt{H}}$ is defined by ${\mathrm{\sigma }}_{\mathtt{H}}\left(\right({\mathcal{x}}_{¡}\left)\right)=\left({\mathcal{x}}_{¡+1}\right)$ . The space ${\mathrm{X}}_{\mathtt{H}}$ is endowed with the metric $\tilde{\mathcal{d}}\left(\right({\mathcal{x}}_{¡}),({\mathcal{y}}_{¡}\left)\right)={\sum }_{¡\in \mathbb{Z}}(\mathcal{d}({\mathcal{x}}_{¡},{\mathcal{y}}_{¡})/2^{|¡|})$ .

For each $¡\in \mathbb{Z}$ , define the projection mapping ${\mathrm{\pi }}_{¡}:{\mathrm{X}}_{\mathtt{H}}\to \mathrm{X}$ by ${\mathrm{\pi }}_{¡}\left({\left\{{\mathcal{x}}_{\mathrm{j}}\right\}}_{\mathrm{j}\in \mathbb{Z}}\right)={\mathcal{x}}_{¡}$ for any ${\left\{{\mathcal{x}}_{\mathrm{j}}\right\}}_{\mathrm{j}\in \mathbb{Z}}\in {\mathrm{X}}_{\mathtt{H}}$ . Then for every $¡\in \mathbb{Z}$ , the mapping ${\mathrm{\pi }}_{¡}$ is continuous. Moreover, it satisfies ${\mathrm{\pi }}_{¡}\circ {\mathrm{\sigma }}_{\mathtt{H}}={\mathrm{\pi }}_{¡+1}$ .

In addition, by the definition of the inverse limit space, we have ${\mathrm{\pi }}_{¡+1}\left(\mathcal{x}\right)\in \mathtt{H}\left({\mathrm{\pi }}_{¡}\left(\mathcal{x}\right)\right)$ for all $\mathcal{x}\in {\mathrm{X}}_{\mathtt{H}}$ .

3. Results

4. Conclusion

In this paper, we present a new definition of the inverse and ergodic shadowing properties for set-valued mappings. We explore the relationship between the inverse shadowing property and the ergodic shadowing property of set-valued mappings, as well as their connection to the shift mapping ${\mathrm{\sigma }}_{\mathtt{H}}$ on the inverse limit space.