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arxiv: 2605.15922 · v1 · pith:NGZU3VBYnew · submitted 2026-05-15 · 🧮 math.DS

Robustly transitive behavior in symplectic dynamics

Jaime Paradela This is my paper

Pith reviewed 2026-05-19 19:13 UTC · model grok-4.3

classification 🧮 math.DS
keywords symplectic dynamicsrobust transitivityblender horseshoesreal-analytic mapsdomination conditionsymplectomorphismsnon-hyperbolic behavior
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The pith

Under a domination condition, real-analytic deformations of the product of a symplectomorphism with a basic set and one with a non-degenerate elliptic equilibrium produce large robustly transitive sets.

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

The paper examines the direct product of two symplectomorphisms, one with a basic set and the other with a non-degenerate elliptic equilibrium. Under a domination condition between these two features, many real-analytic deformations of the product system contain large sets that remain transitive after small perturbations. This approach also produces new examples of real-analytic robustly transitive symplectomorphisms that are not uniformly hyperbolic. The argument rests on new perturbation methods that create blender horseshoes in the analytic category and on control-theory techniques that give those horseshoes a wide domain of influence.

Core claim

The direct product of a symplectomorphism possessing a basic set and a second symplectomorphism possessing a non-degenerate elliptic equilibrium, when deformed by a broad class of real-analytic perturbations under a domination condition, contains large robustly transitive sets realized by blender horseshoes whose domain of influence is made large by control-theoretic arguments.

What carries the argument

Blender horseshoes created by real-analytic perturbations, whose domain of influence is enlarged by ideas imported from control theory.

If this is right

  • New examples appear of real-analytic robustly transitive symplectomorphisms that are not uniformly hyperbolic.
  • A broad class of real-analytic deformations of the product system contains large robustly transitive sets.
  • The same construction works for symplectic systems that combine hyperbolic basic sets with elliptic equilibria.

Where Pith is reading between the lines

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

  • The perturbation methods for creating analytic blender horseshoes could be tested numerically on low-dimensional examples to measure the actual size of the transitive regions.
  • The results suggest that robust transitivity without uniform hyperbolicity may occur more widely in real-analytic conservative systems once suitable domination conditions are identified.
  • Similar control-theoretic enlargement of influence domains might apply to other perturbation problems in which horseshoe-like structures appear.

Load-bearing premise

The domination condition between the basic set and the non-degenerate elliptic equilibrium is required to guarantee that the blender horseshoes created by perturbation have a sufficiently large domain of influence.

What would settle it

A concrete real-analytic deformation that satisfies the domination condition yet produces no large robustly transitive set, or in which the influence of the created blender horseshoe remains localized.

Figures

Figures reproduced from arXiv: 2605.15922 by Jaime Paradela.

Figure 1.1
Figure 1.1. Figure 1.1: Sketch of an orbit gaining momentum from a chain of partially hyperbolic objects (in blue) and leaking between the KAM tori (in red). means of the machinery developed by Kolmogorov, Arnold and Moser. Note that, in view of this phe￾nomenon, a positive measure set of orbits (which indeed becomes of relative measure one as we approach the equilibrium), are confined forever within a neighbourhood of the equi… view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Let Fn “ T n 0 ˝ T1. In the left we show the image of a small rectangle Qcs under the map Fn for n " 1{ε. In the right we show the image of D under Fni , i “ 1, 2, 3 for ni „ 1{ε. In this regime the expansion/contraction is close to one. Moreover, if β is sufficiently irrational it is possible to chose ni such that the union Ť i Fni pQcsq contains Qcs . That is, for any open sets B0, B1 P A d ? ε there e… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: A homoclinic channel Γ i Ă Ws pΛq&Wu pΛq and its orbit. satisfy (Bunching) α κ`1λ¯ ă 1, and (Pinching) α¯ κ`1λ¯ ă 1. (2) Small nonlinear remainder: The functions Q, P are C 2 and at any pq, p, zq P r´2δ0, 2δ0s 2 ˆ N we have Qp0, p, zq “ 0 “ BqQp0, 0, zq Ppq, 0, zq “ 0 “ BpPp0, 0, zq. 3.2. The local map. In the following proposition we give a precise description of the dynamics of F : X Ñ X around Λ. A st… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: The sets Qi δ for i “ 1, . . . , m. The purpose of this section is to study the map F pn´`n`q : X Ñ X when restricted to small neighbourhoods around tΓ i ´ui with i “ 1, . . . , m. We start by chosing suitable neighbourhoods of tΓ i ˘ui Ă X. To do so, observe that from their very definition, we can deduce the existence of C 2 functions (i “ 1, 2) q i ` : N Ñ r0, δ0{2s p i ´ : N Ñ r0, δ0{2s z ÞÑ q i `pzq … view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Sketch of the Markov partition elements for the case where m “ 2, i.e. we only involve two homoclinic channels. To build the partition it is helpful to analyze the preimages (and images) of the connected dashed rectangle and then look at where the left and right shaded subrectangles land. 3.6. The induced return map. Finally, in this section, we use the results obtained above to describe in detail the fi… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Image of D “ r´1, 1s 2d under the coordinate change Ψε,κ,χ. In red and blue we also indicate the expanding and contracting directions. Blender geometry and quantifiers. Based on the results in Lemma 4.3 and Theorem 4.4, we now describe the region Qcs Ă A d in which we will establish the existence of a symbolic cs-blender. To do so, it will be necessary to introduce auxilary constants κ, χ ą 0 which, as w… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: A sketch of the relative position between the sets Qcs and Qcu . which is a hyperbolic fixed point for the map FrNl and such that the pair pP cu, Qcuq where Qcu “ ψRpQcsq is a cu-blender for the IFS generated by tT, Su (see [PITH_FULL_IMAGE:figures/full_fig_p031_4_2.png] view at source ↗
read the original abstract

We consider the direct product of two symplectomorphisms, one of which exhibits a basic set and the other one a non-degenerate elliptic equilibrium. Under a domination condition we show that a broad class of real-analytic deformations of this system display large robustly transitive sets. As a corollary of our construction we also obtain new examples of real-analytic robustly transitive symplectomorphisms which are not uniformly hyperbolic. To establish these results we develop perturbation techniques to create blender horseshoes in the real-analytic setting and import ideas from control theory which show that, typically, these objects have a large domain of influence.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript considers the direct product of two symplectomorphisms, one possessing a basic set and the other a non-degenerate elliptic equilibrium. Under an explicit domination condition between these objects, it shows that a broad class of real-analytic deformations of the product system admit large robustly transitive sets. The argument proceeds by developing new perturbation techniques to produce blender horseshoes in the real-analytic category and importing control-theoretic influence estimates to establish that these objects typically exert a large domain of influence. A corollary yields new examples of real-analytic robustly transitive symplectomorphisms that are not uniformly hyperbolic.

Significance. If the central construction holds, the work supplies concrete, non-generic examples of robust transitivity in the real-analytic symplectic category, which is technically more demanding than the C^∞ or C^0 settings. The combination of analytic blender constructions with control-theoretic domain-of-influence arguments constitutes a methodological advance that may extend to other perturbation problems in conservative dynamics. The explicit hypothesis (domination) and direct-construction approach render the result falsifiable and potentially useful for producing further examples beyond uniform hyperbolicity.

major comments (2)
  1. [Proof of the main theorem (analytic perturbation construction)] The central claim rests on the preservation of the domination condition under the real-analytic deformations and on quantitative control of the domain of influence of the newly created blender horseshoes. The abstract sketches the strategy via analytic perturbations and control-theoretic estimates, but the manuscript must supply the detailed error bounds and verification that domination survives the deformations; without these, the load-bearing step from hypothesis to large transitive sets cannot be assessed.
  2. [Corollary on non-hyperbolic examples] The corollary on new examples of non-uniformly hyperbolic robustly transitive symplectomorphisms follows from the main construction, yet the manuscript should explicitly verify that the resulting maps remain symplectic and that the robust transitivity persists in a C^1-neighborhood; a brief check in the corollary section would confirm this does not introduce additional assumptions.
minor comments (2)
  1. [Abstract and Introduction] The phrase 'broad class of real-analytic deformations' in the abstract and introduction would benefit from a precise characterization (e.g., an open set in a suitable Banach space of analytic maps) to clarify the scope of the result.
  2. [Section on control-theoretic estimates] Notation for the domination condition and the 'domain of influence' should be introduced with a short reminder of the relevant constants or norms when first used in the technical sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance. We have revised the paper to address the two major comments by improving the clarity of the analytic perturbation arguments and by adding an explicit verification in the corollary.

read point-by-point responses

  1. Referee: [Proof of the main theorem (analytic perturbation construction)] The central claim rests on the preservation of the domination condition under the real-analytic deformations and on quantitative control of the domain of influence of the newly created blender horseshoes. The abstract sketches the strategy via analytic perturbations and control-theoretic estimates, but the manuscript must supply the detailed error bounds and verification that domination survives the deformations; without these, the load-bearing step from hypothesis to large transitive sets cannot be assessed.

    Authors: We thank the referee for this observation. The original manuscript contains the relevant estimates in the proofs of the main theorems, but we agree that the presentation of the error bounds and the explicit verification of domination preservation could be made more transparent. In the revised version we have added two new lemmas in Section 3 (Lemmas 3.5 and 3.6) that supply the quantitative C^ω error bounds for the analytic deformations and directly verify that the domination condition between the basic set and the elliptic equilibrium is preserved. These lemmas also incorporate the control-theoretic influence estimates to confirm the large domain of influence of the blender horseshoes. revision: yes

  2. Referee: [Corollary on non-hyperbolic examples] The corollary on new examples of non-uniformly hyperbolic robustly transitive symplectomorphisms follows from the main construction, yet the manuscript should explicitly verify that the resulting maps remain symplectic and that the robust transitivity persists in a C^1-neighborhood; a brief check in the corollary section would confirm this does not introduce additional assumptions.

    Authors: We agree that an explicit check strengthens the exposition. In the revised manuscript we have inserted a short verification paragraph immediately after the statement of the corollary. The paragraph records that the constructed maps remain symplectic because they arise from real-analytic deformations that preserve the symplectic form, and that robust transitivity holds in a C^1-neighborhood by the C^1-robustness of the blender-horseshoe construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes an existence result for large robustly transitive sets in real-analytic deformations of a product of symplectomorphisms, under an explicitly stated domination hypothesis between a basic set and a non-degenerate elliptic equilibrium. It proceeds via direct construction: new analytic perturbation techniques to produce blender horseshoes, combined with control-theoretic estimates on their domain of influence. No step reduces by definition to its own output, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation chain whose verification is internal to the present work. The domination condition functions as an independent hypothesis rather than an implicit or derived assumption, and the argument remains independent of its target conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard background facts from symplectic geometry and hyperbolic dynamics plus one explicit domain assumption; no free parameters or new postulated entities are introduced.

axioms (2)
  • standard math Standard properties of symplectomorphisms, basic sets, and non-degenerate elliptic equilibria in smooth dynamical systems
    Invoked throughout to set up the product system and the objects being perturbed.
  • domain assumption The domination condition between the basic set and the elliptic equilibrium
    Explicitly stated as the hypothesis under which the deformations produce large robustly transitive sets.

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