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Conditions Implying Annular Chaos: Quantitative results and Computer Assisted Proofs

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Pith's one-line read This paper proves that a single orbit whose vertical displacement exceeds a threshold depending only on the map's maximal step forces unbounded diffusion; for area-preserving maps with zero vertical drift it also forces rotational chaos…

desk verdict Useful quantitative thresholds and broad CAPs, but the whole theorem stack rests on the authors' unpublished [31] and a numerical inconsistency needs fixing. read the letter →

arxiv 2506.06608 v1 pith:GLXMKW2F submitted 2025-06-07 math.DS

M. J. Capiński , M. Gröger , A. Passeggi , F.A. Tal This is my paper
classification math.DS MSC 37E3037B40
keywords annularhomeomorphismsrotationalhorseshoeunboundeddiffusionDehntwistrotationsetcomputer-assistedproofintervalarithmeticNon-TwistStandardFamily
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 converts the qualitative theory of annular chaos into a finite numerical test. For a non-wandering homeomorphism of the annulus whose lift behaves like a Dehn twist $(x,y)\mapsto(x+ky,y)$, it proves that a single orbit with vertical displacement at least $M=\min\{3N(f)+2,N(f)+4\}$, where $N(f)$ is the largest one-step vertical displacement, forces unbounded diffusion; if the map preserves an absolutely continuous measure with zero vertical drift, the same displacement forces a rotational horseshoe and total unbounded diffusion. For non-wandering maps homotopic to the identity that have fixed points with rotational difference $\rho\ge1$, the thresholds are $M_1=6N(f)+2$, $M_2=4N(f)+2$, and $M_i=2N(f)+2$ for $i\ge3$, and crossing $M_\rho$ again gives unbounded diffusion and a non-empty rotation-set interior. These bounds give computable control on deviations from a horizontal rotation interval, and the paper demonstrates the method with validated computer-assisted proofs in the Standard Family, its non-twist and dissipative variations, and the Non-Twist Standard Family.

What carries the argument

The load-bearing mechanism is the invariant circloid: a minimal essential continuum that separates the annulus into two ends. The proof shows that bounded diffusion produces such a circloid $C$, and then bounds its vertical diameter $VD(C)$ in terms of $N(f)$ and the rotational difference $\rho$. This is done with prime-end rotation numbers: arcs joining $C$ to its vertical translate are stretched by $f^2$ or $f^3$, and Lemmas 3.3 and 3.7 turn the stretching into inequalities such as $VD(C)<3N(f)+1$ or $VD(C)<2nN(f)+1$. For the computer-assisted proofs, the indispensable tool is parallel shooting: a candidate orbit from $y=-B$ to $y>B$ is encoded as a zero of a finite-dimensional map $F$, whose existence is proved by the Krawczyk interval-Newton method; the method needs only interval enclosures of iterates, not derivatives. In the dissipative case, the imported $N$-disjoint-pair criterion of [31] is verified by building small segments near fixed points and using validated trajectories from them across the map.

What would settle it

Exhibit a non-wandering Dehn-twist homeomorphism of the annulus with $N(f)\le1$ and an orbit satisfying $|pr_2(f^n(z)-z)|\ge5$, while every orbit of the map remains in a bounded vertical strip. That would refute the diffusion part of Theorem 1.

Watch

Extended reading notes

Core claim

On its own terms, the central claim is that unbounded diffusion is forced by a single, quantitatively bounded displacement. For a non-wandering Dehn-twist map of the annulus, Theorem 1 states that an orbit with $|pr_2(f^n(z)-z)|\ge \min\{3N(f)+2,N(f)+4\}$ has unbounded diffusion, and under zero vertical drift it has rotational chaos and total unbounded diffusion. Theorem 2 gives the analogous constants $M_\rho$ for non-wandering maps homotopic to the identity with a rotational difference $\rho\ge1$, and adds that the projected torus map has a rotation set with non-empty interior. Corollary 1 extracts from Theorem 2 a computable bound on how far orbits can deviate from a horizontal rotation interval containing two rational points. The paper also proves for the Non-Twist Standard Family a parameter-dependent bound $M_{a,b}$ such that an orbit crossing from below $-M_{a,b}$ to above $M_{a,b}$ yields total unbounded diffusion, and that every $a>0$, $b\ne0$ parameter pair has a rotational horseshoe.

Load-bearing premise

The whole construction assumes that the topological criterion from [31] is correct: whenever a map has two fixed points differing by a whole-number vertical rotation and a pair of small disjoint neighborhoods whose orbits interlink, a chaotic rotational horseshoe necessarily exists.

Editorial extensions

If this is right

  • If Theorem 1 is right, chaos detection for Dehn-twist-type maps reduces to computing $N(f)$ and locating one orbit with vertical displacement at least $\min\{3N(f)+2,N(f)+4\}$; no invariant curves or derivative estimates are needed.
  • If Theorem 2 is right, then any non-wandering area-preserving map whose rotation set is a horizontal interval containing two rational points has all its vertical deviations inside the explicit strip $[-M_\rho,M_\rho]$; leaving that strip certifies a non-empty rotation-set interior.
  • The Non-Twist Standard Family result implies that every parameter pair $a>0$, $b\ne0$ in that family carries a rotational horseshoe, so the twist-breaking region is not an obstacle to symbolic chaos.
  • The CAP validations establish diffusion and chaos in reproducible instances: for all listed twist and non-twist variations of the Standard Family, for at least 240,359 parameter pairs in the Non-Twist Standard Family mesh, and for at least 95.95% of the area of the dissipative parameter domain considered.
  • Because the CAP validation uses only $C^0$ information, the same algorithm is directly portable to maps whose derivatives are unavailable, such as Poincaré return maps, a direction the paper explicitly says it plans to pursue.

Reading between the lines

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

  • The constants are likely not sharp: the proof fixes the number of iterates used in the prime-end argument, so a finer prime-end count could push the thresholds down without changing the topology.
  • The CAP routine is a natural template for an automated chaos screen: on any parameter mesh it needs only a crossing orbit candidate plus interval arithmetic, and a failure means “no proof found yet” rather than “no chaos.”
  • Zero vertical drift enters only through the final horseshoe step, so the paper proves unbounded diffusion without it but a rotational horseshoe only with it; whether the drift condition is actually necessary for the horseshoe conclusion is left open.
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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

3 major / 6 minor

Summary. The paper derives quantitative sufficient conditions for unbounded diffusion and rotational horseshoes in annular homeomorphisms, building on topological criteria from the companion paper [31] (Passeggi and Tal, 'Conditions implying annular chaos', submitted). Theorem 1 gives the bound M = min{3N(f)+2, N(f)+4} for non-wandering annulus homeomorphisms homotopic to a Dehn twist, and Theorem 2 gives M1 = 6N(f)+2, M2 = 4N(f)+2, and Mi = 2N(f)+2 for i ≥ 3 for maps with rotational difference ρ; Corollary 1 converts these into a computable bound on deviations from a rotation interval. Theorem 3 gives a parameter-uniform horseshoe statement for the Non-Twist Standard Family and a crossing criterion for total unbounded diffusion. The rest of the paper presents computer-assisted proofs for variations of the Standard Family, for the Non-Twist Standard Family on a 1000×1000 mesh, and for the Dissipative Standard Family, using parallel shooting and Krawczyk-verified interval arithmetic, with code at github.com/mcapinsk/cac-cap.

Significance. If the companion criteria in [31] are correct, this is a substantial practical advance: explicit and easily evaluated bounds depending only on N(f) and ρ, no dependence on derivatives, and computer-assisted proofs over large parameter ranges in conservative and dissipative settings. The CAP methodology is a genuine strength: interval arithmetic with Krawczyk-type verification is used, the code is provided, and the theoretical constants are explicit and falsifiable through the stated numerical results. The main caveat is that Theorems 1, 2, Theorem 3(2), and the dissipative CAP inherit their force from the unpublished source [31], whose Corollary B, Proposition 4.9, and Theorem C are imported as black boxes; consequently the significance is conditional on independent verification of that companion work.

major comments (3)
  1. [§2, §3.1, §3.2, §3.3, §6] The central theoretical results depend on unpublished black-box statements from [31]. Specifically, Corollary B of [31] is restated in §2 and applied directly in §6; Lemma 3.3 in §3.1 is disposed of by saying the proof is 'essentially the same as [31]'; the proof of Theorem 3.6 invokes Proposition 4.9 of [31] and Theorem C of [31]; and Lemma 3.11 invokes both Theorem C and Proposition 4.9 of [31]. Since [31] is a submitted manuscript that is not publicly available and is co-authored by two of the present authors, the proofs of Theorems 1, 2, Theorem 3(2), and the dissipative CAP are not independently verifiable as written. If any of these imported statements has a gap, the constants M, M1, M2, Mi and the rotational-horseshoe conclusions collapse. I ask that the authors either prove the needed statements within this paper, or make [31] publicly available and, ideally, have the specific imported results independently checked; otherwise the theorems should be explicitly labeled as conditional on [31].
  2. [§1.4 vs §6.4] Theorem 1.4 states that chaos is obtained 'in at least 95.95% of the area' of the parameter domain, while Section 6.4 reports that the validated area 'exceeds 98.47% of the domain'. Since the quantified area is precisely the content of Theorem 1.4, this is not a cosmetic discrepancy. The authors must determine which number is correct and ensure that the theorem statement, the reported computation, and the code output agree.
  3. [§3.3, proof of point (1) of Theorem 3.8] The proof of Theorem 3.8(1) begins by assuming that the map 'has no topological entropy' and then asserts 'from the considerations of the previous subsection' that the map has bounded diffusion and that every orbit stays in a horizontal strip of uniformly bounded width. The previous subsection establishes only a sufficient condition for unbounded diffusion (a crossing from below -Ma,b to above Ma,b), not an equivalence, so the implication 'no entropy ⇒ bounded diffusion' needs a proof or a precise citation. The argument then invokes Theorem A of [25], Theorem 1.4 of [20], and Ushiki's theorem as cited in [16]; please state the exact hypotheses and verify explicitly that they apply for all a ∈ (0,∞) and b ≠ 0.
minor comments (6)
  1. [§3.1, Lemma 3.3] The statement of Lemma 3.3 contains a redundancy: after the case p = 2, the clause 'if p ≥ 3 then we can show the same for σ ∪ f^2(σ)' should presumably read 'σ ∪ f(σ)' or should be corrected; in the proof, 'the cases where p = 1 or p + 2 are similar' appears to be a typo for 'p = 1 or p = 2'.
  2. [§3.3, Lemma 3.10] The proof of Lemma 3.10 writes fa,b(1/4, y) = (x', y + |b|), but for b > 0 the second coordinate is y - b = y - |b|; the argument works after choosing x = 3/4 instead for b > 0. Please make the sign choice explicit.
  3. [§4.1, paragraph after Eq. (8)] The sentence claiming that the parametrization of the initial and final points 'ensures that there can be only one solution for (8)' overstates the matter; uniqueness is guaranteed by the Krawczyk verification on the chosen interval box, not by the parametrization alone. I suggest rephrasing.
  4. [Throughout] The manuscript contains numerous typos and infelicities that should be cleaned up in revision, for example 'copared' in the caption of Figure 1, 'supremmun' in §3, 'employees' in the proof of Theorem 3.2, 'estasblish' in §4.1, and 'the taks' in the footnote of §5.2.
  5. [Code availability [1]] For reproducibility of the computer-assisted proofs, the repository should be pinned to a specific commit or release, and the computational environment (compiler/interpreter and interval arithmetic library versions) should be stated.
  6. [References] Reference [31] should indicate its current status and availability; the citation to Ushiki's theorem through [16, p.289] would be easier to verify if the original source or a more precise statement were given.

Circularity Check

4 steps flagged · score 4.0 of 10

The quantitative theorems and CAP conclusions import their core criteria from unpublished [31] by two of the present authors; no independent proof is supplied, though the numerics themselves are not fitted.

  1. self citation load bearing [Lemma 3.3, proof (Section 3.1)]
    "The proof is essentially the same as that of [31]. One can go to the prime ends compactification of A, and obtain a homeomorphism h : A → T × (0, 1) such that g = h ◦ f ◦ h−1 extends to a continuous homeomorphism of T × [0, 1], which we still denote g, and that has a lift ˜g : R × [0, 1] → R × [0, 1] such that the rotation number for ˜g of T1 × {0} is ρ and of the upper boundary is ρ + p."

    Lemma 3.3 is the central estimate-producing tool in the proof of Theorem 3.2: it turns a crossing σ with f^3(σ) into an essential compact set, which then yields the N(f) bound on vertical diameter. The paper explicitly delegates the proof to [31], an unpublished submitted paper by Passeggi and Tal, two members of the present author list. Thus the quantitative diffusion criterion M = min{3N(f)+2, N(f)+4} is not derived in this paper independently; it inherits its validity from a self-citation that is not publicly verifiable.

  2. self citation load bearing [Proof of Theorem 3.6, Section 3.2]
    "If this was not the case, then there would exist a neighborhood U of σ such that S n i=0 f i(U ) is inessential and contains x0. By Proposition 4.9 of [31] this implies that one of the prime ends rotation number for C lies in the interval [−1/n, 1/n] a contradiction since this interval does not intersect [ρ/2, ∞)."

    In the proof of Theorem 3.6, the non-wandering homeomorphism case, Proposition 4.9 of [31] is used as a black box to convert the absence of an essential intersection σ ∪ f^n(σ) into a prime-end rotation number constraint. This is precisely the step that produces the constants M1, M2, Mi. Since [31] is by two of the present authors and not publicly available, the theorem's conclusion rests on an unverified self-citation; no proof of Proposition 4.9 is included.

2 more flagged steps
  1. uniqueness imported from authors [Proof of Theorem 3.6, Section 3.2]
    "As a consequence, since every circloid for a nonwandering homeomorphism of A has the same rotation number for ˜f (see, for instance, Theorem C of [31]), every point of C has the same rotation number."

    This step invokes Theorem C of [31] as an external mathematical fact to force all circloid points to have a single rotation number. The uniqueness-type assertion is from the same authors' submitted work and is load-bearing: it excludes other prime-end rotation behaviors and allows the dichotomy leading to the Mρ bound. Treating it as an established theorem is an example of importing a uniqueness result from the authors' own unpublished manuscript.

  2. self citation load bearing [Section 1.4 and Section 6.1 (Dissipative Standard Family CAP)]
    "Applying Corollary B of [31] as described in Section 6, we obtained a computer-assisted proof of the existence of rotational horseshoes for the following range of parameters, which is taken under the following consideration: the twist parameter a is taken large enough so to have fixed points with a rotational difference, avoiding to consider powers of the maps."

    The dissipative CAP validates disjoint neighborhoods and orbit visits numerically, but the theorem that these data imply a rotational horseshoe is exactly Corollary B of [31], restated in Section 6.1 as 'the adapted version of Corollary B'. The numerical validation is rigorous, but its dynamical interpretation as chaos is conditional on [31], a self-citation by Passeggi and Tal; if Corollary B fails, Theorem 1.4's 'chaos' conclusion fails with it.

full rationale

The paper's new quantitative content is real: N(f) is computed, the constants M, M1, M2, Mi and Ma,b are derived, and the CAP validations use Krawczyk/interval arithmetic rather than fitting. No fitted input is renamed as a prediction. However, the load-bearing step converting fixed points plus disjoint neighborhoods into a rotational horseshoe, and the prime-end lemmas producing the M bounds, are imported from [31] (Passeggi-Tal, submitted, not publicly available and authored by two of the present authors). Theorems 1 and 2, Corollary 1, Lemmas 3.3 and 3.7, and the dissipative CAP would stand or fall with [31]. This is a self-citation chain rather than an equivalence by construction, so the score is 4 rather than higher. The numerical discrepancy between 95.95% (Theorem 1.4) and 98.47% (Section 6.4) is a correctness/reporting issue, not additional circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The quantitative bounds M and M_{a,b} are derived, not fitted; there are no free scientific parameters. The central theoretical claims rest on the unpublished topological criteria of [31] by two of the same authors, plus several deep external theorems such as prime-end rotation theory, [15], [25] and Ushiki's theorem, and the CAPs rest on the Krawczyk and interval-arithmetic framework. No invented entities are introduced.

assumptions (7)
  • domain assumption The topological criteria of [31], N-disjoint pairs of neighborhoods and Birkhoff-related fixed points imply rotational horseshoe or unbounded diffusion, are correct.
    Section 2 states these as the theoretical starting point; Corollary B, Theorem C and Proposition 4.9 of [31] are used in proofs of Theorems 3.2, 3.6 and in Section 6.
  • domain assumption Prime-end compactification and rotation number estimates for invariant annuli, Lemma 3.3, similar to Lemma 8.3 of [31], are valid.
    Used to bound the vertical diameter of invariant circloids and derive M = min{3N+2, N+4}.
  • domain assumption Bounded diffusion in Homeo_{0,nw,rho} implies existence of an invariant circloid disjoint from its vertical translates, Theorem 5.1 of [15].
    Used in Lemma 3.7 to set up the proof of Theorem 3.6.
  • domain assumption In the absence of a topological horseshoe, rotation numbers exist and are continuous, Theorem A of [25].
    Used in the proof of Theorem 3(1) to derive a contradiction via Ushiki's theorem.
  • standard math Ushiki's theorem: no saddle connections between hyperbolic saddles for biholomorphic maps.
    Final step of Theorem 3(1) for the non-twist standard family.
  • standard math Birkhoff's curve theorem for conservative twist maps.
    Lemma 3.12 assumes invariant circloids in twist regions are graphs.
  • standard math Krawczyk interval-arithmetic validation theorem [6] correctly certifies zeros of C^1 maps.
    Basis for all computer-assisted proofs in Sections 4, 5 and 6.

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Pith. "Pith review of Conditions Implying Annular Chaos: Quantitative results and Computer Assisted Proofs." pith.science (2026). https://pith.science/paper/GLXMKW2F

@misc{pith2026250606608,
  author       = {Pith},
  title        = {Pith review of: Conditions Implying Annular Chaos: Quantitative results and Computer Assisted Proofs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GLXMKW2F}},
  note         = {Machine review of arXiv:2506.06608}
}
read the original abstract

We derive quantitative sufficient conditions for rotational chaos and diffusion in annular homeomorphisms, building on the topological criteria established in [31]. These conditions depend only on basic properties of the maps, making their implementation straightforward. To demonstrate the effectiveness of the method, we provide computer-assisted proofs of rotational chaos and diffusion for classical families of annular maps and their variations, in both conservative (twist and non-twist) and dissipative settings.

Figures

Figures reproduced from arXiv: 2506.06608 by the authors.

Figure 1
Figure 1. Parameter pairs for which we have validated the ex￾istence of unbounded diffusion in the Non-Twist Standard Family (in red), from a mesh of 1000 × 1000 points (a, b) ∈ [0, 1]2 . This can be copared with [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. A 2-d.p.n. for the pair of fixed points x0, x1. The annulus is represented as the plane minus a single point denoted by a cross. Finally, we say that two such fixed points x0, x1 are N-Birkhoff related whenever there exists a N-d.p.n. U0, U1 such that both the forward orbit of U0 intersects U1 and the forward orbit of U1 intersects U0. Let us briefly explain how the results of [31] are applied in the present work. I… view at source ↗
Figure 3
Figure 3. The trajectories for the twist maps from [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Sketch of the configuration of fixed points as well as of the lines y = B and y = −B together with their forward images (dotted lines) [PITH_FULL_IMAGE:figures/full_fig_p028_4.png]
Figure 5
Figure 5. Figure 5: Sketch of the boxes Bκ ℓ for the iterations of (xκ, yκ). (III) S + 1 starts at (x−κ, y−κ) and ends at a point (x + 1 , y+ 1 ) which goes above the line y = B after some number of steps. Again, one needs to find such a point (x + 1 , y+ 1 ) first and then work with the …
Figure 6
Figure 6. Figure 6: Sketch of the boxes B −κ ℓ for the iterations of (x−κ, y−κ). The next 4 conditions for the segments S − 0 and S − 1 , which are needed, are just symmetric to those in (1)-(4) but this time we need to reach the line y = −B. (V) S − 0 starts at (xκ, yκ) and finishes at a…
Figure 7
Figure 7. Figure 7: In the left and middle plots we have the segments in the original coordinates, in the right plot we see the segments in the local coordinates at our fixed points. 6.3. CAP-setup. In this section we address the issue of how to validate the conditions (I)-(VIII). The dir…
Figure 8
Figure 8. Figure 8: An illustration for Lemma 6.3. Let us start with a simple fact. Lemma 6.2. For every N > 0 there exists a δ such that f ℓ a,b S ± 0 (δ)  ⊂ B κ ℓ and f ℓ a,b S ± 1 (δ)  ⊂ B −κ ℓ for ℓ = 1, . . . , N. Proof. This follows from the continuity of fa,b. □ The above lemma m…
Figure 8
Figure 8. Figure 8: The [Dfi(U˜)] and [−L, L] are compact, so there exists a λ > 1 such that if |y| ≤ L and C ∈ [Dfi(U˜)] then |πxC (1, y)| > λ. For every q1, q2 ∈ U˜ fi (q1) − fi (q2) = Z 1 0 d dsfi (q2 + s (q1 − q2)) ds = Z 1 0 Dfi (q2 + s (q1 − q2)) ds (q1 − q2) = C (q1 − q2) for C = C…
Figure 9
Figure 9. Figure 9: Establishing a point which goes up from a segment at (xκ, yκ). This means that starting from two points q1, q2 such that πx (q1 − q2) ̸= 0 and |πy (q1 − q2)| ≤ L |πx (q1 − q2)| we can iterate the argument as long as f m−1 i (q1), f m−1 i (q2) ∈ U˜ to obtain |πx (f m i …

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Pith tools

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