Stability Theorems in Pointwise Dynamics

Abdul Gaffar Khan; Tarun Das

REVIEW 3 major objections 5 minor 19 references

Stability Theorems in Pointwise Dynamics

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A minimally expansive shadowable point of a homeomorphism is topologically stable and GH-stable.

desk verdict The measure-stability theorem is vacuous as stated—the empty set-valued map satisfies Definition 3.7—so the paper needs major revision, though the pointwise GH-stability half has some substance. read the letter →

arxiv 1908.09536 v1 pith:G5HGJXFB submitted 2019-08-26 math.DS

Abdul Gaffar Khan , Tarun Das This is my paper · ORCID

classification math.DS MSC 54H2037C7537C50

keywords minimallyexpansivepointGH-stableshadowabletopologicalstabilityBorelmeasureexpansivityshadowingstrongpointwisedynamics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that two classical stability theorems survive when their hypotheses are weakened from a whole system to a single point. It introduces minimally expansive points and GH (Gromov–Hausdorff) stable points for homeomorphisms, together with $\mu$-uniformly expansive, $\mu$-shadowable, and strong $\mu$-topologically stable points for Borel measures. The main results are that on a compact metric space, a minimally expansive shadowable point of a homeomorphism is topologically stable and GH-stable, and a $\mu$-uniformly expansive $\mu$-shadowable point is strong $\mu$-topologically stable. If correct, these theorems extend the classical topological stability theorem for expansive homeomorphisms with shadowing, and its measure analogue, to the pointwise setting: a system need not be globally expansive or globally shadowing for one orbit to be stable.

What carries the argument

The load-bearing construction is a semiconjugacy obtained by tracing pseudo-orbits: for a perturbation $g$ of $f$, start with a pseudo-orbit through $x$, let $z$ be the point that shadows it, and define $h(g^n(y))=f^n(z)$. Minimal expansivity makes $h$ well defined: if two iterates of $y$ under $g$ agree, the shadowing inequality and the common expansivity constant force the corresponding iterates of $z$ to agree. Lemma 2.6 supplies the compactness step that turns 'close for all times along a long segment' into 'close at time zero', giving uniform continuity of $h$ and permitting extension to the orbit closure. In the measure case, the same tracing is packaged as the set-valued map $H(u)=\bigcap_{n\in\mathbb{Z}} f^{-n}(B[g^n(u),\eta])$, whose nonempty fibers are the traced points; the proof shows its domain is closed, hence measurable, and that its fibers have $\mu$-measure zero.

What would settle it

Construct a homeomorphism on a compact metric space with a point $x$ whose orbit is not closed but which is minimally expansive and shadowable, and test whether the conclusion holds. A concrete test is a saddle-connection type system in which the orbit of $x$ accumulates on a periodic orbit without reaching it; if the domain of the set-valued map $H$ in Theorem 3.9 has a sequence $x_k$ with limit in $\overline{O_g(x)} \setminus O_g(x)$, the proof's closedness assertion fails, and whether $x$ is actually strong $\mu$-topologically stable would decide the theorem.

Watch

Extended reading notes

Core claim

The central discovery is that expansivity and shadowing can be localized to a point and still imply stability. A point $x$ is minimally expansive when some $c>0$ makes the homeomorphism expansive on the orbit closure of every $y$ within distance $c$ of $x$, with the same constant $c$; $x$ is shadowable when every sufficiently small pseudo-orbit that starts at $x$ is traced by a real orbit. Theorem 2.7 asserts that such a point is topologically stable and GH-stable: every homeomorphism $g$ sufficiently close to $f$ in the relevant distance admits a continuous semiconjugacy $h$ defined on the orbit (or orbit closure) of $x$ under $g$, with $h$ close to the identity. Theorem 3.9 moves the same conclusion to Borel measures: a $\mu$-uniformly expansive $\mu$-shadowable point is strong $\mu$-topologically stable, and the $\mu$-shadowability hypothesis can be dropped if one only wants $\mu$-topological stability. These are the pointwise versions of the global stability theorems, and they reduce to the classical statements when every point of the space satisfies the pointwise hypotheses.

Load-bearing premise

The load-bearing premise is that the orbit of the perturbed point can be treated as closed (or replaced by its closure), because the proof writes $O_g(x)$ and then uses 'since $O_g(x)$ is closed' to extend the stability map; if a literal orbit is meant, that step is unjustified for points whose orbits are not closed.

Editorial extensions

If this is right

On a compact metric space, one minimally expansive shadowable point of a homeomorphism is enough to make that point topologically stable and GH-stable; the whole system need not be expansive or shadowing.
If the homeomorphism itself is expansive, every shadowable point is topologically stable and GH-stable, recovering the pointwise form of the classical stability theorem.
On compact manifolds of dimension at least two, a minimally expansive point is shadowable if and only if it is topologically stable.
For every Borel measure μ, a μ-uniformly expansive point is μ-topologically stable, and adding μ-shadowability upgrades it to strong μ-topologically stable.
When the measure is non-atomic, any topologically stable point is automatically strong μ-topologically stable.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because stability of a point is certified by data on a single orbit and its neighborhood, numerical shadowing of a finite pseudo-orbit could be used to certify topological stability of that orbit in simulations, without verifying the shadowing property on the whole space.
The measure version suggests an almost-everywhere stability: the semiconjugacy's domain carries full μ-measure on the relevant orbit fragment, so nearby systems could be compared for μ-almost every point rather than for every point.
The definitions are stated for homeomorphisms; a natural test is whether the same pointwise implications hold for continuous non-invertible maps, where pseudo-orbits are one-sided and orbit closures behave differently.
The GH-stability condition at a point depends on the geometry of the orbit through the epsilon-isometry; one could try to localize the Gromov-Hausdorff distance to the pair of orbit closures, which may remove the compactness of the whole space from the hypothesis.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read this one if you want a case study in how a vacuous definition can sink a theorem. The main measure-theoretic result, Theorem 3.9, is content-free as stated: Definition 3.7 of a µ-topologically stable point is satisfied by the empty set-valued map. For any x, any g within δ, take H(z)=∅ for all z. Conditions (i)–(iii) hold vacuously. So every point is µ-topologically stable, without any µ-uniform expansivity or µ-shadowing. The "strong" variant adds condition (iv), which rules out the empty map on a full-measure set, but the first assertion of Theorem 3.9 is just true by definition.

That said, the paper isn't worthless. The pointwise notions—minimally expansive points, GH-stable points—are natural, and the proof of the GH-stability half of Theorem 2.7 is mostly coherent. The discussion around minimally expansive points (rational rotation example) is informative, and the idea of extending Walters and Lee-Morales to pointwise assumptions is reasonable. The Section 2 machinery (Lemma 2.6, the uniform convergence theorem) looks plausible.

The soft spots beyond the vacuity: (1) The topological stability half of Theorem 2.7 is not proved; "proof of first case can be done on similar lines" is not acceptable for a central result. (2) In the proof of Theorem 3.9, O_g(x) is treated as closed. A literal orbit isn't closed (irrational rotation). The intended domain is presumably the orbit closure; Definition 3.7 and the proof need to be consistent on this. (3) Minor notational confusions—e.g., T S in Prop 3.8(i) vs Ts elsewhere.

Who is this for? People working in shadowing and pointwise stability. They might find the minimally expansive point notion useful, and the GH-stability proof is worth a look. But as it stands, the main advertised result (ii) is vacuous and the other main theorem is half-proved. I wouldn't cite this in its current form.

Recommendation: send it to a serious referee rather than desk-reject, but expect heavy revision. A referee should catch the vacuous definition and the missing proof; if the authors fix both, there may be a decent paper here.

Referee Report

3 major / 5 minor

Summary. The paper introduces pointwise versions of expansivity, shadowing, topological stability, and GH-stability for homeomorphisms on compact metric spaces, together with measure-theoretic analogues. The main theorems are Theorem 2.7, asserting that every minimally expansive shadowable point is both topologically stable and GH-stable, and Theorem 3.9, asserting that every μ-uniformly expansive μ-shadowable point is strong μ-topologically stable. The paper also proves invariance and sequence-characterization results and derives corollaries for pointwise and measure-theoretic stability.

Significance. If the results are correct after appropriate repairs, they would extend the classical Walters stability theorem and the Lee–Morales measure-stability theorem from global assumptions to pointwise assumptions, and the GH-stability part would extend the Arbieto–Rojas result to points. The paper is self-contained and proof-based, with no fitted parameters or circular derivations. However, the current manuscript contains a vacuous definition in Section 3 and a load-bearing orbit-versus-orbit-closure ambiguity, so the main theorems as stated are not yet established.

major comments (3)

[Definition 3.7 and Theorem 3.9] Definition 3.7 of μ-topological stability is vacuous. The empty set-valued map H(z)=∅ for every z∈Og(x) satisfies all three conditions: Dom(H)=∅ is Borel, ∅ is compact and upper semi-continuous, condition (i) holds vacuously, condition (ii) holds because ∅⊂B[z,ε], and condition (iii) holds because both compositions are empty. Consequently every point of every homeomorphism is μ-topologically stable in the sense of Definition 3.7, and the first assertion of Theorem 3.9 is content-free rather than a genuine stability statement. This also conflicts with the global definition earlier in Section 3, which explicitly requires μ(X\Dom(H))=0. The proof of Theorem 3.9 cannot repair this by saying 'Former case follows by choosing δ=η', because without a nonempty-domain or full-measure-domain condition there is nothing to prove. The definition needs to be repaired, for example by adding μ(X\Dom(H))=0 or a nonempty-valued/full-measure condition, and then the first assertion of Theorem 3.9 needs a new proof.
[Theorem 3.9, proof of Dom(H) closed] The proof states 'Since Og(x) is closed, z∈Og(x)' when a sequence xk∈Dom(H) converges to z. The literal orbit Og(x) need not be closed in a compact metric space, e.g. for an irrational rotation on the circle. If the intended set is the orbit closure \overline{Og(x)}, the notation must be changed consistently throughout Definition 3.7 and Theorem 3.9, and the text should explicitly say that H is defined on the closed invariant set \overline{Og(x)} rather than on the raw orbit. This is not a presentation quibble: the closedness of the domain is used to conclude that the limit z lies in the domain and to make the compactness argument for measurability of Dom(H) valid. As written, the proof fails at this step if Og(x) denotes the literal orbit.
[Theorem 2.7] The proof of Theorem 2.7 only treats the GH-stable case; the topological-stability half is deferred with 'Proof of first case can be done on similar lines.' This is a central theorem of the paper, so the omitted half should be written out in full. Moreover, the GH-stable proof contains an extension step whose wording is problematic: h is defined on the orbit Og(y) and shown uniformly continuous, and then the text says 'Since Y is compact and dX(j(y1),j(y2))<δ+dY(y1,y2) ... we can extend h continuously to a function H:Og(y)→X.' If Og(y) is the literal orbit, no extension is needed; if it is the orbit closure, the extension follows from uniform continuity and completeness of X, not from the displayed inequality involving j. The manuscript should clarify the intended domain in Definition 2.5 and give the extension argument explicitly.

minor comments (5)

[Throughout] The notation Og(x) is used inconsistently for the orbit and the orbit closure. The paper should define both O_g(x) and \overline{O_g(x)} and use them carefully; in particular Definition 2.5, Definition 3.7, and Theorem 3.9 all need this distinction.
[Theorem 2.4(2)] The statement of item (2) has a grammatical/formal issue: 'x∈Mf(X) if and only if there exists a δ>0 such that for each y∈B(x,δ), if for every pair of distinct points u,v∈Of(y), ∪...≠∅' contains an extra 'if' and is hard to parse. It should be rephrased, e.g. '... such that for each y∈B(x,δ) and every pair of distinct points u,v∈Of(y), we have ...'.
[Proposition 2.2(6)] In the proof of item (6), the line 'f^{n_i^u}∘h^{-1}(w)→h^{-1}(u) and f^{n_i^u}∘h^{-1}(w)→h^{-1}(u)' should presumably read '...→h^{-1}(v)' in the second convergence. This is a typographical error but should be corrected.
[Section 3, Definition 3.2 and Proposition 3.4] The term 'Borelian' is used without definition; it should say 'Borel set' or define 'Borelian' explicitly.
[Theorem 3.6] In the statement of Theorem 3.6, the set Cz is written as 'Cz={y∈B(x,δ): ∪...=φ}=0', which mixes the set definition with the measure-zero assertion. It should be defined as a set and then asserted to have μ(Cz)=0.

Circularity Check

1 steps flagged · score 6.0 of 10

Definition 3.7 of µ-topological stability is vacuous, so Theorem 3.9's first implication is true by definition for every point and does not follow from µ-uniform expansivity.

self definitional [Definition 3.7 and Theorem 3.9, Section 3]
"Definition 3.7 ... there exists an upper semi-continuous compact valued map H : Og(x) → 2X with the measurable domain such that: (i) µ(H(z)) = 0, for each z ∈ B(x, δ/4) ∩ Og(x). (ii) d(H, Id) ≤ ǫ. (iii) f ◦ H = H ◦ g. ... Theorem 3.9. If x ∈ X is a µ-uniformly expansive point of f , then x is a µ-topologically stable point of f ."

The empty set-valued map H(z)=∅ satisfies (i)–(iii): Dom(H)=∅ is measurable, ∅ is compact, upper semicontinuity is vacuous, (i) is vacuous, (ii) holds since ∅⊂B[z,ǫ], and (iii) holds since f(∅)=∅=H(g(z)). Hence every point is µ-topologically stable under Definition 3.7, without µ-uniform expansivity or µ-shadowing; Theorem 3.9's first assertion is an immediate consequence of the definition. The earlier global definition, quoted in Section 3, includes µ(X\Dom(H))=0 as condition (i); the pointwise definition dropped that nonempty/full-domain condition. The strong clause (iv) is similarly weakened: for µ-null U the empty map also satisfies (iv).

full rationale

The paper is a self-contained proof-based work with no fitted parameters and no predictions that reduce to empirical inputs. The Section 2 stability theorem for minimally expansive shadowable points is argued directly from the definitions and the shadowing property; its self-citations are illustrative and not load-bearing. The serious issue is confined to Section 3: Definition 3.7 defines µ-topological stability without requiring nonempty values or a full-measure domain condition, so Theorem 3.9's first assertion is true for every point by construction, independent of µ-uniform expansivity. This makes that theorem's first claim vacuous rather than derived; the 'strong' version retains some content when U has positive measure, but the same missing-condition flaw weakens it for µ-null orbit closures. Overall, this is a partial definitional circularity: one theorem's conclusion is an immediate consequence of the definition of that conclusion, not of the stated hypothesis.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on compact metric space assumptions, homeomorphisms, Borel measures, and standard compactness facts. There are no fitted numerical parameters; the constants c, delta, and eta in the proofs are existential or chosen small in the usual epsilon-delta manner. No new physical objects or unexplained entities are postulated. The main unresolved load-bearing assumption is the intended use of orbit closures versus literal orbits in Theorem 3.9.

assumptions (4)

domain assumption The phase space is a compact metric space and the dynamics are given by a homeomorphism.
Stated at the start of Section 2 and used throughout; compactness is needed for Lebesgue numbers, sequential compactness, and extension of uniformly continuous maps to orbit closures.
domain assumption For the measure results, mu is a nontrivial Borel measure, and in some statements mu is assumed non-atomic.
Introduced in Section 3 before Definition 3.1; non-atomicity is used in Proposition 3.8(iii). The main Theorem 3.9 is stated for a general Borel measure, so the central result does not require non-atomicity.
standard math Standard facts about compactness, Lebesgue numbers, upper semicontinuous compact-valued set-valued maps, and extension of uniformly continuous functions to closures are used without proof.
These background facts are invoked in the proofs of Proposition 3.4, Theorem 3.6, and Theorem 3.9, especially the Lebesgue number argument and the compactness argument for upper semicontinuity.
standard math The GH-stability definition assumes the existence of delta-isometries i and j and uses the resulting inequalities on C0 distances and metric distortions.
This is part of the definition transcribed from Arbieto-Rojas [3] and is used in the proof of Theorem 2.7 to build the pseudo-orbit and the semi-conjugacy.

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Cite this review

Pith. "Pith review of Stability Theorems in Pointwise Dynamics." pith.science (2026). https://pith.science/paper/G5HGJXFB

@misc{pith2026190809536,
  author       = {Pith},
  title        = {Pith review of: Stability Theorems in Pointwise Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G5HGJXFB}},
  note         = {Machine review of arXiv:1908.09536}
}

abstract

We introduce minimally expansive and GH-stable points for homeomorphisms on metric spaces and $\mu$-uniformly expansive, $\mu$-shadowable and strong $\mu$-topologically stable points for Borel measures (with respect to a homeomorphism on a metric space). We prove that: (i) minimally expansive shadowable point of a homeomorphism on a compact metric space is topologically stable and GH-stable. (ii) $\mu$-uniformly expansive $\mu$-shadowable point for a Borel measure $\mu$ (with respect to a homeomorphism on a compact metric space) is strong $\mu$-topologically stable.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 7 canonical work pages

[1]

Nonlinear Analysis: Theory, Methods & Applications

Abu-Saris, R., Al-Hami, K.: Uniform convergence and chaotic beha vior. Nonlinear Analysis: Theory, Methods & Applications. 65(4), 933-937(2006)

2006
[2]

Positive Entropy Through Pointwise Dynamics

Arbieto, A., Rego, E.: Positive Entropy Through Pointwise Dynamic s. Preprint. arXiv:1805.07503 (2018)

work page Pith review arXiv 2018
[3]

Discrete & Continuous Dynamical Systems-A

Arbieto, A., Rojas, C.A.: Topological Stability from Gromov-Hausd orﬀ Viewpoint. Discrete & Continuous Dynamical Systems-A. 37, 3531-3544(2017)

work page 2017
[4]

Cordiero, W.: N-expansive Homeomorphisms with the Shadowing Property

Carvalho, B. Cordiero, W.: N-expansive Homeomorphisms with the Shadowing Property. J. Diﬀerential Equations. 261, 3734-3755(2016)

work page 2016
[5]

Bulletin of the Brazilian Mathematical Society, New Series(2019)

Das, P., Khan, A.G., Das, T., Measure Expansivity and Speciﬁcation for Pointwise Dynamics. Bulletin of the Brazilian Mathematical Society, New Series(2019). ht tps://doi.org /10.1007/s00574- 019-00134-3

work page doi:10.1007/s00574- 2019
[6]

Fedeli, A., Donne A.L.: A note on the uniform limit of transitive dynamic al systems. Bull. Belgian Math. Soc.-Simon Stevin. 16(1), 59-66(2019)

2019
[7]

Proc eedings of the Edinburgh Mathematical Society

Koo, N., Lee, K., Morales, C.A.: Pointwise Topological Stability. Proc eedings of the Edinburgh Mathematical Society. 61(4), 1179-1191(2018)

2018
[8]

Bulletin of the Brazilian Mathematical Society, New Series

Kawaguchi, N.: Properties of shadowable points: Chaos and equic ontinuity. Bulletin of the Brazilian Mathematical Society, New Series. 48(4), 599-622(2017 )

2017

Show all 19 references

[9]

Dynamical Syste ms

Kawaguchi, N.: Quantitative shadowable points. Dynamical Syste ms. 32(4), 504-518(2017)

2017
[10]

Chaos, S olitons & Fractals

Li, R.: A note on uniform convergence and transitivity. Chaos, S olitons & Fractals. 45(6),759- 764(2012)

2012
[11]

Diﬀerential Equations, 262, 3467-3487(2017)

Lee, K., Morales, C.A.: Topological stability and pseudo-orbit tra cing property for expansive measures, J. Diﬀerential Equations, 262, 3467-3487(2017)

2017
[12]

Topology Applications

Moothathu, T.K.S.: Implications of Pseudo-orbit Tracing Proper ty for Continuous Maps on Compacta. Topology Applications. 158, 2232-2239(2011)

2011
[13]

IMPA D0 83, (2011)

Morales, C.A.: Measure-expansive systems, Preprint. IMPA D0 83, (2011)

2011
[14]

Dynamical Systems

Morales, C.A.: Shadowable points. Dynamical Systems. 31(3),34 7-56(2016)

2016
[15]

Reddy, W.L.: Pointwise expansion homeomorphisms. J. London Ma th. Soc. 2, 232–236 (1970)

1970
[16]

Journal of Applied Mathematics and Physics

Sharma, P.: Uniform Convergence and dynamical behavior of a d iscrete dynamical system. Journal of Applied Mathematics and Physics. 3(07), 766-770(201 5)
[17]

Utz, W.R.: Unstable homeomorphisms. Proc. Amer. Math. Soc. 1 , 769-774(1950)

1950
[18]

P. Walters. Anosov Diﬀeomorphisms are Topologically Stable. Top ology 9, 71-78(1970)

1970
[19]

In the structure of attractors in dynamical systems

Walters, P.: On the pseudo orbit tracing property and its relatio nship to stability. In the structure of attractors in dynamical systems. Springer, Berlin, Heidelberg. 231–244(1978)

1978

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Nonlinear Analysis: Theory, Methods & Applications

Abu-Saris, R., Al-Hami, K.: Uniform convergence and chaotic beha vior. Nonlinear Analysis: Theory, Methods & Applications. 65(4), 933-937(2006)

2006

[2] [2]

Positive Entropy Through Pointwise Dynamics

Arbieto, A., Rego, E.: Positive Entropy Through Pointwise Dynamic s. Preprint. arXiv:1805.07503 (2018)

work page Pith review arXiv 2018

[3] [3]

Discrete & Continuous Dynamical Systems-A

Arbieto, A., Rojas, C.A.: Topological Stability from Gromov-Hausd orﬀ Viewpoint. Discrete & Continuous Dynamical Systems-A. 37, 3531-3544(2017)

work page 2017

[4] [4]

Cordiero, W.: N-expansive Homeomorphisms with the Shadowing Property

Carvalho, B. Cordiero, W.: N-expansive Homeomorphisms with the Shadowing Property. J. Diﬀerential Equations. 261, 3734-3755(2016)

work page 2016

[5] [5]

Bulletin of the Brazilian Mathematical Society, New Series(2019)

Das, P., Khan, A.G., Das, T., Measure Expansivity and Speciﬁcation for Pointwise Dynamics. Bulletin of the Brazilian Mathematical Society, New Series(2019). ht tps://doi.org /10.1007/s00574- 019-00134-3

work page doi:10.1007/s00574- 2019

[6] [6]

Fedeli, A., Donne A.L.: A note on the uniform limit of transitive dynamic al systems. Bull. Belgian Math. Soc.-Simon Stevin. 16(1), 59-66(2019)

2019

[7] [7]

Proc eedings of the Edinburgh Mathematical Society

Koo, N., Lee, K., Morales, C.A.: Pointwise Topological Stability. Proc eedings of the Edinburgh Mathematical Society. 61(4), 1179-1191(2018)

2018

[8] [8]

Bulletin of the Brazilian Mathematical Society, New Series

Kawaguchi, N.: Properties of shadowable points: Chaos and equic ontinuity. Bulletin of the Brazilian Mathematical Society, New Series. 48(4), 599-622(2017 )

2017