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Characteristic flows on contact Hamiltonian manifolds are never Anosov, and convexity of hypersurfaces is decided by the dynamics of those flows.

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2026-07-12 05:24 UTC pith:JJHGW6K4

load-bearing objection Clean survey that packages the dynamical dictionary for convex hypersurfaces, with one self-contained new obstruction and a useful open-problem list; soft only where surveys are allowed to be soft.

arxiv 2607.03019 v1 pith:JJHGW6K4 submitted 2026-07-03 math.SG

Conformally symplectic topology from a dynamical viewpoint

Julian Chaidez This is my paper
classification math.SG MSC 53D1053D3537C1037D20
keywords contact Hamiltonian manifoldcharacteristic foliationconvex hypersurfaceconformally symplectic dynamicsMorse-SmaleAnosov flowsdividing setLiouville halves
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This survey treats hypersurfaces in contact manifolds as intrinsic contact Hamiltonian manifolds and studies the characteristic foliation that lives on them. That foliation is a conformally symplectic dynamical system: its flow scales a Liouville form rather than preserving a symplectic form. The paper shows that such a flow on a closed manifold can never be Anosov. Convexity of the hypersurface is then characterised by dynamical conditions on the same foliation. When the foliation is Morse-Smale the hypersurface is convex; convex hypersurfaces are C0-dense in every dimension, yet in dimension five and higher there exist hypersurfaces that remain non-convex under every C2-small perturbation. The survey therefore turns classical questions about contact convexity into questions about recurrence, hyperbolicity and robust cycles in a natural class of conformally symplectic flows.

Core claim

A contact Hamiltonian manifold is convex precisely when its characteristic foliation admits a splitting of the non-wandering set into positive and negative Liouville pieces with no retrograde connections; in particular every Morse-Smale characteristic foliation is convex, convex hypersurfaces are C0-dense, yet C2-robustly non-convex examples exist in every contact manifold of dimension at least five, and no characteristic flow on a closed contact Hamiltonian manifold can be Anosov.

What carries the argument

The characteristic foliation of a contact Hamiltonian form, defined by the singular line field spanned by any vector field Z satisfying ι_Z μ = λ ∧ (dλ)^{n-1}. Its divergence and stable/unstable manifolds decide which orbits are positive or negative Liouville and therefore which hypersurfaces can be convex.

Load-bearing premise

The reconstruction of positive and negative halves from the non-wandering set works only if Liouville forms can be extended handle-by-handle across ordinary and round handles; the survey sketches this step and relies on earlier detailed arguments.

What would settle it

Exhibit a closed contact Hamiltonian manifold whose characteristic flow is Anosov, or a C2-open set of hypersurfaces in a five-dimensional contact manifold that are all convex, or a Morse-Smale characteristic foliation that cannot be framed so that the halves become Liouville domains.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. This survey packages the interaction between conformally symplectic dynamics and contact topology via contact Hamiltonian manifolds (intrinsic models for hypersurfaces in contact manifolds) and their characteristic foliations. It develops elementary structure theory (rescaling invariance, well-defined singular line field, contact transversals, contactization germs, Liouville subsets via divergence, isotropic stable/unstable manifolds of hyperbolic orbits/singularities), proves that closed characteristic flows are never Anosov (Theorem 2.48), and surveys the Morse-Smale criterion for convexity (Theorem 3.11, Giroux/Breen), C0-density of convex hypersurfaces (Honda-Huang, Theorem 3.15), and C2-robust non-convexity in dimensions ≥5 (Chaidez, Theorem 3.21). It closes with open problems on existence/tightness, doubles, chain-convexity criteria, C1-genericity, and generic Liouville flows.

Significance. The paper gives a clean dynamical language for convex hypersurface theory and makes the recent C0-density / C2-robust non-convexity dichotomy accessible. The self-contained non-Anosov theorem (generalizing Asaoka-Mitsumatsu) and the elementary lemmas on characteristic foliations and Liouville subsets are useful reference material. The open-problem section (chain convexity, Conjecture 4.19 on positive-negative heterodimensional cycles, Axiom A for Liouville flows) is well-posed and likely to stimulate further work. As a survey for conference proceedings it succeeds in organizing the state of the art without claiming new theorems beyond the non-Anosov result and the forthcoming chain-convexity criterion.

minor comments (4)
  1. [§3.3] Proof sketch of Theorem 3.11 (Morse-Smale criterion) defers the inductive handle-by-handle (and round-handle) extension of Liouville forms to Breen [13] and Honda-Huang [43]. A one-sentence pointer to the precise sections of those papers would help readers who want the full argument.
  2. [Exercise 2.11 / Figure 1] Figure 1 caption refers to “Figure 6” for the singular line fields; the cross-reference appears to be off by several figures.
  3. Scattered typos: “singuar” (Def. 2.15), “charactertic” (Lemma 2.25), “satisifes” (Lemma 2.21), “auxilliary” (Lemma 2.29), “Lioiville” (§4.2), “result result” (before Theorem 4.17).
  4. [§2.1, Remark 2.5] Definition 2.1 of contact Hamiltonian form is slightly more general than the author’s earlier work [16]; a brief remark on the relationship to even-contact structures would clarify the scope for readers coming from that paper.

Circularity Check

0 steps flagged

No significant circularity: survey packages attributed results and elementary lemmas without definitional or self-citation reduction of its claims.

full rationale

This is a survey of conformally symplectic / convex hypersurface theory. Core objects (contact Hamiltonian form Def. 2.1, characteristic foliation Def. 2.16, convexity Def. 3.1, dividing set Def. 3.6) are defined independently of the dynamical conclusions they support; characterizations such as the divergence criterion (Lemma 2.36) and isotropic stable/unstable manifolds (Lemma 2.44) are explicit equivalences or corollaries, not predictions forced by fitted inputs. Theorem 2.48 (no Anosov characteristic flows) is proved in-line from the isotropic-dimension bound and Anosov leaf structure. The Morse-Smale convexity criterion (Thm 3.11), C0-density (Thm 3.15), and C2-robust non-convexity (Thm 3.21) are attributed to Giroux/Breen, Honda-Huang, and the author’s prior arXiv:2406.05979 respectively; the survey sketches strategies and defers details, which is standard for a survey and does not make the packaging circular. Self-citation of [16] is ordinary attribution of a surveyed result, not a load-bearing uniqueness theorem or ansatz smuggled as external fact. No fitted-parameter-as-prediction, no self-definitional loop, and no renaming of an empirical pattern as a first-principles derivation. Score 0 is the honest finding.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 2 invented entities

As a pure-mathematics survey the paper rests on standard differential topology and on previously published theorems of Giroux, Honda-Huang, Breen et al. No free parameters are fitted. The only invented entities are definitional re-packagings (contact Hamiltonian manifold, framing) that coincide with existing notions of even-contact structures and convex hypersurfaces. Load-bearing external results are treated as domain assumptions.

axioms (5)
  • standard math Kupka-Smale theorem and Peixoto genericity of Morse-Smale flows on surfaces
    Used to deduce C∞-genericity of convex surfaces in dimension three (Theorem 3.13).
  • domain assumption Honda-Huang C0-density of gradient-like characteristic foliations (Theorem 3.16) and existence of contact Hamiltonian plugs
    Taken as established; the survey only sketches the Eliashberg-Pancholi simplification.
  • standard math Stable Manifold Theorem for hyperbolic orbits and singularities
    Invoked for isotropic stable/unstable manifolds (Lemma 2.44) and index constraints.
  • domain assumption Borman-Eliashberg-Murphy h-principle for overtwisted contact structures
    Used for the formal existence result (Proposition 4.3).
  • domain assumption Existence of blenders and robustly transitive contactomorphisms on unit cosphere bundles of hyperbolic manifolds
    Taken from the author’s earlier work [16] to produce C2-robust non-convexity.
invented entities (2)
  • contact Hamiltonian manifold / form independent evidence
    purpose: Intrinsic model for hypersurfaces in contact manifolds and for even-contact structures
    Definitional packaging of already-studied objects; coincides with even-contact manifolds when non-singular.
  • framing (u,θ) and contactization independent evidence
    purpose: Produce a germ of contact structure on a thickening of the even-dimensional manifold
    Standard construction already implicit in Giroux convexity; contractible space of framings is elementary.

pith-pipeline@v1.1.0-grok45 · 26590 in / 2775 out tokens · 19767 ms · 2026-07-12T05:24:24.397683+00:00 · methodology

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read the original abstract

This survey article discusses the emerging interaction between conformally symplectic topology and dynamics, with a focus on recent developments in convex hypersurface theory.

Figures

Figures reproduced from arXiv: 2607.03019 by Julian Chaidez.

Figure 1
Figure 1. Figure 1: Examples of contact Hamiltonian structures on the 2-torus and the disk. Here are the motivating examples of contact Hamiltonian manifolds that arise in practice. Example 2.6 (Hypersurfaces). Fix a hypersurface Σ in a contact manifold p𝑌, 𝜉q with contact form 𝛼. Then Σ is a contact Hamiltonian manifold with contact Hamiltonian structure 𝜂 “ 𝜉 X 𝑇Σ defined by the contact Hamiltonian form 𝜆 “ 𝛼|Σ Indeed, note… view at source ↗
Figure 2
Figure 2. Figure 2: The characteristic foliation along with contact transversals mapped to each other, and small neighborhoods isomorphic to their symplectizations. Lemma 2.25 (Dimension). Any isotropic sub-manifold Λ in a contact Hamiltonian manifold pΣ, 𝜂q has (2.5) dimpΛq ď 1 2 dimpΣq The isotropic Λ is tangent to the characteristic foliation Σ𝜂 (or equivalently, invariant under any charac￾terstic flow) if the inequality (… view at source ↗
Figure 3
Figure 3. Figure 3: The different types of singularities and closed orbits in dimension two. The index one singularities are distinguished by the sign of the divergence. The two lemmas follow immediately from Lemma 2.44, Lemma 2.25 and Definition 2.43. Remark 2.47 (Symplectic Case). In the case of symplectic diffeomorphisms and flows on a symplectic manifold p𝑋, Ωq, any hyperbolic fixed point or closed orbit must have index g… view at source ↗
Figure 4
Figure 4. Figure 4: Giroux’s strategy for constructing contact Morse functions used an analysis of the family of level sets given by a Morse function and specifically the analysis (and modification) of their characteristic foliations. strategy. Their work also lead to further applications, such as a well developed handlebody theory for convex hypersurfaces, including a higher dimensional theory of bypasses, and the completed … view at source ↗
Figure 5
Figure 5. Figure 5: Characterstic foliations on the torus with dividing sets in green. Lemma 3.7 (Dividing Set Is Contact). The dividing set Γ Ă Σ is a contact sub-manifold transverse to the characteristic foliation that is independent of the framing function up to isotopy. Proof. Let 𝑢 be any framing function. By Definition 2.27, the differential form (2.6) is a volume form 𝜇. It follows that 𝑑𝑢 ‰ 0 along the subset Γ “ 𝑢 ´1… view at source ↗
Figure 6
Figure 6. Figure 6: The splitting of a contact Hamiltonian 2-manifold with Morse-Smale characteristic foliation, reconstructed from critical points. Precisely, consider the simplified setting where Σ𝜂 is gradient-like and there are no critical points of middle index 𝑛. In this case, NW`pΣ𝜂q and NW´pΣ𝜂q is simply the set of all of the critical points of index below 𝑛 and above 𝑛, respectively. Since Σ𝜂 satisfies the Smale prop… view at source ↗
Figure 7
Figure 7. Figure 7: The three types of non-wandering components. We can apply the Kupka-Smale theorem [46, 53] to 𝐶 8-approximate the flow by a Kupka-Smale flow where all of the singularities and periodic orbits are non-degenerate, and where the unstable and stable manifolds of the singularities intersect transversely. This prevents any heteroclinic connections between saddle points, leaving only singularities and closed orbi… view at source ↗
Figure 8
Figure 8. Figure 8: A 2-dimensional plug for small trapping size. Given a blocking collection Γ thickened to an embedding r0, 1s ˆ Γ Ñ Σ, one may replace each component with a gradient-like plug to get a plugged singular line field. One may show that, if the trapping size is small enough, then every trajectory of the plugged line field must begin and end on a non-degenerate singularity in one of the plugs. After a further 𝐶 8… view at source ↗
Figure 9
Figure 9. Figure 9: A cartoon of a blender. Here the two points in black are the interacting periodic points and the red curve is one of the heteroclinics. Theorem 3.25. [16] Let 𝑆Λ be the unit cosphere bundle of a hyperbolic manifold and let Φ : 𝑆Λ Ñ 𝑆Λ be the time-𝑇 map of the (hyperbolic) geodesic flow where 𝑇 is the period of a closed orbit. Then there is a 𝐶 8-small, 𝐶 1 -robustly transitive contact perturbation Ψ contai… view at source ↗

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Works this paper leans on

54 extracted references · 13 linked inside Pith

  1. [1]

    On the weinstein conjecture in higher dimensions.Commentarii Mathematici Helvetici, 84(2):429–436, 2009

    Peter Albers and Helmut W Hofer. On the weinstein conjecture in higher dimensions.Commentarii Mathematici Helvetici, 84(2):429–436, 2009

    2009

  2. [2]

    Ergodic properties of geodesic flows on closed riemannian manifolds of negative curvature

    Dmitry Victorovich Anosov. Ergodic properties of geodesic flows on closed riemannian manifolds of negative curvature. InDoklady Akademii Nauk, volume 151, pages 1250–1252. Russian Academy of Sciences, 1963

    1963

  3. [3]

    A𝐶 8-closing lemma for Hamiltonian diffeomorphisms of closed surfaces

    Masayuki Asaoka and Kei Irie. A𝐶 8-closing lemma for Hamiltonian diffeomorphisms of closed surfaces. Geometric and Functional Analysis, 26(5):1245–1254, 2016

    2016

  4. [4]

    There exist no contact anosov diffeomorphisms.Journal of Fixed Point Theory and Applications, 27(4):1–5, 2025

    Masayuki Asaoka and Yoshihiko Mitsumatsu. There exist no contact anosov diffeomorphisms.Journal of Fixed Point Theory and Applications, 27(4):1–5, 2025

    2025

  5. [5]

    An algebraic generalization of giroux’s criterion.arXiv preprint arXiv:2307.09068, 2023

    Russell Avdek. An algebraic generalization of giroux’s criterion.arXiv preprint arXiv:2307.09068, 2023

    Pith/arXiv arXiv 2023

  6. [6]

    What is

    Christian Bonatti, Sylvain Crovisier, Lorenzo Diaz, and Amie Wilkinson. What is... a blender?arXiv preprint arXiv:1608.02848, 2016

    Pith/arXiv arXiv 2016

  7. [7]

    ChristianBonattiandLorenzoJ.Diaz.Persistentnonhyperbolictransitivediffeomorphisms.AnnalsofMathemat- ics, 143(2):357–396, 1996

    1996

  8. [9]

    Robust heterodimensional cycles and-generic dynamics.Journal of the Institute of Mathematics of Jussieu, 7(3):469–525, 2008

    Christian Bonatti and Lorenzo J Díaz. Robust heterodimensional cycles and-generic dynamics.Journal of the Institute of Mathematics of Jussieu, 7(3):469–525, 2008

    2008

  9. [10]

    A c1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources.Annals of Mathematics, pages 355–418, 2003

    Christian Bonatti, Lorenzo J Díaz, and Enrique R Pujals. A c1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources.Annals of Mathematics, pages 355–418, 2003

    2003

  10. [11]

    Springer Science & Business Media, 2004

    Christian Bonatti, Lorenzo J Díaz, and Marcelo Viana.Dynamics beyond uniform hyperbolicity: A global geometric and probabilistic perspective, volume 3. Springer Science & Business Media, 2004

    2004

  11. [12]

    Existence and classification of overtwisted contact structures in all dimensions.Acta Mathematica, 215(2):281–361, 2015

    Matthew Strom Borman, Yakov Eliashberg, and Emmy Murphy. Existence and classification of overtwisted contact structures in all dimensions.Acta Mathematica, 215(2):281–361, 2015

    2015

  12. [13]

    JosephBreen.Morse-smalecharacteristicfoliationsandconvexityincontactmanifolds.ProceedingsoftheAmerican Mathematical Society, 149(9):3977–3989, 2021

    2021

  13. [14]

    Folded symplectic forms in contact topology.Journal of Geometry and Physics, 201:105213, 2024

    Joseph Breen. Folded symplectic forms in contact topology.Journal of Geometry and Physics, 201:105213, 2024

    2024

  14. [15]

    The giroux correspondence in arbitrary dimensions.arXiv preprint arXiv:2307.02317, 2023

    Joseph Breen, Ko Honda, and Yang Huang. The giroux correspondence in arbitrary dimensions.arXiv preprint arXiv:2307.02317, 2023

    Pith/arXiv arXiv 2023

  15. [16]

    Robustly non-convex hypersurfaces in contact manifolds.arXiv preprint arXiv:2406.05979, 2024

    Julian Chaidez. Robustly non-convex hypersurfaces in contact manifolds.arXiv preprint arXiv:2406.05979, 2024

    arXiv 2024

  16. [17]

    Contact homology and higher dimensional closing lemmas.arXiv preprint arXiv:2206.04738, 2022

    Julian Chaidez, Ipsita Datta, Rohil Prasad, and Shira Tanny. Contact homology and higher dimensional closing lemmas.arXiv preprint arXiv:2206.04738, 2022

    Pith/arXiv arXiv 2022

  17. [18]

    KaiCieliebak,YEliashberg,andNMishachev.Introductiontotheℎ-Principle,volume239.AmericanMathematical Society, 2024

    2024

  18. [19]

    First steps in stable hamiltonian topology.Journal of the European Mathematical Society, 17(2):321–404, 2015

    Kai Cieliebak and Evgeny Volkov. First steps in stable hamiltonian topology.Journal of the European Mathematical Society, 17(2):321–404, 2015

    2015

  19. [20]

    On the coarse classification of tight contact structures.arXiv preprint math/0305186, 2003

    Vincent Colin, Emmanuel Giroux, and Ko Honda. On the coarse classification of tight contact structures.arXiv preprint math/0305186, 2003

    Pith/arXiv arXiv 2003

  20. [21]

    VincentColin,EmmanuelGiroux,andKoHonda.Finitudehomotopiqueetisotopiquedesstructuresdecontact tendues.Publications mathématiques, 109:245–293, 2009

    2009

  21. [22]

    Number 38

    Charles C Conley.Isolated invariant sets and the Morse index. Number 38. American Mathematical Soc., 1978

    1978

  22. [23]

    Classification of tight contact structures on surgeries on the figure-eight knot

    James Conway and Hyunki Min. Classification of tight contact structures on surgeries on the figure-eight knot. Geometry & Topology, 24(3):1457–1517, 2020. CONFORMALLY SYMPLECTIC TOPOLOGY FROM A DYNAMICAL VIEWPOINT 25

    2020

  23. [24]

    DanCristofaro-Gardiner,RohilPrasad,andBoyuZhang.PeriodicFloerhomologyandthesmoothclosinglemma for area-preserving surface diffeomorphisms.arXiv preprint arXiv:2110.02925, 2021

    Pith/arXiv arXiv 2021

  24. [25]

    PFH spectral invariants and𝐶 8 closing lemmas.arXiv preprint arXiv:2110.02463, 2021

    Oliver Edtmair and Michael Hutchings. PFH spectral invariants and𝐶 8 closing lemmas.arXiv preprint arXiv:2110.02463, 2021

    Pith/arXiv arXiv 2021

  25. [26]

    Convex symplectic manifolds.Several complex variables and complex geometry, pages 135–162, 1991

    Yakov Eliashberg and Mikhael Gromov. Convex symplectic manifolds.Several complex variables and complex geometry, pages 135–162, 1991

    1991

  26. [27]

    Honda-huang’s work on contact convexity revisited.arXiv preprint arXiv:2207.07185, 2022

    Yakov Eliashberg and Dishant Pancholi. Honda-huang’s work on contact convexity revisited.arXiv preprint arXiv:2207.07185, 2022

    arXiv 2022

  27. [28]

    On the nonexistence of tight contact structures.Annals of Mathematics, pages 749–766, 2001

    John B Etnyre and Ko Honda. On the nonexistence of tight contact structures.Annals of Mathematics, pages 749–766, 2001

    2001

  28. [29]

    Tight contact structures with no symplectic fillings.Inventiones Mathematicae, 148(3):609, 2002

    John B Etnyre and Ko Honda. Tight contact structures with no symplectic fillings.Inventiones Mathematicae, 148(3):609, 2002

    2002

  29. [30]

    Feral curves and minimal sets.Annals of Mathematics, 197(2):533–738, 2023

    Joel Fish and Helmut Hofer. Feral curves and minimal sets.Annals of Mathematics, 197(2):533–738, 2023

    2023

  30. [31]

    Todd Fisher and Boris Hasselblatt.Hyperbolic flows. 2019

    2019

  31. [32]

    Morse theory for lagrangian intersections.Journal of differential geometry, 28(3):513–547, 1988

    Andreas Floer. Morse theory for lagrangian intersections.Journal of differential geometry, 28(3):513–547, 1988

    1988

  32. [33]

    The unregularized gradient flow of the symplectic action.Communications on Pure and Applied Mathematics, 41(6):775–813, 1988

    Andreas Floer. The unregularized gradient flow of the symplectic action.Communications on Pure and Applied Mathematics, 41(6):775–813, 1988

    1988

  33. [34]

    Transversality in elliptic morse theory for the symplectic action

    Andreas Floer, Helmut Hofer, and Dietmar Salamon. Transversality in elliptic morse theory for the symplectic action. 1995

    1995

  34. [35]

    The weinstein conjecture in pˆc.Mathematische Zeitschrift, 203(1):469–482, 1990

    Andreas Floer, Helmut Hofer, and Claude Viterbo. The weinstein conjecture in pˆc.Mathematische Zeitschrift, 203(1):469–482, 1990

    1990

  35. [36]

    Cambridge University Press, 2008

    Hansjörg Geiges.An introduction to contact topology, volume 109. Cambridge University Press, 2008

    2008

  36. [37]

    The conley conjecture.Annals of mathematics, pages 1127–1180, 2010

    Viktor L Ginzburg. The conley conjecture.Annals of mathematics, pages 1127–1180, 2010

    2010

  37. [38]

    Geometrie de contact de la dimension trois vers les dimensions superior

    E GIROUX. Geometrie de contact de la dimension trois vers les dimensions superior. InProceedings of the ICM, Beijing 2002, volume 2, pages 405–414, 2002

    2002

  38. [39]

    Convexity in contact topology.Comment

    Emmanuel Giroux. Convexity in contact topology.Comment. Math. Helv, 66(4):637–677, 1991

    1991

  39. [40]

    On the classification of tight contact structures i.arXiv preprint math/9910127, 1999

    Ko Honda. On the classification of tight contact structures i.arXiv preprint math/9910127, 1999

    Pith/arXiv arXiv 1999

  40. [41]

    On the classification of tight contact structures ii.Journal of Differential Geometry, 55(1):83–143, 2000

    Ko Honda. On the classification of tight contact structures ii.Journal of Differential Geometry, 55(1):83–143, 2000

    2000

  41. [42]

    Bypass attachments in higher-dimensional contact topology.arXiv preprint arXiv:1803.09142, 2018

    Ko Honda and Yang Huang. Bypass attachments in higher-dimensional contact topology.arXiv preprint arXiv:1803.09142, 2018

    Pith/arXiv arXiv 2018

  42. [43]

    Convex hypersurface theory in contact topology.arXiv preprint arXiv:1907.06025, 2019

    Ko Honda and Yang Huang. Convex hypersurface theory in contact topology.arXiv preprint arXiv:1907.06025, 2019

    Pith/arXiv arXiv 1907

  43. [44]

    The weinstein conjecture for stable hamiltonian structures

    Michael Hutchings and Clifford Henry Taubes. The weinstein conjecture for stable hamiltonian structures. Geometry & Topology, 13(2):901–941, 2009

    2009

  44. [45]

    KeiIrie.DenseexistenceofperiodicreeborbitsandECHspectralinvariants.JournalofModernDynamics,9(01):357, 2015

    2015

  45. [46]

    Contributiona la théorie des champs génériques.Contributions to differential equations, 2:457–484, 1963

    Ivan Kupka. Contributiona la théorie des champs génériques.Contributions to differential equations, 2:457–484, 1963

    1963

  46. [47]

    McDuff and D

    D. McDuff and D. Salamon.Introduction To Symplectic Topology, 3rd Ed. Oxford University Press, 2017

    2017

  47. [48]

    AtsuhideMori.Reebfoliationsonsˆ5andcontact5-manifoldsviolatingthethurston-bennequininequality.arXiv preprint arXiv:0906.3237, 2009

    Pith/arXiv arXiv 2009

  48. [49]

    On the violation of thurston-bennequin inequality for a certain non-convex hypersurface.arXiv preprint arXiv:1111.0383, 2011

    Atsuhide Mori. On the violation of thurston-bennequin inequality for a certain non-convex hypersurface.arXiv preprint arXiv:1111.0383, 2011

    Pith/arXiv arXiv 2011

  49. [50]

    Stanford University, 2012

    Emmy Murphy.Loose Legendrian embeddings in high dimensional contact manifolds. Stanford University, 2012

    2012

  50. [51]

    Getting a handle on contact manifolds.arXiv preprint arXiv:1905.11965, 2019

    Kevin Sackel. Getting a handle on contact manifolds.arXiv preprint arXiv:1905.11965, 2019

    Pith/arXiv arXiv 1905

  51. [52]

    On the hofer-zehnder conjecture.Annals of Mathematics, 195(3):775–839, 2022

    Egor Shelukhin. On the hofer-zehnder conjecture.Annals of Mathematics, 195(3):775–839, 2022

    2022

  52. [53]

    Differentiable dynamical systems.Bulletin of the American mathematical Society, 73(6):747–817, 1967

    Stephen Smale. Differentiable dynamical systems.Bulletin of the American mathematical Society, 73(6):747–817, 1967

    1967

  53. [54]

    The seiberg–witten equations and the weinstein conjecture.Geometry & Topology, 11(4):2117–2202, 2007

    Clifford Henry Taubes. The seiberg–witten equations and the weinstein conjecture.Geometry & Topology, 11(4):2117–2202, 2007

    2007

  54. [55]

    The classification of tight contact structures on the 3-torus.Communications in Analysis and Geometry, 5(3):413–438, 1997

    Kanda Yutaka. The classification of tight contact structures on the 3-torus.Communications in Analysis and Geometry, 5(3):413–438, 1997. Department of Mathematics, University of Southern California, Los Angeles, CA, 90007, USA Email address:julian.chaidez@usc.edu

    1997