Gromov-Hausdorff distance and stability of dynamical systems

E. Abdullaev

REVIEW 3 major objections 4 minor 16 references

Gromov-Hausdorff distance and stability of dynamical systems

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Lyapunov stability passes to Gromov–Hausdorff limits under a new pointed continuous distance.

desk verdict Useful F-space machinery and mostly sound stability-transfer theorems, but the paper's two-sided stability definition disqualifies its own contracting-flow examples, and there are a few concrete numerical/construction errors to fix. read the letter →

arxiv 2608.02906 v1 pith:5P4ZRTA4 submitted 2026-08-03 math.MG

E. Abdullaev This is my paper · ORCID

classification math.MG MSC 53C2337C75

keywords Gromov-HausdorffdistanceLyapunovstabilityasymptoticF-spacepointeddynamicalsystemscontinuousHausdorffmaptorustranslations

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

The paper introduces a new way to measure how close two dynamical systems are: it views each system as an F-space, a set carrying a family of time-shifted pseudometrics and, for pointed systems, a marked point, and defines a Gromov–Hausdorff distance on these objects. The central result is that Lyapunov stability survives limit transitions: if a sequence of Lyapunov stable pointed systems converges to a system under the pointed continuous version of this distance, then the limit is Lyapunov stable. Asymptotic stability is also preserved, provided all approximating systems share the same radius of attraction; the paper gives an example showing this radius condition cannot be dropped. A second, modified distance takes into account not only how distances between points change over time but also the geometry of whole trajectories, and it yields the exact distance between non-resonant torus translations. This matters because local stability properties could previously be lost under coarse Gromov–Hausdorff approximations; the new distance keeps the topology of orbits intact.

What carries the argument

The central object is the F-space: a set equipped with a family of generalized pseudometrics indexed by some set $I$ and, in the pointed case, a distinguished marked point. Each dynamical system is turned into an F-space by adding the time-shifted pseudometrics $d_{X,g}(x_1,x_2)=d_X(gx_1,gx_2)$ for every $g\in G$, so the distance sees how the whole arrangement of points evolves at every moment of time. The Gromov–Hausdorff distance between two such F-spaces is half the infimum, over pairs of structure-preserving maps, of the largest distortion of distances and codistortion; the pointed continuous variant restricts both maps to be continuous and to send the marked point to the marked point. The proofs work through distortion inequalities: if an approximating system has small distortion relative to the limit, then a ball around the marked point in the limit maps into a ball in the approximating system, and the stability estimate transfers back with an error controlled by the distortion. The common radius of attraction $\delta_0$ enters as the uniform scale on which this transfer works.

What would settle it

A concrete counterexample would be a sequence of pointed systems, each Lyapunov stable, that converges to a non-stable system in the pointed continuous Gromov–Hausdorff distance $d^G_{\mathrm{GH},p,c}$; Theorem 2.5 says this is impossible, so exhibiting one would disprove the paper's main claim. The paper's Example 2.6 shows only that the non-continuous pointed distance allows this failure, which is why the continuity condition is part of the theorem.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is a pair of preservation theorems for local stability under a carefully chosen pointed continuous Gromov–Hausdorff distance $d^G_{\mathrm{GH},p,c}$ between dynamical systems. Theorem 2.5 states that if a sequence of Lyapunov stable pointed dynamical systems converges in this distance to a pointed dynamical system, then the limit system is Lyapunov stable. Theorem 2.8 states that if a sequence of quasi-asymptotically stable pointed systems shares a common radius of attraction $\delta_0>0$ and converges in the (not necessarily continuous) pointed Gromov–Hausdorff distance, then the limit is quasi-asymptotically stable with the same radius $\delta_0$; combining the two gives preservation of asymptotic stability under the pointed continuous distance. The paper also proves that the stable systems form a closed, nowhere dense subset of the space of compact dynamical systems, while asymptotically stable systems form a nowhere dense $F_\sigma$-set, and it shows that the Hausdorff map sending a space to its space of closed bounded subsets is 1-Lipschitz for F-spaces and for uniformly continuous dynamical systems with bounded action.

Load-bearing premise

The asymptotic-stability result depends on all approximating systems pulling an entire neighborhood of the marked point toward it with one common positive radius; without that shared radius, a limit can end up stable but not asymptotically stable, as Example 2.9 shows.

Editorial extensions

If this is right

If the paper is right, local stability properties can be studied through pointed continuous Gromov–Hausdorff limits: a converging sequence of stable systems cannot suddenly become unstable in the limit.
Asymptotic stability is stable under limits only when the approximating systems attract a common-size neighborhood of the base point; Example 2.9 shows that radii shrinking to zero can destroy asymptotic stability even when the limit is stable.
The class of Lyapunov stable compact dynamical systems is closed and nowhere dense, so stable systems form a topologically small subset of the space of all compact systems.
The Hausdorff map is 1-Lipschitz in this theory, meaning that passing from a system to its system of closed bounded subsets does not increase the Gromov–Hausdorff distance between systems.
The modified trajectory-based distance distinguishes torus translations: for a non-resonant pair $\omega,\omega'$, the distance equals $\pi\sqrt{n}$, which the primary distance cannot detect because it is blind to the geometry of whole orbits.

Reading between the lines

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

By the same distortion-transfer mechanism, one might expect Theorem 2.5 to extend to parametrized families of systems, such as random or time-dependent systems, because the proof only uses continuity of the maps and a Lipschitz transfer of balls; the paper does not state this.
The modified distance $\hat{d}^G_{\mathrm{GH},K}$ records whole-trajectory geometry, so it should be able to separate ergodic rotations with different rotation vectors beyond the non-resonant case; the paper only computes the non-resonant pair.
The nowhere-density result suggests that Lyapunov stability is fragile in a topological sense: in the Baire-category sense, most compact dynamical systems are unstable; quantifying how often stability appears in concrete parameter families would be a natural next step.
Example 2.6 indicates that continuity of the approximating maps is essential: the non-continuous pointed distance allows a sequence of stable systems to converge to an unstable limit, so any numerical or applied use of the distance must check that the identified correspondences are continuous.

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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

Two things to know upfront. The F-space construction and the categorical Gromov–Hausdorff distance are a real extension of Bogaty–Tuzhilin, and the stability-preservation theorems (2.5, 2.8, 2.14, 2.25) are internally consistent and look provable as stated. But the paper's advertised link to classical Lyapunov stability for flows is broken: Definition 2.1 quantifies over all g ∈ G, so for G = R it demands stability under backward time as well as forward time. A contracting flow ẋ = -Ax is unstable by that definition. Example 2.9 claims exactly such systems are asymptotically stable, so it is false under the paper's own definition. The theorems survive—they are about this two-sided notion—but the motivation and the main worked example are not measuring what the introduction says they measure. This needs to be fixed by restricting G to a semigroup or by changing the stability quantifier, and all examples must be checked against that.

What is genuinely new: F-spaces with families of pseudometrics and marked points; the categorical GH distance; transfer of Lyapunov stability and asymptotic stability under pointed continuous GH convergence with a common radius of attraction; nowhere-density of stable systems; the Hausdorff map 1-Lipschitz extension; and the new trajectory-aware distance bdG that distinguishes torus translations. The Hausdorff map results look clean, and Lemma 2.18's l1-product argument is neat.

Beyond the definition mismatch, three concrete errors. Corollary 4.5 states π√n, but Theorem 4.4 gives half the maximum diameter, which is π√n/2. In Theorem 2.19, the action on I_ε sends nonzero points to 1, which lies outside [0,ε] when ε < 1, so the nowhere-density construction needs a fix (sending to ε instead). Example 2.9's convergence via identity maps is asserted rather than proved, and with the stability definition issue the example cannot stand as written. None of these kill the central theorems, but they need correcting before acceptance.

The citation pattern looks appropriate; the novelty claims against earlier GH-for-dynamical-systems work are plausible. Who is this for? People working on GH distances for dynamical systems will find a useful language and some transfer theorems. As it stands I would not cite it without caveats, but it deserves a serious referee: the core ideas are sound enough to warrant a revision cycle, and the errors are local. Recommendation: send to peer review, asking for correction of the stability-definition/example issue, the factor of two, and the I_epsilon construction.

Referee Report

3 major / 4 minor

Summary. The paper develops a categorical Gromov–Hausdorff distance for F-spaces, i.e. sets equipped with a family of generalized pseudometrics and marked points, and applies it to pointed dynamical systems by associating to each system the family of pseudometrics d_g(x,x') = d(gx,gx'). The main results are: preservation of Lyapunov stability under pointed continuous Gromov–Hausdorff convergence (Theorem 2.5); preservation of quasi-asymptotic stability under a common radius-of-attraction assumption (Theorems 2.8 and 2.14, generalized in Theorem 2.25); closedness and nowhere density of the stable and asymptotically stable classes (Theorems 2.19–2.20); a 1-Lipschitz property of the Hausdorff map for F-spaces and for continuous compact dynamical systems (Theorems 3.6 and 3.13); and a modified trajectory-aware distance for which the distance between ergodic torus translations is computed (Theorem 4.4 and Corollary 4.5). The authors argue that the ordinary Gromov–Hausdorff distance cannot distinguish qualitative dynamical differences such as ergodicity, and that the new distances address this limitation.

Significance. Assuming the repairs described below, this is a genuinely useful framework. The distance satisfies the triangle inequality, the main stability-preservation theorems are non-trivial and internally coherent, and the Hausdorff-map estimates extend known results to F-spaces. The torus calculation, once the factor error is corrected, provides a clean example in which the modified distance distinguishes ergodic translations, and the paper makes its hypotheses explicit, in particular the common radius of attraction. The self-contained categorical setup, the explicit treatment of morphism classes, and the presence of counterexamples to sharpness are notable strengths. The main limitation is the mismatch between Definition 2.1 and classical Lyapunov stability for flows, which currently undermines the motivating examples rather than the internal logic of the stability proofs.

major comments (3)

[Section 2.1, Definition 2.1 and Example 2.9] Definition 2.1 quantifies stability over all g in G. When G=R and the action is a flow, this includes arbitrarily negative times. Consequently the linear system x' = -A_N x, which is classically asymptotically stable in forward time, is not stable in the sense of Definition 2.1: for fixed N, epsilon = A_N/2 and any delta>0, a point x in (0,min{delta,A_N}) satisfies d(x,0)<delta, but the backward trajectory reaches distance A_N from 0 in finite time, so sup_{g in R} d(gx,0) >= A_N > epsilon. Thus Example 2.9 does not exhibit a sequence of asymptotically stable systems under the paper's own definition, and its advertised conclusion that the common-radius condition in Theorem 2.8 cannot be dropped is not established. The proof of Theorem 2.8 itself is unaffected, but the interpretation of the stability theorems as statements about classical Lyapunov stability of ODE flows requires either restricting G to a forward semigroup in Definition 2.1 or replacing Example 2.9 with an example that satisfies the definition as written.
[Section 2.3, Theorem 2.19] The constructed pointed system I_epsilon = ([0,epsilon],0) is used for arbitrary epsilon>0, but its action sends (g,x), for g != e and x != 0, to the point 1, which belongs to [0,epsilon] only when epsilon >= 1. For epsilon < 1 the map is not a self-map of I_epsilon, so the construction is invalid on neighborhoods of radius smaller than 1. The proof can likely be repaired by sending those points to epsilon rather than 1, but as written the nowhere-density argument does not cover all neighborhoods.
[Section 4, Corollary 4.5] Theorem 4.4 concludes bd^G_GH,K(X,Y) = (1/2) max{diam(X),diam(Y)}. Since the flat n-torus has diameter pi*sqrt(n), Corollary 4.5 must conclude pi*sqrt(n)/2. The displayed equality in the proof of the corollary omits the factor 1/2 and therefore states a value twice the one forced by Theorem 4.4.

minor comments (4)

[Definition 1.2] In the definition of Mor_F, the condition is written as f(x_j)=f(y_j); it should be f(x_j)=y_j, since y_j is the marked point in Y, not an element of X.
[Example 1.20] The assertion that dis_{d_{X,f},d_{Y,f}}(f)=0 appears false as written: the graph distortion is sup_{x,x'} ||x-x'| - |f(f(x))-f(f(x'))||, which is not generally zero for an arbitrary bijection f. If a different map or metric was intended, the notation should be clarified.
[Section 3.2, Theorem 3.13] The proof begins 'Since the dynamical system X is continuous', but Definition 3.9 only introduces uniformly continuous maps; spelling out that uniform continuity gives the closure identity phi(g, closure(A)) subset closure(phi(g,A)) would improve readability.
[Section 4, Theorem 4.4] The notation cdis^G and [codis^G] is introduced with an odd bracket and is not consistently defined; the definitions should be displayed cleanly, and the bracket should be removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the stability-preservation theorems and the torus distance calculation are derived from the paper's own definitions with external classical inputs; nothing reduces to a fit, a self-citation chain, or a definitional identity.

full rationale

The paper's main derivation chain is self-contained. The F-space and pointed continuous Gromov–Hausdorff distance are defined from the dynamics in a direct way (d_{X,g}(x1,x2)=d_X(gx1,gx2), with distorsion and codistorsion computed over all g in G), and Theorems 2.5, 2.8, and 2.14 are proved by explicit epsilon-delta estimates: the stability or quasi-asymptotic stability of the approximating systems is an assumption, the smallness of the dynamical distortion of the comparison maps is the convergence input, and the stability of the limit is deduced rather than assumed. There is no fitted parameter renamed as a prediction: the common radius of attraction delta_0 is a hypothesis, not a fitted quantity, and Example 2.9 is used to show that this hypothesis cannot be dropped. The nowhere-density results (Theorems 2.19 and 2.20) follow from Theorem 2.5, Lemma 2.18, and a constructed unstable perturbation, none of which quotes the conclusion as an input. The Hausdorff-map Lipschitz theorem is proved from the definitions of the Hausdorff metric and the F-space distortion, with the classical metric-space case cited only for background. The torus calculation in Section 4 uses the classical non-resonance/ergodicity/dense-orbit theorem of Cornfeld–Fomin–Sinai as an external input and computes the distance from the general diameter formula; that external theorem is not the target conclusion, and the computation is not a renamed version of the input. Citations to Bogaty–Tuzhilin, Ivanov–Nikolaeva–Tuzhilin, and Mikhailov are background or external results and are not load-bearing in the stability theorems. The only concerns that arise are mathematical-validity issues rather than circularity: Definition 2.1 quantifies over all g in G, so for G=R it is a two-sided stability condition, which would make the contracting systems in Example 2.9 fail the paper's own stability definition; and Corollary 4.5 appears to be off by a factor of two relative to Theorem 4.4's diameter formula. Even if these concerns are sustained, they do not show that the paper's claims are equivalent to their inputs by construction.

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

No numerical free parameters are fitted. The paper's inputs are classical background results (ODE theory, ergodic minimality, Hausdorff metric properties); none are ad hoc to this paper or used to define the conclusion into existence. The F-space is a new structural object, but not introduced to force a target result.

assumptions (4)

standard math Classical existence, uniqueness, and continuous dependence for ODE flows (used in Examples 1.23, 2.6, 2.9).
The examples rely on standard ODE theory as background; not proved in the paper.
domain assumption For a non-resonant vector omega, the translation T_omega on the torus is ergodic and every orbit is dense (cited to [16]).
Used in Section 4 to satisfy the dense-trajectory hypothesis of Theorem 4.4 and to compute the torus distance.
standard math The Hausdorff distance is a metric on closed bounded subsets and preserves completeness, boundedness, total boundedness, and compactness (Proposition 3.2, cited to [4]).
Used as the basis for the Hausdorff map results in Section 3.
standard math Continuous maps satisfy f(cl A) is a subset of cl f(A), and Hausdorff distances between sets equal those between closures (Remark 3.8).
Used in the proof of Theorem 3.13 to handle the closure in the definition of H(X).

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Cite this review

Pith. "Pith review of Gromov-Hausdorff distance and stability of dynamical systems." pith.science (2026). https://pith.science/paper/5P4ZRTA4

@misc{pith2026260802906,
  author       = {Pith},
  title        = {Pith review of: Gromov-Hausdorff distance and stability of dynamical systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5P4ZRTA4}},
  note         = {Machine review of arXiv:2608.02906}
}

read the original abstract

We introduce the notion of an F-space - a set equipped with a family of generalized pseudometrics and marked points, and construct the categorical Gromov-Hausdorff distance for F-spaces. Based on this, we propose a new definition of the distance between dynamical systems, understood in a broad sense as parameterized families of maps. The main focus is on local dynamics: we prove the preservation of Lyapunov stability and asymptotic stability under limit transitions with respect to the new distance (in the latter case, assuming a common radius of attraction). It is established that the class of stable systems is a closed and nowhere dense subset in the space of compact dynamical systems. Next, we investigate the properties of the Hausdorff map, which induces an F-space structure on the family of subsets. In the final part, we introduce another modified version of the Gromov-Hausdorff distance for dynamical systems and use it to calculate the exact distance between torus translations for a non-resonant vector.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages

[1]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Hausdorff.Grundz¨ uge der Mengenlehre, Leipzig, Veit, 1914.[reprinted by Chelsea],1949

F. Hausdorff.Grundz¨ uge der Mengenlehre, Leipzig, Veit, 1914.[reprinted by Chelsea],1949

work page 1914

[2] [2]

D. Edwards. The Structure of SuperspaceStudies in Topology, ed. by N. M. Stavrakas, K. R. Allen, New York, London, San Francisco, Academic Press, Inc., 1975

work page 1975

[3] [3]

M. Gromov. Groups of Polynomial growth and Expanding Maps,Publ. Math. Inst. Hautes ´Etudes Sci.,1981,53

work page 1981

[4] [4]

Burago, Yu

D. Burago, Yu. Burago, S. Ivanov.A Course in Metric Geometry, Moscow–Izhevsk, Institute of Computer Science, 2004

work page 2004

[5] [5]

A. O. Ivanov, N. K. Nikolaeva, A. A. Tuzhilin. The Gromov–Hausdorff metric on the space of compact metric spaces is strictly intrinsic,Math. Notes,2016,100, no. 6, pp. 883–885

work page 2016

[6] [6]

M´ emoli, Z

F. M´ emoli, Z. T. Smith. Embedding-projection correspondences for the estimation of the Gromov–Hausdorff distance,arXiv,2024, preprint arXiv:2407.03295 [math.MG]

arXiv 2024

[7] [7]

S. Lim, F. M´ emoli, Z. Smith. The Gromov–Hausdorff distance between spheres,Geometry & Topology,2023,27, no. 9, pp. 3733–3800

work page 2023

[8] [8]

S. A. Bogaty, A. A. Tuzhilin. Fundamentals of Theory of Continuous Gromov–Hausdorff distance,arXiv,2025, preprint arXiv:2512.02611

work page arXiv 2025