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2046-1402

F1000 Research Limited

London, UK

10.12688/f1000research.173380.1

Research Article

Articles

On Ergodic and Inverse Shadowing Properties of Set-Valued Mapping

[version 1; peer review: 1 approved, 1 approved with reservations, 1 not approved]

Wattan Kamil

Farah

Writing – Original Draft Preparation Writing – Review & Editing a 1 AL-Shara'a

Iftichar M.T.

Supervision 1 1Department of Mathematics, College of Education for Pure Sciences, University of Babylon, Babil, Iraq

a farah.kamil.pure571@student.uobabylon.edu.iq

No competing interests were disclosed.

20 2 2026

2026

299

24 12 2025

2026

This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Background

Shadowing-type properties play a fundamental role in the qualitative theory of dynamical systems, as they describe the relationship between approximate trajectories and exact orbits. In recent years, increasing attention has been given to extending these concepts to set-valued mappings, which naturally arise in various areas of mathematics and applied sciences. However, several shadowing-related notions for such mappings remain insufficiently explored.

Methods

In this work, we introduce precise definitions of the inverse shadowing property and the ergodic shadowing property for set-valued mappings. We analyse these properties within a general topological framework and examine their behaviour under the shift mapping on the inverse limit space. The relationships between inverse shadowing and ergodic shadowing are investigated using tools from topological dynamics.

Results

We establish connections between the inverse shadowing property and the ergodic shadowing property for set-valued mappings. In particular, we show how these properties interact when considered together with the shift mapping on the inverse limit space, and we identify conditions under which one property implies the other.

Conclusions

The results provide a clearer understanding of shadowing phenomena for set-valued mappings and highlight the role of inverse limit spaces in studying their dynamical behavior. This work contributes to the development of shadowing theory beyond single-valued dynamics and offers a foundation for further investigations in this direction.

set–valued mappings; shift mapping σ_H; inverse shadowing property.

The author(s) declared that no grants were involved in supporting this work.

1. Introduction

The phenomenon of shadowing approximation trajectories (pseudo trajectories) of dynamical systems by exact trajectories is one of the most extensively examined issues in contemporary global dynamical systems theory (for example, ^{1,
2}). On the other hand, there has begun an equally intense development at the inverse shadowing problem, where a class of methods generating pseudo trajectories is specified and the question is studied as to if it is possible to approximate any precise trajectory by a trajectory of any method from this class (see ^{3,
4}).

Román-Flores, ⁵ in 2003, researched the relationship of transitivity with the associated system. After his research, numerous scientists examined the characteristics of set-valued discrete systems. For instance, the transitivity and mixing. ^{6,
7} The ergodic shadowing property is shown by Shabani and Ahmadi ⁸ to be a feature of chain mixing in non-autonomous discrete-time dynamical systems.

Koo and Lee, ⁹ in 2024 proved that the uniform ergodic shadowing characteristic holds for sequences of homeomorphisms on non-compact metric spaces. Al-Sharaa and Russl A., ¹⁰ in 2023, study generic non-autonomous discrete dynamical systems’ shadowing and w-expansive properties and the relationship between them. In the recent work, ¹¹ in 2025, Koo and Lee introduced the notions of dynamical systems on non-compact metric spaces and their ergodic shadowing property.

In this research, we focus on the inverse shadowing property with the ergodic shadowing property in set-valued systems and its relation with inverse limits. We introduce a new definition of the inverse and ergodic shadowing properties of set-valued mappings. This means we will create a space whose elements are sets. We will structure examples that satisfy the new definition in the future research. In the following section, we introduce some important concepts that we need:

1.1 Non–autonomous discrete system

Let Ⲭ be represented by any compact metric space, where d is the symbol for the metric on Ⲭ . Let H η : Ⲭ ⟶ Ⲭ , η ∈ ℕ be a series of mappings represented by H 1 , ∞ = ( H 1 , H 2 , … ) . The above sequence indicates a non–autonomous discrete system ( Ⲭ , H 1 , ∞ ) . According to this mapping sequence. The point S ∈ Ⲭ ’s trajectory can be defined as orb ( S , H 1 , ∞ ) = ( H 1 η ( S ) ) , η ∈ ℕ , where H 1 η = H η ∘ … ∘ H 1 , and H 1 0 represents the identity mapping similarly H η k = H η + k − 1 ∘ … ∘ H η + 1 ∘ H η .

K ( Ⲭ ) is the hyperspace of Ⲭ . The space of compact nonempty subsets of Ⲭ where the Hausdorff metric d H is defined as follows: d H ( Α , Β ) = max { sup S ∈ Α inf T ∈ Β d ( S , T ) , sup T ∈ Β inf S ∈ Α d ( S , T ) } for any Α , Β ∈ K ( Ⲭ ) . ( K ( Ⲭ ) , d H ) is a compact metric space and H : K ( Ⲭ ) → K ( Ⲭ ) is a set–valued mapping on Ⲭ . We call the pair ( Ⲭ , H ) a set–valued system.

The product topology is applied to Ⲭ Ζ , which is the compact metric space of all bi–infinite sequences ઇ = { S k : k ∈ ℤ } in Ⲭ . Let H : K ( Ⲭ ) → K ( Ⲭ ) be a set–valued mapping for a constant value δ > 0 , where Φ H ( δ ) represents the set of all δ ▁ pseudo trajectories of H . We will provide definitions of inverse shadowing property and ergodic shadowing property for set–valued mapping.

A mapping ⱷ : Ⲭ → Φ H ( δ ) ⊆ Ⲭ Ζ that satisfies ⱷ 0 ( S ) = S , S ∈ Ⲭ is known as a δ ▁ method for H . For any positive ε , there is a positive δ such that for every S in Ⲭ , and any δ ▁ method ⱷ : Ⲭ → Ⲭ Ζ , there exists T in Ⲭ , such that d ( H k ( S ) , ⱷ k ( T ) < ε ) for any k in ℤ , we say that H has inverse shadowing property, denoted as ISP.

Considering a sequence ઇ = { S ¡ } ¡ ∈ ℤ , take D ( ઇ , H , δ ) = { ¡ ∈ ℤ : d ( H ( S ¡ ) , S ¡ + 1 ) ≥ δ } D η ( ઇ , H , δ ) = D ( ઇ , H , δ ) ∩ { − η , … , − 2 , − 1 , 0 , 1 , 2 , … , η }

In consideration of a sequence ઇ = { S ¡ } ¡ ∈ ℤ and a point S ∈ Ⲭ put D S ( ઇ , S , H , δ ) = { ¡ ∈ ℤ : d ( H ¡ ( S ) , S ¡ ) ≥ δ } , D S η ( ઇ , S , H , δ ) = D S ( ઇ , S , H , δ ) ∩ { − η , … , − 2 , − 1 , 0 , 1 , 2 , … , η } .

Let ઇ = { S ¡ } ¡ ∈ ℤ be defined as an δ ▁ ergodic pseudo trajectories for H if the following limit holds: lim | η | → ∞ ( card ( D η ( ઇ , H , δ ) ) / η ) = 0 .

An δ ▁ ergodic pseudo trajectory is considered ϵ ▁ ergodic shadowing with a point S ∈ Ⲭ denoted by ESP if lim | η | → ∞ ( card ( D S η ( ઇ , S , H , δ ) ) / η ) = 0 .

We propose that an H ▁ inveriant set Λ possesses an ergodic shadowing property when, for every ϵ > 0 , there exists a δ > 0 that means all δ ▁ ergodic pseudo trajectories in Λ can be ergodic shadowed by a single point S ∈ Ⲭ .

A closed subspace Ⲭ H = { ( S η ) : S ∈ Ⲭ , H ( S η ) = S η + 1 , η ∈ ℤ } of Ⲭ Ζ in addition to the corresponding shift mapping σ H : Ⲭ H → Ⲭ H , which is defined as σ H ( ( S η ) ) = ( T η ) where T η = S η + 1 for all η ∈ ℤ is referred to as the inverse limit space of H . It is important to observe that the compact metric space Ⲭ H is determined by the metric d ∼ , which is defined as d ∼ ( ( S ¡ ) , ( T ¡ ) ) = ∑ ¡ ∈ ℤ ( d ( S ¡ , T ¡ ) / 2 | ¡ | ) .

For each ¡ , j ∈ ℤ , consider ¡ the projection mapping π ¡ : Ⲭ H → K ( Ⲭ ) by π ¡ ( S ) = S ¡ for any S ∼ ∈ Ⲭ H where S ∼ = { … , S ¡ − 1 , S ¡ , S ¡ + 1 , … } , for any ¡ ∈ ℤ , the mapping π i satisfies that H ∘ π ¡ = π ¡ ∘ σ H as it is an open continuous mapping.

2. Results

Theorem 2.1.

A continuous surjective set–valued mapping on a compact metric space Ⲭ is denoted by H : K ( Ⲭ ) → K ( Ⲭ ) . Assuming H has ESP, the corresponding shift mapping σ H on the inverse limit space Ⲭ H also has ESP.

Proof:

Let ϵ be greater than zero; then diam ( Ⲭ ) = α we can select N from ℕ such that ( α 2 N − 1 ) < ( ϵ 8 ) . Since d ( S , T ) < γ , the uniform continuity of H ensures that d ( H ¡ ( S ) , H ¡ ( T ) ) < ( ϵ 8 ) for 0 ≤ ¡ ≤ 2 N . Given that H has ESP. Any τ ▁ ergodic pseudo trajectory is γ ▁ ergodic shadowed by a point in Ⲭ for any γ > 0 there is τ > 0 . Get δ > 0 so that 0 < δ 2 N < τ .

Suppose { S η } η ∈ ℤ ⊆ Ⲭ H be a δ ▁ ergodic pseudo trajectory with σ H . Because { η \ d ( H ( S − N η ) , S − N η + 1 ) ≥ τ } ⊆ { η \ d ∼ ( σ H ( S η ) , S η + 1 ) ≥ δ } the sequence of { S − N η } forms a τ ▁ ergodic pseudo trajectory for H . However, there exists T ∈ Ⲭ that means lim | m | → ∞ ( card ( { η \ d ( H η ( T ) , S − N η ) ≥ γ } ∩ { − m + 1 , … , m − 1 } ) ) × m − 1 = 0 .

Suppose T ¡ − N = H ¡ ( T ) for any ¡ ≥ 0 and T ¡ − N ∈ H − 1 ( T ¡ + 1 − N ) for ¡ < 0 , the T ∼ = ( T ¡ ) ∈ Ⲭ H . If η ∉ { η / d ( H η ( T ) , S η ) ≥ γ } then η ∉ { η / d ∼ ( σ η ( T ∼ ) , S η ) ≥ ϵ } which is to say { η / d ∼ ( σ η ( T ∼ ) , S η ) ≥ ϵ } ⊆ { η / d ( H η ( T ) , S − N η ) ≥ γ } . Consequently, { S η } ⊆ Ⲭ H could display ϵ ▁ ergodic shadowed for T ∼ ∈ Ⲭ H . ■

Theorem 2.2.

A continuous surjective set–valued mapping on a compact metric space Ⲭ is denoted by H : K ( Ⲭ ) → K ( Ⲭ ) . Assuming H has ISP, the corresponding shift mapping σ H on the inverse limit space Ⲭ H also has ISP.

Proof:

Let ϵ be greater than zero; then diam ( Ⲭ ) = α we can select N from ℕ so that ( α 2 N − 1 ) < ( ϵ 8 ) . Since d ( S , T ) < γ , the uniform continuity of H ensures that d ( H ¡ ( S ) , H ¡ ( T ) ) < ( ϵ 8 ) for 0 ≤ ¡ ≤ 2 N .

Given that H has ISP, for each γ > 0 ,there is τ > 0 which means that with each S ∈ Ⲭ and every τ ▁ method ⱷ , it exists T ∈ Ⲭ satisfying d ( H k ( S ) , ⱷ k ( T ) ) < γ , for all k ∈ ℤ .

Select δ > 0 such that 0 < δ 2 N < τ . suppose that ⱷ ∼ : Ⲭ H → Φ ∼ σ is a δ ▁ method for σ H . We construct a τ ▁ method of H this way: let S ∈ Ⲭ , choose a point S ∼ = ( S ¡ ) ∈ Ⲭ H where π − N ( S ∼ ) = S . If ⱷ ∼ constitutes a δ ▁ method of σ H , therefore, S ∼ ∈ Ⲭ H , d ( σ H ( ⱷ ∼ η ( S ∼ ) ) , ⱷ ∼ η + 1 ( S ∼ ) ) < δ .

Let ⱷ : Ⲭ → Ⲭ Ζ be defined as ⱷ ( S ) = { ⱷ η ( S ) } η ∈ ℤ , where ⱷ η ( S ) = π − N ( ⱷ ∼ η ( S ∼ ) ) . It is evident that ⱷ ( S ) forms a τ ▁ pseudo trajectory for H ; hence, ⱷ behaves as a τ ▁ method for H . Since H is ISP, every π − N ( S ∼ ) ∈ Ⲭ , where S ∼ ∈ Ⲭ H it has T ∈ Ⲭ that gives d ( H η ( π − N ( S ∼ ) ) , ⱷ η ( T ) ) < γ . If T ∼ ∈ Ⲭ H where π − N ( T ∼ ) = T ; then d ( σ H η ( S ∼ ) , ⱷ ∼ η ( T ∼ ) ) < ϵ ; this means that σ H has ISP. ■

Theorem 2.3.

A local homeomorphism set–valued mapping on a compact metric space Ⲭ is denoted by H : K ( Ⲭ ) → K ( Ⲭ ) . Assuming the shift mapping σ H on Ⲭ H has ESP, then H also has ESP.

Proof:

For each ϵ > 0 , according to the ESP of σ H , it has a τ > 0 for every τ ▁ ergodic pseudo trajectory in σ H a point in Ⲭ H that forms ϵ ▁ ergodic shadowed. Let α = diam ( Ⲭ ) , and choose N ∈ ℕ so that ( α 2 N − 1 ) < ( τ 4 ) .

H be the local homeomorphism; it has a γ such that 0 < γ < ( τ / 4 ) and H | U : U → U forms a homeomorphism when U defines the γ ▁ neighborhood of S ∈ Ⲭ . H being uniformly continuous, for τ / 8 , there is 0 < Γ < ( τ / 8 ) where d ( S , T ) < Γ means d ( H j ( S ) , H j ( T ) ) < ( τ / 8 ) for every j , | j | ≤ N .

Let { S η } η ∈ ℤ be a Γ ▁ ergodic pseudo trajectory for H ; define S 0 η = S η and S η = ( S ¡ η ) ∈ Ⲭ H . If η ∉ { η | d ( H ( S η ) , S η + 1 ) ≥ Γ } , then η ∉ { η | d ∼ ( σ H ( S η ) , S η + 1 ) ≥ τ } . This shows that { η | d ∼ ( σ H ( S η ) , S η + 1 ) ≥ τ } ⊆ { η | d ( H ( S η ) , S η + 1 ) ≥ Γ } , and hence { S η } defines a τ ▁ ergodic pseudo trajectory for σ H . Since σ H has ESP, there is T ∼ = ( T i ) in Ⲭ H such that lim | m | → ∞ ( { η | d ∼ ( σ H η ( T ∼ ) , S η ) ≥ ϵ } ⋂ { − m + 1 , … , m − 1 } / m ) = 0 .

If η ∉ { η | d ∼ ( σ H η ( T ∼ ) , S η ) ≥ ϵ } , then η ∉ { η | d ( H η ( T 0 ) , S 0 η ) ≥ ϵ } , and we get { η | d ( H η ( T 0 ) , S η ) ≥ ϵ } , ⊆ { η | d ∼ ( σ H η ( T ) , S η ) ≥ ϵ } . It is now evident by definition that H has ESP.■

Theorem 2.4.

A local homeomorphism set–valued mapping on a compact metric space Ⲭ is denoted by H : K ( Ⲭ ) → K ( Ⲭ ) . Assuming the shift mapping σ H on Ⲭ H has ISP, then H also has ISP.

Proof:

For each ϵ > 0 by ISP of σ H , there is τ > 0 such that, for each of S ∼ ∈ Ⲭ H and for every τ ▁ method ⱷ ∼ of σ H , there is T ∼ ∈ Ⲭ H so that d ∼ ( σ H η ( S ∼ ) , ⱷ ∼ η ( T ∼ ) ) < ϵ .

Let α = diam ( Ⲭ ) , select N ∈ ℕ so that ( α 2 N − 1 ) < ( τ 4 ) . H be a local homeomorphism; U represents the γ ▁ neighborhood of S ∈ Ⲭ , and H | U : U → U is a homeomorphism if there is a γ such that 0 < γ < ( τ / 4 ) . Since H represents uniform continuity at τ / 8 ; for every j where | j | ≤ N there exists 0 < Γ < ( τ / 8 ) such that d ( S , T ) < Γ and d ( H j ( S ) , H j ( T ) ) < ( τ / 8 ) .

Let S be an element of Ⲭ , and define ⱷ : Ⲭ → Φ H ( δ ) as a γ ▁ method of H . Let ⱷ ∼ : Ⲭ H → Ⲭ H ℤ so that ⱷ ∼ ( S ∼ ) = { ⱷ ∼ η ( S ∼ ) } η ∈ ℤ with π 0 ( ⱷ ∼ ( S ∼ ) ) = ⱷ η ( S ) we have d ∼ ( σ H ( ⱷ ∼ η ( S ∼ ) ) , ⱷ ∼ η + 1 ( S ∼ ) ) < δ . Therefore ⱷ ∼ : Ⲭ H → Ⲭ H ℤ represents a δ ▁ method for σ H , since σ H has the ISP; thus, given S ∼ ∈ Ⲭ H there is T ∼ ∈ Ⲭ H , which means d ∼ ( σ H η ( S ∼ ) , ⱷ ∼ η ( T ∼ ) ) < ϵ , this implies that d ( H η ( S ) , ⱷ η ( T ) ) < ϵ ; hence, H has the ISP. ■

3. 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 σ H on the inverse limit space.

Data availability

Our article type does not require data.

References 1

Pilyugin

: Shadowing in Dynamical Systems. Berlin: Springer;1999. 10.1007/BFb0093184

Palmer

: Shadowing in Dynamical Systems. Theory and Applications. 2000. 10.1007/978-1-4757-3210-8

Corless

Pilyugin

: Approximate and Real Trajectories for Generic Dynamical Systems. J. Math. Anal. Appl. 1995;189:409–423. 10.1006/jmaa.1995.1027

Kloeden

Ombach

: Hyperbolic Homeomorphisms and Bishadowing. Ann. Polon. Math. 1997;65:171–177. 10.4064/ap-65-2-171-177

Román-Flores

: A Note on Transitivity in Set-Valued Discrete Systems. Chaos, Solitons Fractals. 2003;17:99–104. 10.1016/S0960-0779(02)00406-X

Sánchez

Sanchis

Villanueva

: Chaos in Hyperspaces of Non autonomous Discrete Systems. Chaos, Solitons and Fractals: The Interdisciplinary Journal of Nonlinear Science, and Non equilibrium and Complex Phenomena. 2017;94:68–74. 10.1016/j.chaos.2016.11.009

Shao

Zhu

: Chaos in Non-Autonomous Discrete Systems and Their Induced Set-Valued Systems. Chaos. 2019;29:033117. 30927855

10.1063/1.5054867

Shabani

Ahmadi

: On Ergodic Shadowing and Specification Properties of Non autonomous Discrete Dynamical Systems. Maha. Math. Rese. Cent. 2021;2645–4505. 10.22103/jmmrc.2021.15567.1114

Russl

Al-Sharaa

: The w-Expansive With The Shadowing Property in g-Non autonomous Discrete Dynamical Systems. Kufa for Mathematics and Computer. 2023;10(2):125–131. 10.31642/JoKMC/2018/100221%20

Koo

Lee

: On The Ergodic Shadowing Property Through Uniform Limits. Chun. Math. Soc. 2024;37(2):75–80. 10.14403/jcms.2024.37.2.75

Koo

Lee

: Totally Ergodic Shadowing Property on Non-compact Metric Spaces. Bull. Korean Math. Soc. 2025;62(1):187–199. 10.4134/BKMS.b240118

10.5256/f1000research.191190.r461381

Reviewer response for version 1

Dzul-Kifli

Syahida Che

1 Referee https://orcid.org/0000-0002-1677-2983 1Universiti Kebangsaan Malaysia, Bangi, Malaysia

Competing interests: No competing interests were disclosed.

13 3 2026

2026

This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

recommendation

approve-with-reservations

1) The abstract should clearly indicate the principal results obtained regarding the relationship between the inverse shadowing property and the ergodic shadowing property for set-valued mappings.

2) In the Introduction section, several sentences lack clarity and the logical flow between them is weak. The connection between some statements and the main objective of the paper is not clearly established. Furthermore, the discussion of the cited works is rather limited, and their relevance to the present study is not sufficiently explained.

3) The subsection 1.1 Non–autonomous discrete system mainly introduces terminology and basic concepts. It would be more appropriate to restructure this part as a separate section, e.g., Section 2: Terminology and Basic Concepts (or Preliminaries)

4) In 1.1, the statement that the mapping is a set-valued mapping on the space may require clarification, since the mapping is actually defined on the hyperspace consisting of compact subsets of the space rather than directly on the space itself.

5) In Section 1.1, the exposition in this section would benefit from clearer explanation and notation. For example, several concepts are introduced without sufficient description, such as the set of pseudo-trajectories, the index sets used to measure deviations along a sequence, and the mapping that generates sequences of points. The authors should explain these objects more clearly, indicate their roles in the definitions of pseudo-trajectories and inverse shadowing and cite appropriate references for the definitions and terminology introduced in this section.

6) The Section 2 on results would benefit from a short introductory paragraph summarizing the main results. Moreover, if Section 1.1 is reorganized as a preliminaries section as suggested above, the numbering of the sections should be adjusted accordingly (e.g., the current Section 2 may become Section 3).

7) The assumptions of the Theorem 2.1 require clarification. In particular, the notions of continuity and surjectivity for the set-valued mapping should be specified, and an appropriate reference for the construction of the inverse limit space in this context would be helpful.

8) The proof of Theorem 2.1 seems to omit some intermediate steps, particularly in the passage from pseudo-trajectories in the inverse limit space to pseudo-trajectories for the original mapping, and in the final construction of the shadowing sequence. These arguments should be justified more carefully.

9) Similar comments to those for Theorem 2.1 apply to Theorem 2.2, particularly regarding the clarity of the theorem statement, the density of the notation, and the need for more detailed justification of several steps in the proof. If similar arguments relating shadowing properties and inverse limit spaces appear in the literature, appropriate references should be provided.

10) Similar comments to those for Theorems 2.1 and 2.2 apply to Theorem 2.3 regarding the clarity of the statement, the notation used in the proof, and the justification of several intermediate steps. In addition, the notion of a local homeomorphism for the set-valued mapping should be clarified.

11) Similar comments to those for Theorems 2.1–2.3 apply to Theorem 2.4, particularly concerning the clarity of the assumptions, the density of the notation in the proof, and the need for clearer explanation of several intermediate steps.

12) The novelty and significance of the theorems should be clarified in relation to existing results on shadowing properties and inverse limit spaces. In addition, some assumptions (such as continuity, surjectivity, and the notion of a local homeomorphism for the set-valued mapping) should be more clearly defined or supported by appropriate references.

13) If the results are correct, they could provide a useful extension of known relationships between shadowing properties and inverse limit spaces to the setting of set-valued dynamical systems. However, the authors should clarify more explicitly how their results differ from or extend the existing theory for single-valued dynamical systems.

14) The conclusion section is relatively brief and mainly restates the objectives of the paper. It would be helpful if the authors summarize the main results more clearly and emphasize the novelty and significance of the findings, as well as possible directions for future research.

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Partly

Reviewer Expertise:

Dynamical systems, topology, and fuzzy mathematics.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

wattan

farah

University of Babylon, Iraq

Competing interests: No competing interests were disclosed.

27 3 2026

We thank the reviewer for the constructive and helpful comments.

The abstract and introduction have been improved to clearly reflect the main contributions.

The structure of the paper has been reorganized for better clarity.

Definitions and notation have been refined and clarified.

The proofs have been strengthened with additional details and explanations.

The assumptions and conditions used in the theorems have been clearly stated and justified.

We believe that the revised manuscript fully addresses the reviewer’s comments and meets the expected scientific standards.

10.5256/f1000research.191190.r461379

Reviewer response for version 1

Khan

Abdul Gaffar

1 Referee 1University of Delhi, Delhi, New Delhi, India

Competing interests: No competing interests were disclosed.

12 3 2026

2026

recommendation

reject

The results seem to be publishable at first glance, but the quality of the writing and structure needs to be improved a lot. Authors are highly recommended to do the careful reading and address the comments below. Note that these comments are just few examples that needs to be addressed throughout the paper and so after careful reading you might find many more such changes. I am just writing

1.English sentences and grammar are poorly structured. For eg. "in Introuduction section i.e. [Line 2, Page 3/7 of pdf] 'shadowing approximation trajectories -> shadowing by approximated trajectories", "in section 1.1 According to this mapping sequence . - > fullstop needs to be addressed" etcetera"

2. Some references are not properly cited for example see Section 1 - Para 2 and Para 3

3. Section 1 claims that authors are going to provide examples but the paper does not contain any example

4. Standard sentences and necessary definitions are not even written properly in standard form mathematics. for example 'k(X) is the hyperspace of X" is improper sentence structuring.

5. Use of non-standard notations should be avoided. For example, use x, y,z etc for elements of $X$ instead of curl letters, do not use greek letters for integeres etc

5. The paper does not study Non-autonomous systems and thus it is absoulutely absurd to see this section in paper.

6. The work is done on the induced dynamical systems and not on the set-valued systems. Set-valued systems correspond to a map which takes one element of a space to a subset but the systems author studied takes one element to one element of space. Thus term set-valued should be dropped.

7. Check definition of density

8. Definition of ESP is not what it should be. it is written with no care.

I am skipping a lot of observations that need to be addressed by the authors after reading the paper with extreme care. Only after that it will be easier to decide the quality of the results as in this stage this paper is not reader friendly.

Is the work clearly and accurately presented and does it cite the current literature?

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Partly

Reviewer Expertise:

Topological Dynamics

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

wattan

farah

University of Babylon, Iraq

Competing interests: No competing interests were disclosed.

27 3 2026

We thank the reviewer for the careful evaluation and valuable remarks.

The notation has been standardized throughout the manuscript to improve consistency and readability.

The definition of set-valued mapping has been corrected and clarified as a mapping of the form H: X → K(X).

The role of the hyperspace K(X) and the Hausdorff metric has been clearly explained.

The definitions of pseudo-trajectories, δ-methods, and ergodic shadowing have been rewritten with precise mathematical formulation.

All proofs have been revised to include detailed justifications and to ensure mathematical rigor.

These revisions have significantly improved the clarity and correctness of the manuscript.

10.5256/f1000research.191190.r461372

Reviewer response for version 1

Mangang

Khundrakpam Binod

1 Referee 1Manipur University, Imphal, Manipur, India

Competing interests: No competing interests were disclosed.

9 3 2026

2026

recommendation

approve

This paper explores the intersection of set-valued dynamical systems and shadowing theory, focusing on the Ergodic Shadowing Property (ESP) and the Inverse Shadowing Property (ISP).

Recommendation: Accept / Highly Favourable

1. Theoretical Significance

The research successfully bridges a gap in contemporary global dynamical systems theory by extending ergodic shadowing concepts to hyperspaces (sets of compact subsets). By proving that the ESP is preserved under the shift mapping of the inverse limit space (Theorem 2.1), the authors provide a vital link between the internal dynamics of a set-valued mapping and its asymptotic topological structure.

2. Innovation in Definitions

The paper introduces a rigorous new framework for $\delta$-methods and ergodic pseudo-trajectories within set-valued systems. This is a timely contribution, following the trajectory of recent work by Koo and Lee (2024, 2025), and moves the field forward by shifting the focus from single-point trajectories to set-valued evolutions.

3. Mathematical Rigour

The proof for Theorem 2.1 demonstrates a sophisticated use of the Hausdorff metric ($d_H$) and the product topology metric on the inverse limit space. The authors correctly handle the complexities of uniform continuity and density limits, ensuring the results are robust.

4. Future Potential

The introduction of a space where elements are themselves sets opens new doors for structuring examples in non-autonomous discrete-time systems. This provides a strong foundation for future research into the stability and "inverse" behavior of complex systems.

Conclusion:

This is a well-structured and technically sound contribution to the field. It provides both a theoretical synthesis of existing literature (Román-Flores, Al-Sharaa, etc.) and original results that are essential for researchers studying the stability of discrete dynamical systems.

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Topological Dynamics

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

wattan

farah

University of Babylon, Iraq

Competing interests: No competing interests were disclosed.

27 3 2026

We thank the reviewer for the detailed and insightful comments.

The abstract has been revised to clearly state the main results and highlight the relationship between inverse shadowing and ergodic shadowing properties for set-valued mappings.

The introduction has been rewritten to improve clarity and logical flow and to better explain the relevance of the cited works to our study.

The preliminary material has been reorganized into a separate section titled "Preliminaries."

All definitions and notation have been clarified, especially those related to pseudo-trajectories, methods, and ergodic shadowing.

The assumptions in all theorems (continuity, surjectivity, and local homeomorphism) are now explicitly stated.

All proofs have been expanded by adding missing intermediate steps and improving explanations.

The novelty and contribution of the results have been clarified, emphasizing the extension from single-valued to set-valued dynamical systems.

The conclusion has been expanded to better summarize the results and highlight their significance.