Topologically Stable Equicontinuous Non-Autonomous Systems

Abdul Gaffar Khan; Pramod Kumar Das; Tarun Das

arxiv: 1906.09815 · v1 · pith:WC4UBLW5new · submitted 2019-06-24 · 🧮 math.DS

Topologically Stable Equicontinuous Non-Autonomous Systems

Abdul Gaffar Khan , Pramod Kumar Das , Tarun Das This is my paper

Pith reviewed 2026-05-25 17:03 UTC · model grok-4.3

classification 🧮 math.DS

keywords topological stabilitynon-autonomous dynamical systemsequicontinuityexpansivenessshadowing propertycommutative mapsmetric spacesmean shadowing

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The pith

Commutative non-autonomous systems on metric spaces are topologically stable when equicontinuous, expansive, and possessing shadowing.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper supplies sufficient conditions under which commutative non-autonomous dynamical systems on metric spaces become topologically stable. It establishes three results: mean equicontinuous mean expansive systems with strong average shadowing are topologically stable, equicontinuous recurrently expansive systems with almost shadowing are topologically stable, and equicontinuous expansive systems with shadowing are topologically stable. A reader would care because topological stability ensures that small changes to the sequence of maps do not alter the system's long-term orbit structure. The commutativity condition lets the sequence of maps act coherently, allowing these classical stability criteria to carry over from single-map systems.

Core claim

The authors prove that every equicontinuous, expansive commutative non-autonomous system with the shadowing property is topologically stable. Parallel statements hold when equicontinuity and expansiveness are replaced by their mean versions together with strong average shadowing, and when expansiveness is replaced by recurrent expansiveness together with almost shadowing.

What carries the argument

The central mechanism is the combination of equicontinuity (or mean equicontinuity), expansiveness (or mean or recurrent expansiveness), and a shadowing property (or strong average or almost shadowing) that forces the non-autonomous system to remain topologically stable under small perturbations of the maps.

If this is right

Mean equicontinuous mean expansive systems with strong average shadowing property are topologically stable.
Equicontinuous recurrently expansive systems with almost shadowing property are topologically stable.
Equicontinuous expansive systems with shadowing property are topologically stable.
The long-term behavior of these systems persists when the defining maps are replaced by nearby maps.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Verification of the listed properties supplies a practical test for topological stability that avoids checking every possible perturbation directly.
The same pattern of conditions may extend to sequences on spaces equipped with uniform structures weaker than metrics, provided the continuity and expansiveness notions remain well-defined.
Relaxing commutativity would likely require additional uniformity assumptions on the maps to recover comparable stability conclusions.

Load-bearing premise

The maps in the sequence must commute so that the order of application leaves the dynamics unchanged.

What would settle it

A single commutative non-autonomous system on a metric space that is equicontinuous and expansive, satisfies the shadowing property, yet changes its orbit structure under an arbitrarily small perturbation of the maps would disprove the claim.

read the original abstract

We find sufficient conditions for commutative non-autonomous systems on certain metric spaces to be topologically stable. In particular, we prove that (i) Every mean equicontinuous, mean expansive system with strong average shadowing property is topologically stable. (ii) Every equicontinuous, recurrently expansive system with almost shadowing property is topologically stable. (iii) Every equicontinuous, expansive system with shadowing property is topologically stable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper states three sufficient conditions for topological stability in commutative non-autonomous systems by extending the usual autonomous ones, with no visible contradictions in the claims.

read the letter

The main thing to know is that the paper lists three explicit theorems: mean equicontinuous plus mean expansive plus strong average shadowing implies stability; equicontinuous plus recurrently expansive plus almost shadowing implies stability; and equicontinuous plus expansive plus shadowing implies stability. All are for commutative non-autonomous systems on metric spaces. This is the direct extension of the classical autonomous result to the non-autonomous commutative setting, and the abstract states the commutativity and metric-space hypotheses clearly at the start. That counts as new for this narrow corner of topological dynamics. The paper does well by keeping the statements short and by separating the three cases instead of forcing everything into one theorem. The stress-test finds no internal contradictions or unsupported steps in the stated claims, which matches what the abstract shows. The soft spots are proportionate and not severe. These are only sufficient conditions, not characterizations, and the commutativity assumption restricts how far the results reach. No examples or downstream uses appear in the abstract, so the work stays inside the subfield without broadening its appeal. Proof details are not visible here, but nothing in the setup looks circular or invented. This paper is for readers already working on equicontinuity, expansiveness, and shadowing in non-autonomous systems. Someone in that group would find the three listed combinations useful to have on record. It deserves a serious referee to check the actual proofs and the handling of the non-autonomous definitions. I would send it to peer review rather than desk reject.

Referee Report

1 major / 0 minor

Summary. The paper claims to establish sufficient conditions for commutative non-autonomous dynamical systems on metric spaces to be topologically stable. It proves three results: (i) every mean equicontinuous, mean expansive system with the strong average shadowing property is topologically stable; (ii) every equicontinuous, recurrently expansive system with the almost shadowing property is topologically stable; and (iii) every equicontinuous, expansive system with the shadowing property is topologically stable.

Significance. If the stated implications hold with the indicated hypotheses, the work extends the classical autonomous result on equicontinuous expansive systems with shadowing to the non-autonomous commutative setting. This would be a modest but useful contribution to the literature on topological stability and shadowing in non-autonomous dynamics.

major comments (1)

The provided manuscript text consists solely of the abstract; no definitions of the key notions (mean equicontinuity, recurrent expansiveness, strong average shadowing, etc.), no statements of the theorems with precise hypotheses, and no proofs are supplied. Without these, the central claims cannot be verified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The single major comment identifies a clear issue with the review materials provided, which we address directly below. We are happy to supply the complete manuscript to allow verification of the claims.

read point-by-point responses

Referee: The provided manuscript text consists solely of the abstract; no definitions of the key notions (mean equicontinuity, recurrent expansiveness, strong average shadowing, etc.), no statements of the theorems with precise hypotheses, and no proofs are supplied. Without these, the central claims cannot be verified.

Authors: We apologize for the oversight in the review package. Only the abstract appears to have been forwarded. The full manuscript on arXiv:1906.09815 contains the required definitions of all notions (mean equicontinuity, mean expansiveness, recurrent expansiveness, strong average shadowing, almost shadowing, etc.), the precise statements of the three theorems with their hypotheses on metric spaces, and the complete proofs. We will attach the full PDF to the resubmission so that the referee can verify the arguments. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states three direct implications from combinations of equicontinuity/mean equicontinuity, expansiveness/mean expansiveness/recurrent expansiveness, and shadowing/strong average/almost shadowing properties to topological stability for commutative non-autonomous systems on metric spaces. These are presented as theorems proved from the listed hypotheses; no equations reduce a claimed prediction to a fitted input by construction, no self-definitional loops appear, and no load-bearing self-citations or uniqueness theorems imported from the authors' prior work are referenced in the abstract or stated claims. The derivation chain consists of standard mathematical implications without the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on standard definitions and background results from topological dynamics (metric spaces, continuity of maps, shadowing notions) that are assumed known; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The systems are commutative non-autonomous dynamical systems on metric spaces.
Invoked in the first sentence of the abstract as the setting for all three results.

pith-pipeline@v0.9.0 · 5596 in / 1151 out tokens · 24828 ms · 2026-05-25T17:03:16.486804+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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work page 1970
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N. Aoki, K. Hiraide, Topological theory of dynamical systems: re cent advances, Elsevier, (1994)

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A. Z. Bahabadi, Shadowing and average shadowing properties fo r iterated function systems, Georgian Math. J., 22 (2015), 179–184

work page 2015
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M. L. Blank, Discreteness and continuity in problems of chaotic dy namics, Amer. Math. Soc., (1997)

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Cull, Diﬀerence equations as biological models, Scientiae Mathem aticae Japonicae, 64 (2006), 217–234

P. Cull, Diﬀerence equations as biological models, Scientiae Mathem aticae Japonicae, 64 (2006), 217–234

work page 2006
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E. M. Coven, M. Keane, Every compact metric space that suppo rts a positively expansive homeomorphism is ﬁnite, IMS Lecture Notes Monogr. Ser., Dynamics & Stochastics, 48 (2006), 304–305

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R. Gu, Y. Sheng, Z. Xia, The average shadowing property and tr ansitivity for continuous ﬂows, Chaos, Solitons and Fractals, 23 (2005), 989-995

work page 2005
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Kwietniak, P

D. Kwietniak, P. Oprocha, A note on the average shadowing prop erty for expansive maps, Topology Appl., 159 (2012), 19–27

work page 2012
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Mandelkern, Metrization of the one-point compactiﬁcation, P roc

M. Mandelkern, Metrization of the one-point compactiﬁcation, P roc. Amer. Math. Soc., 107 (1989), 1111–1115

work page 1989
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Marcon, F.B

D. Marcon, F.B. Rodrigues, Topological properties of discrete non-autonomous dynamical sys- tems; 2016 preprint

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Thakkar, R

D. Thakkar, R. Das, Topological stability of a sequence of maps on a compact metric space, Bull. Math. Sci., 4 (2014), 99–111

work page 2014
[12]

W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 6 (1950), 769–774

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[13]

Walters, Anosov diﬀeomorphisms are topologically stable, Top ology 9 (1970), 71–78

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work page 1970
[14]

Walters, On the pseudo orbit tracing property and its relatio nship to stability, Springer, Berlin, Heidelberg, (1978), 231–244

P. Walters, On the pseudo orbit tracing property and its relatio nship to stability, Springer, Berlin, Heidelberg, (1978), 231–244

work page 1978

[1] [1]

D. V. Anosov, On a class of invariant sets of smooth dynamical sy stems, Proc. 5th Int. Conf. on Nonlin. Oscill., 2, Kiev (1970), 39–45

work page 1970

[2] [2]

N. Aoki, K. Hiraide, Topological theory of dynamical systems: re cent advances, Elsevier, (1994)

work page 1994

[3] [3]

A. Z. Bahabadi, Shadowing and average shadowing properties fo r iterated function systems, Georgian Math. J., 22 (2015), 179–184

work page 2015

[4] [4]

M. L. Blank, Discreteness and continuity in problems of chaotic dy namics, Amer. Math. Soc., (1997)

work page 1997

[5] [5]

Cull, Diﬀerence equations as biological models, Scientiae Mathem aticae Japonicae, 64 (2006), 217–234

P. Cull, Diﬀerence equations as biological models, Scientiae Mathem aticae Japonicae, 64 (2006), 217–234

work page 2006

[6] [6]

E. M. Coven, M. Keane, Every compact metric space that suppo rts a positively expansive homeomorphism is ﬁnite, IMS Lecture Notes Monogr. Ser., Dynamics & Stochastics, 48 (2006), 304–305

work page 2006

[7] [7]

R. Gu, Y. Sheng, Z. Xia, The average shadowing property and tr ansitivity for continuous ﬂows, Chaos, Solitons and Fractals, 23 (2005), 989-995

work page 2005

[8] [8]

Kwietniak, P

D. Kwietniak, P. Oprocha, A note on the average shadowing prop erty for expansive maps, Topology Appl., 159 (2012), 19–27

work page 2012

[9] [9]

Mandelkern, Metrization of the one-point compactiﬁcation, P roc

M. Mandelkern, Metrization of the one-point compactiﬁcation, P roc. Amer. Math. Soc., 107 (1989), 1111–1115

work page 1989

[10] [10]

Marcon, F.B

D. Marcon, F.B. Rodrigues, Topological properties of discrete non-autonomous dynamical sys- tems; 2016 preprint

work page 2016

[11] [11]

Thakkar, R

D. Thakkar, R. Das, Topological stability of a sequence of maps on a compact metric space, Bull. Math. Sci., 4 (2014), 99–111

work page 2014

[12] [12]

W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 6 (1950), 769–774

work page 1950

[13] [13]

Walters, Anosov diﬀeomorphisms are topologically stable, Top ology 9 (1970), 71–78

P. Walters, Anosov diﬀeomorphisms are topologically stable, Top ology 9 (1970), 71–78

work page 1970

[14] [14]

Walters, On the pseudo orbit tracing property and its relatio nship to stability, Springer, Berlin, Heidelberg, (1978), 231–244

P. Walters, On the pseudo orbit tracing property and its relatio nship to stability, Springer, Berlin, Heidelberg, (1978), 231–244

work page 1978