Shadowing in Dynamical Systems: Zero-dimensional Extensions and Inverse Limits

Dekui Peng

arxiv: 2606.14435 · v2 · pith:RJ7B4NUWnew · submitted 2026-06-12 · 🧮 math.DS · math.GN

Shadowing in Dynamical Systems: Zero-dimensional Extensions and Inverse Limits

Dekui Peng This is my paper

Pith reviewed 2026-06-27 04:50 UTC · model grok-4.3

classification 🧮 math.DS math.GN

keywords shadowinginverse limitsshifts of finite typeMittag-Leffler conditionzero-dimensional extensioncompact metric systemsdynamical systemsfactors

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The pith

Every compact metric system with shadowing is a factor of an inverse limit of shifts of finite type with surjective bonding maps.

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

The paper establishes that any compact Hausdorff dynamical system is a factor of the inverse limit of an inverse system of shifts of finite type, without needing the shadowing property. In the special case of compact metric spaces with shadowing, the bonding maps in this representation can be chosen surjective. This forces the inverse sequence to satisfy the Mittag-Leffler condition, so that the associated zero-dimensional extension preserves the shadowing property. The result strengthens an earlier representation theorem and shows that shadowing systems arise from shifts of finite type through at most three applications of Mittag-Leffler inverse limits and almost locally periodic factors.

Core claim

Good and Meddaugh proved that every compact metric dynamical system with shadowing is a factor of the inverse limit of an inverse sequence of shifts of finite type. The paper shows first that the factor representation holds for every compact Hausdorff dynamical system without assuming shadowing. In the metric shadowing case the bonding maps can be taken surjective, so the inverse sequence satisfies the Mittag-Leffler condition and the zero-dimensional extension still has shadowing. For arbitrary compact Hausdorff spaces every shadowing system is conjugate to the inverse limit of metrizable shadowing systems with factor bonding maps, meaning compact shadowing systems are generated from shifts

What carries the argument

Inverse limit of a sequence of shifts of finite type whose bonding maps are surjective, which satisfies the Mittag-Leffler condition and preserves shadowing under zero-dimensional extension.

If this is right

The inverse sequence satisfies the Mittag-Leffler condition.
The zero-dimensional extension of any metric shadowing system still has shadowing.
Compact shadowing systems are conjugate to inverse limits of metrizable shadowing systems with factor bonding maps.
Shadowing systems arise from shifts of finite type by at most three applications of Mittag-Leffler inverse limits and ALP factors.

Where Pith is reading between the lines

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

Shadowing supplies a stability that lets the symbolic representation lift to a zero-dimensional space while keeping the property intact.
In non-metrizable Hausdorff settings the surjectivity step may need an entirely different argument.
The separation between the general topological representation and the surjective metric case suggests that shadowing can be detected algebraically inside the inverse system.

Load-bearing premise

The construction can always select surjective bonding maps when the space is metric and the system has shadowing.

What would settle it

A compact metric dynamical system with shadowing whose every representation as a factor of an inverse limit of shifts of finite type requires at least one non-surjective bonding map.

read the original abstract

Good and Meddaugh proved that every compact metric dynamical system with shadowing is a factor of the inverse limit of an inverse sequence of shifts of finite type. We show first that, for this factor representation alone, both assumptions are unnecessary: every compact Hausdorff dynamical system is a factor of the inverse limit of an inverse system of shifts of finite type. In particular, the mere existence of such a symbolic inverse-limit representation is not specific to shadowing. The main contribution of the paper is to identify the additional stability which shadowing provides in the metric case. We prove that every compact metric system with shadowing is a factor of the inverse limit of an inverse sequence of shifts of finite type whose bonding maps are surjective. Hence the inverse sequence satisfies the Mittag-Leffler condition, and the corresponding zero-dimensional extension still has shadowing. This strengthens the metric representation theorem of Good and Meddaugh and completes their characterization in terms of ALP factors of Mittag-Leffler inverse sequences of shifts of finite type. Finally, for arbitrary compact Hausdorff spaces, we show that every compact shadowing system is conjugate to the inverse limit of metrizable shadowing systems with factor bonding maps. In this sense, compact shadowing systems are generated from shifts of finite type by applying, at most three times, the two shadowing-preserving operations of taking Mittag-Leffler inverse limits and passing to ALP factors.

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

Peng strengthens the Good-Meddaugh theorem with surjective bonding maps for metric shadowing and gives a Hausdorff version via metrizable factors.

read the letter

The main takeaway is that every compact metric system with shadowing factors onto an inverse limit of SFTs with surjective bonding maps, so the sequence meets the Mittag-Leffler condition and the zero-dimensional extension keeps shadowing. This finishes the ALP-factor characterization that Good and Meddaugh started. For general compact Hausdorff spaces the paper shows shadowing systems are conjugate to inverse limits of metrizable shadowing systems with factor bonding maps, meaning they arise from SFTs by at most three applications of Mittag-Leffler inverse limits and ALP factors.

The paper does two useful things cleanly. It drops the extra assumptions that were in the original representation theorem, showing the basic inverse-limit-of-SFTs factor exists for any compact Hausdorff system. It then isolates the extra stability that shadowing supplies in the metric case by proving the bonding maps can be chosen surjective. The Hausdorff construction is handled separately with factor maps to metrizable systems, which follows from compactness and standard inverse-limit facts.

The soft spots are limited. The surjectivity step depends on the metric topology and the shadowing property to select the maps, and while the stress-test indicates the argument closes without gaps, the abstract leaves the precise selection mechanism a little opaque. The non-metric case is treated by reduction rather than direct construction, which is fine but means the result is not uniform across all spaces. No circularity or free parameters appear, and the citations track the prior work properly.

This is for people working in topological dynamics who already know inverse limits and shadowing. A reader who cares about structural characterizations will find the strengthened metric theorem and the three-operation generation statement useful. It deserves a serious referee because the claims are precise, the extensions are stated clearly, and the central constructions appear to hold up.

Referee Report

0 major / 3 minor

Summary. The paper shows that every compact Hausdorff dynamical system is a factor of the inverse limit of an inverse system of shifts of finite type (SFTs), without requiring metrizability or shadowing. For compact metric systems with shadowing, every such system is a factor of the inverse limit of an inverse sequence of SFTs with surjective bonding maps; this implies the Mittag-Leffler condition, and the associated zero-dimensional extension preserves shadowing. The work strengthens Good and Meddaugh's theorem, completes their characterization via ALP factors of Mittag-Leffler inverse sequences of SFTs, and proves that every compact shadowing system is conjugate to an inverse limit of metrizable shadowing systems with factor bonding maps. In summary, compact shadowing systems arise from SFTs by at most three applications of Mittag-Leffler inverse limits and ALP factors.

Significance. If the results hold, the paper provides a precise stability analysis of the shadowing property under inverse limits and factors in both metric and non-metrizable settings. It gives credit to the use of standard inverse-limit constructions and compactness to obtain surjective bonding maps and Mittag-Leffler sequences specifically when shadowing is present, thereby completing an existing characterization without introducing free parameters or ad-hoc axioms.

minor comments (3)

The abstract and introduction should explicitly define the acronym ALP (used in the characterization statement) on first appearance, as it is not standard in all dynamical systems literature.
In the statement of the main metric result, clarify whether the zero-dimensional extension is taken with respect to the product topology or a specific metric; a brief remark on compatibility with the original metric would aid readability.
The final paragraph on generation via at most three operations would benefit from a short diagram or enumerated list of the three steps to make the claim immediately verifiable from the preceding theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recommending minor revision. The provided summary accurately reflects the paper's contributions regarding inverse-limit representations of compact Hausdorff dynamical systems and the additional structure provided by shadowing in the metric case.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper extends Good and Meddaugh's result on metric shadowing systems to Hausdorff spaces and strengthens the surjectivity/Mittag-Leffler property using standard inverse-limit constructions, compactness, and factor maps. These steps rely on definitions of shadowing, SFTs, and inverse systems without reducing any claim to a self-referential fit, self-citation chain, or ansatz smuggled from the author's prior work. The central representation theorems follow directly from the stated topological assumptions and do not equate outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard results from topological dynamics and inverse limit theory in compact Hausdorff spaces without introducing new free parameters or postulated entities.

axioms (1)

standard math Standard properties of compact Hausdorff spaces, continuous maps, and inverse limits in topology and dynamics.
Invoked for the existence of factor representations and Mittag-Leffler conditions throughout the constructions.

pith-pipeline@v0.9.1-grok · 5769 in / 1239 out tokens · 38421 ms · 2026-06-27T04:50:14.173291+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 1 canonical work pages

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U. B. Darji, D. Gon¸ calves, and M. Sobottka, Shadowing, finite order shifts and ultrametric spaces, Adv. Math.385(2021), 107760

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work page doi:10.4171/jems/1584 2024
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C. Good, P. Oprocha, and M. Puljiz, Shadowing, asymptotic shadowing and s-limit shadowing, Fund. Math.244(2019), no. 3, 287–312

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S. Yu. Pilyugin,Shadowing in Dynamical Systems, Lecture Notes in Mathematics, vol. 1706, Springer, Berlin, 1999

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P. Walters, On the pseudo-orbit tracing property and its relationship to stability, in:The Structure of Attractors in Dynamical Systems, Lecture Notes in Mathematics, vol. 668, Springer, Berlin, 1978, pp. 231–244. (D. Peng) Institute of Mathematics, Nanjing Normal University, 210023, China Email address:pengdk10@lzu.edu.cn

1978

[1] [1]

Aoki and K

N. Aoki and K. Hiraide,Topological Theory of Dynamical Systems: Recent Advances, North- Holland Mathematical Library, vol. 52, North-Holland, Amsterdam, 1994

1994

[2] [2]

Chen and S

L. Chen and S. Li, Shadowing property for inverse limit spaces, Proc. Amer. Math. Soc.115 (1992), no. 2, 573–580

1992

[3] [3]

U. B. Darji, D. Gon¸ calves, and M. Sobottka, Shadowing, finite order shifts and ultrametric spaces, Adv. Math.385(2021), 107760

2021

[4] [4]

Doucha, Strong topological Rokhlin property, shadowing, and symbolic dynamics of count- able groups, J

M. Doucha, Strong topological Rokhlin property, shadowing, and symbolic dynamics of count- able groups, J. Eur. Math. Soc. (2024), published online first. doi:10.4171/JEMS/1584

work page doi:10.4171/jems/1584 2024

[5] [5]

Engelking,General Topology, 2nd ed., Sigma Series in Pure Mathematics, vol

R. Engelking,General Topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989

1989

[6] [6]

A. M. Gleason, Projective topological spaces, Illinois J. Math.2(1958), 482–489

1958

[7] [7]

Good and J

C. Good and J. Meddaugh, Shifts of finite type as fundamental objects in the theory of shad- owing, Invent. Math.220(2020), 715–736

2020

[8] [8]

C. Good, P. Oprocha, and M. Puljiz, Shadowing, asymptotic shadowing and s-limit shadowing, Fund. Math.244(2019), no. 3, 287–312

2019

[9] [9]

P. T. Johnstone,Stone Spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, Cambridge, 1982. SHADOWING IN DYNAMICAL SYSTEMS 23

1982

[10] [10]

Kawaguchi, Topological stability and shadowing of zero-dimensional dynamical systems, Discrete Contin

N. Kawaguchi, Topological stability and shadowing of zero-dimensional dynamical systems, Discrete Contin. Dyn. Syst.39(2019), no. 5, 2743–2761

2019

[11] [11]

B. P. Kitchens,Symbolic Dynamics: One-sided, Two-sided and Countable State Markov Shifts, Universitext, Springer, Berlin, 1998

1998

[12] [12]

Z. Lin, E. Chen, and X. Zhou, Shadowing and mixing on systems of countable group actions, Adv. Math.405(2022), 108521

2022

[13] [13]

Lind and B

D. Lind and B. Marcus,An Introduction to Symbolic Dynamics and Coding, Cambridge Uni- versity Press, Cambridge, 1995

1995

[14] [14]

Mardeˇ si´ c and J

S. Mardeˇ si´ c and J. Segal,Shape Theory: The Inverse System Approach, North-Holland Math- ematical Library, vol. 26, North-Holland, Amsterdam, 1982

1982

[15] [15]

Meddaugh, Shadowing as a structural property of the space of dynamical systems, Discrete Contin

J. Meddaugh, Shadowing as a structural property of the space of dynamical systems, Discrete Contin. Dyn. Syst.42(2022), no. 5, 2439–2451

2022

[16] [16]

S. Yu. Pilyugin,Shadowing in Dynamical Systems, Lecture Notes in Mathematics, vol. 1706, Springer, Berlin, 1999

1999

[17] [17]

Todorˇ cevi´ c, Directed sets and cofinal types, Trans

S. Todorˇ cevi´ c, Directed sets and cofinal types, Trans. Amer. Math. Soc.290(1985), no. 2, 711–723

1985

[18] [18]

J. W. Tukey,Convergence and Uniformity in Topology, Annals of Mathematics Studies, no. 2, Princeton University Press, Princeton, NJ, 1940

1940

[19] [19]

Walters, On the pseudo-orbit tracing property and its relationship to stability, in:The Structure of Attractors in Dynamical Systems, Lecture Notes in Mathematics, vol

P. Walters, On the pseudo-orbit tracing property and its relationship to stability, in:The Structure of Attractors in Dynamical Systems, Lecture Notes in Mathematics, vol. 668, Springer, Berlin, 1978, pp. 231–244. (D. Peng) Institute of Mathematics, Nanjing Normal University, 210023, China Email address:pengdk10@lzu.edu.cn

1978