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C0-Stability for actions implies shadowing property

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Every C0-stable action of the two-generator free group on a compact manifold of dimension at least two has the shadowing property.

desk verdict A worthwhile but incomplete paper: the F2 version of Walters' theorem is plausible and the stable non-expansive example has real content, but the proof of Theorem A has an unproved distinctness step that is load-bearing. read the letter →

arxiv 1908.05299 v2 pith:2HHEWR7E submitted 2019-08-14 math.DS

Jorge Iglesias , Aldo Portela This is my paper · ORCID
classification math.DS MSC 37C5037C8537B05
keywords C0-stabilityshadowingpropertyfreegroupactionspseudo-orbittracingexpansivitycompactmanifoldssemiconjugacy
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 proves Theorem A: for an action of the free group on two generators on a compact manifold of dimension at least two, C0-stability implies the shadowing property. That is, if every sufficiently small perturbation of the action is semiconjugate to it by a near-identity continuous surjection, then every sufficiently fine approximate orbit lies uniformly close to a genuine orbit. This extends to group actions a classical property of single homeomorphisms, and it gives a necessary condition that any stable group action must satisfy. Along the way the authors construct a non-expansive action that is both C0- and C1-stable, showing that expansivity is not required for stability.

What carries the argument

The load-bearing mechanism is the coset reduction of a group-action pseudotrajectory to pseudotrajectories of one homeomorphism. Writing the free group F2 with generators a and b, each index g is equivalent to g′ when g=($b^{{-1}}$a)^n g′; along such a class the action's δ-error becomes a δ+δ1-error for Φ_{$b^{{-1}}$}Φ_a, transferring the classical shadowing theorem for single maps into the group setting. The second mechanism is the interpolation lemma: on a compact manifold of dimension at least two, a small diffeomorphism can send finitely many prescribed points to nearby prescribed points, provided the points are pairwise distinct; this is what allows the corrected points to be woven into a perturbed action. The third mechanism is an extension lemma ensuring that a semiconjugacy built on a dense subset extends to the whole space.

What would settle it

The theorem would be refuted by any C0-stable action of F2 on a compact surface and a sequence of δ-pseudotrajectories with δ→0 that are not uniformly close to any true orbit. Short of that, a concrete calculation is to run the proof's construction on the paper's own stable action and test whether the corrected points ~x_g can be kept pairwise distinct for all g of length at most n; failure there would block Lemma 4.2.

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Extended reading notes

Core claim

On its own terms, the paper's discovery is the implication C0-stability implies shadowing for actions of the free group F2 on compact manifolds of dimension at least two, stated as Theorem A. The proof shows that a δ-pseudotrajectory of the action can be organized along the cosets of the cyclic subgroup generated by the group element $b^{{-1}}$a: for each coset, the restricted sequence is a δ+δ1-pseudotrajectory of the single homeomorphism Φ_{$b^{{-1}}$}Φ_a. Since C0-stability of the action makes that homeomorphism C0-stable, and C0-stable homeomorphisms on compact manifolds of dimension at least two have the shadowing property, each coset can be approximated by a true orbit of Φ_{$b^{{-1}}$}Φ_a. A point-interpolation lemma then builds a small diffeomorphism f so that the action generated by fΦ_a and fΦ_b is C0-close to Φ; the near-identity semiconjugacy supplied by C0-stability converts the corrected finite pseudo-orbit into a genuine orbit of Φ, giving the desired ε-shadowing.

Load-bearing premise

The proof assumes that the shadowing point chosen for each group element class can be picked so that the finitely many corrected points are all different from each other; the paper gives no argument for this, and the interpolation lemma requires it.

Editorial extensions

If this is right

  • Every C0-stable F2-action on a compact manifold of dimension at least two has the shadowing property, so shadowing belongs to the list of necessary conditions for stability of such actions.
  • The single homeomorphism Φ_{b^{-1}}Φ_a is C0-stable whenever the action is, so one generator combination carries the shadowing information for the whole action.
  • Expansivity is not a necessary condition for stability, because the constructed action is C0- and C1-stable and non-expansive.
  • A finite pseudo-orbit of bounded word length can be ε-shadowed, and the finite-to-infinite passage in Lemma 4.1 upgrades this to the full shadowing property.
  • The proof identifies a concrete route from stability of group actions to shadowing: reduce to a cyclic subgroup, shadow a single homeomorphism, and interpolate the corrected points by a small diffeomorphism.

Reading between the lines

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

  • The coset-reduction argument should extend to free groups on k generators: pick any two generators to define the cosets and treat the remaining generators as perturbations, likely yielding the same implication for F_k-actions.
  • The interpolation step is the only place where the manifold hypothesis really enters, so a similar theorem may hold on any compact space that admits a finite point-placement lemma of the same kind.
  • Combining the theorem with the reverse implication, that expansive actions with shadowing are C0-stable, points toward a two-sided characterization of C0-stability for F2-actions on compact manifolds of dimension at least two, possibly with a weaker recurrence condition in place of full expansivity.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper has two main parts. First, it constructs an action Φ of the free group F2 on the two-sphere, claimed to be both C0-stable and C1-stable and to have a Cantor minimal set, thereby showing that expansivity is not necessary for stability of actions. Second, it proves Theorem A: every C0-stable action of F2 on a compact manifold of dimension at least two has the shadowing property. The proof reduces the action to a single homeomorphism T = Φ_{b^{-1}}Φ_a, uses Walters' theorem that C0-stable homeomorphisms have shadowing to correct each equivalence class of a pseudotrajectory, then constructs a C0-close action via the Nitecki–Shub perturbation lemma and uses a semiconjugacy to produce a shadowing point.

Significance. If the proof can be completed, Theorem A is a genuine extension of Walters' classical result from Z-actions to actions of a free group, and the example would be a useful counterpoint to the fact that expansive actions with shadowing are C0-stable. The strategy is natural and the use of the Nitecki–Shub lemma is apt. However, the manuscript in its present form contains several unproved assertions at load-bearing points, so the significance is conditional on repairs.

major comments (4)
  1. [Section 4, proof of Theorem A, before (4.1)] The assertion that y_{g0} can be chosen so that all ~x_g for |g| ≤ n are pairwise distinct is not proved and is load-bearing. It must rule out two sources of collision: within a single coset, where T^{n-m}(y_{g0}) = y_{g0} for the map T = Φ_{b^{-1}}Φ_a, and between different cosets, where T^n(y_{g0}) = T^m(y_{g0'}) for two independently chosen shadowing points. Remark 5 only excludes an open set of points with one fixed period and does not by itself supply the needed Baire/nowhere-dense argument; cross-coset collisions are not addressed at all. Moreover, because equation (4.1) involves ~x_{a^{-1}g} and ~x_{b^{-1}g}, whose lengths can be n+1, distinctness is actually needed for indices of length up to n+1, not only up to n. Without distinctness of both the domain points and the target points, Lemma 4.2 cannot be applied and the perturbation f cannot be constructed. The gap appears repairable via a finite Baire-category argument, but the proof must be written.
  2. [Lemma 4.4 and proof of Theorem A] The proof invokes 'As the map Φ_{b^{-1}}Φ_a is C0-stable', but Lemma 4.3 proves only that Φ_{a^{-1}}Φ_b is C0-stable. C0-stability of a homeomorphism is not automatically preserved under taking inverses, because inversion is not continuous in the C0 topology. A separate argument is required, for example the same proof as Lemma 4.3 with the roles of a and b exchanged. This is a missing step in the derivation, not an outright falsehood, but it must be supplied.
  3. [Section 2, Definition 2.1, property 4] Property 4 asserts that for d0_S(~Φ, Φ) < δ one has d(~Φ_s^{-1}Φ_s(x), x) < ε/2 and d(Φ_s^{-1}~Φ_s(x), x) < ε/2. This does not follow from C0-closeness alone, since the map f ↦ f^{-1} is not continuous in the uniform topology on the space of homeomorphisms of a compact space. The existence of a δ satisfying all five properties, including property 4, is asserted without proof. The authors should prove that for the specific Φ constructed in Section 2, and with the contraction structure, such a δ exists; as written this is an unproved assertion in the C0/C1-stability example.
  4. [Section 2, construction of Φ_a and Φ_b] The paper states without proof that Φ_a and Φ_b can be constructed to satisfy simultaneously the five listed contraction and intersection properties in the paragraph after the definition of Φ_b. While plausible from the figure, the existence of such maps with the required derivative bounds and the disjointness of the intervals is asserted rather than demonstrated. Since the stable-action example is a stated contribution, this construction should be made precise or at least a proof sketch should be given.
minor comments (4)
  1. [Abstract and Introduction] The abstract and the introduction say the example is a C0- and C1-stable action on S1, but Section 2 constructs the action on the two-sphere S2. The target space should be made consistent throughout.
  2. [Section 4, proof of Theorem A] Lemma 4.4 is stated for a full δ-pseudotrajectory, but the proof of Theorem A speaks of applying it to a finite δ-n pseudotrajectory. The text should clarify that Lemma 4.4 is applied to the full pseudotrajectory and the result is then restricted to |g| ≤ n.
  3. [Equation (4.1)] Equation (4.1) is written as 'f(Φ_a(~x_{a^{-1}g})) = f(Φ_b(~x_{b^{-1}g})) = ~x_g', which is confusing because it suggests f is applied to two different inputs. Since Φ_a(~x_{a^{-1}g}) and Φ_b(~x_{b^{-1}g}) coincide by Lemma 4.4, the notation should be clarified, for instance by introducing p_g as the common point.
  4. [Throughout] There are numerous typographical and grammatical issues, including 'Considerer', 'diffeomorphisms', 'parte', inconsistent spacing in d0_S(~Φ,Φ), and the undefined notation X1\X1 in Lemma 2.5. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem A's derivation is self-contained modulo external theorems; the pairwise-distinctness gap is a rigor issue, not circular reasoning.

full rationale

The derivation chain in Theorem A is not circular. It starts from the assumption that the action Phi is C0-stable, transfers this to the single homeomorphism Phi_b^{-1}Phi_a in Lemma 4.3 by a standard semiconjugacy argument, and then invokes Walters' theorem that C0-stable homeomorphisms have shadowing. This is an external, parameter-free result, not a restatement of the target. Lemma 4.4 converts an action pseudotrajectory into a pseudotrajectory for Phi_b^{-1}Phi_a and uses that homeomorphism's shadowing to produce approximating points. The perturbation in the final step uses Lemma 4.2 from Nitecki-Shub, an external interpolation lemma, to construct a C0-close action, whose semiconjugacy to Phi is supplied again by the assumed C0-stability of Phi. Each use of C0-stability is as a hypothesis yielding a semiconjugacy, exactly as in the standard stability-implies-shadowing proof; no equation is defined in terms of the shadowing conclusion. The only self-citation in the bibliography, [IP], is never cited in the body and plays no role in the arguments. The assertion that y_g0 can be chosen so that the ~x_g are pairwise distinct is indeed unproved and could be a genuine rigor gap, but it is a missing existence argument about avoiding coincidences, not a reduction of the theorem to its own premise. The paper's central claim therefore has independent content and does not exhibit circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The main theorem relies on standard results from the literature (Walters, Nitecki-Shub). The construction part postulates without proof that the maps Phi_a, Phi_b can be chosen to satisfy the contraction properties in Remark 1, and that a C0-neighborhood exists satisfying all five properties of Definition 2.1. These ad hoc assumptions are load-bearing for the claimed stability example.

assumptions (5)
  • standard math ZFC set theory and standard topology facts
    The paper uses standard results from general topology and dynamical systems without proof.
  • standard math Nitecki-Shub perturbation lemma (Lemma 4.2 in the paper)
    The proof of Theorem A uses this cited lemma to construct a diffeomorphism interpolating finite sets of points; it is accepted as a black box.
  • standard math Walters' theorem that C0-stable homeomorphisms on compact manifolds of dimension at least two have the shadowing property
    Used in Lemma 4.4 to deduce shadowing for Phi_{b^{-1}}Phi_a.
  • ad hoc to paper There exist diffeomorphisms Phi_a, Phi_b on the two-sphere satisfying the contraction properties in Remark 1
    The construction in Section 2 asserts these maps exist, but no explicit definition is given; the proof of C0-stability of the example depends on them.
  • ad hoc to paper There exists delta > 0 satisfying all five properties of Definition 2.1
    The construction of the semiconjugacy hinges on this unproved assertion, which is not a consequence of C0 closeness because inversion is not continuous in the uniform topology.

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Pith. "Pith review of C0-Stability for actions implies shadowing property." pith.science (2026). https://pith.science/paper/2HHEWR7E

@misc{pith2026190805299,
  author       = {Pith},
  title        = {Pith review of: C0-Stability for actions implies shadowing property},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2HHEWR7E}},
  note         = {Machine review of arXiv:1908.05299}
}
abstract

We will construct an action $\Phi$, C0 and C1-stable and we will prove that every C0-stable action acting in a manifold of dimensions greater or equal to two, have the shadowing property.

Figures

Figures reproduced from arXiv: 1908.05299 by the authors.

Figure 1
Figure 1. PSfrag replacements fa f −1 a Sa Ib Ib−1 0 Ja Ja−1 +∞ Note that Ia−1 = Φ −1 a (Ia) c . It is clear that Φa : S 2 → S2 is a north-south pole diffeomorphism, with Ω(Φa) = {Na, Sa}, ( Sa ∈ Ia, Na ∈ Ia−1 ) and verifies • ||Φ −1 a (x) − Φ −1 a (y)|| > 2||x − y|| for all x, y ∈ Ia (and consequently ||Φa(x) − Φa(y)|| < 1/2||x − y|| for all x, y ∈ Ia ). • ||Φ −1 a (x) − Φ −1 a (y)|| < 1/2||x − y|| for all x, y ∈ Ia−1 . Now … view at source ↗
Figure 2
Figure 2. PSfrag replacements Ia Ib Ia−1 Ib−1 Φ −1 a (Ia−1 ) Φ −1 a Φ − (Ib) 1 a (Ib−1 ) Φ −1 b (Ib−1 ) Φ −1 b (Ia−1 ) Φ −1 b (Ia) Φb(Ib) Φb(Ia−1 ) Φb(Ia) Φa(Ia) Φa(Ib−1 ) Φa(Ib) (a) Remark 1 If s, s ′ ∈ {a, a−1 , b, b−1} then ||Φs(x) − Φs(y)|| < 1/2||x − y|| ∀x, y ∈ I s ′ with s ′ 6= s −1 . 1.2 If s 6= s ′−1 then Φs(I s ′) ⊂ Is. 3. Φs(Is) ∩ Is−1 = ∅ for all s ∈ {a, b, a−1 , b−1}. Let {An} be such that A0 = Ia ∪ Ia−1 ∪ Ib ∪ I… view at source ↗
Figure 3
Figure 3. PSfrag replacements fe a J Note that if Φea(r, θ) = (fe a(r), θ) is a C 0 -perturbed of Φa, where fe a is as figure 3, then KΦe is not a Cantor set. For the following lemmas, the action Φ satisfies the five properties above. e The following remark will be very helpful. Remark 2 1. Φes(Ies) ∩ Ie s−1 = ∅ for all s ∈ {a, b, a−1 , b−1}. Some of our proofs are by induction in the length of the elements g ∈ G. Thus we nee… view at source ↗

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Works this paper leans on

7 extracted references · 7 canonical work pages

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