Papers
Topics
Authors
Recent
Assistant
AI Research Assistant
Well-researched responses based on relevant abstracts and paper content.
Custom Instructions Pro
Preferences or requirements that you'd like Emergent Mind to consider when generating responses.
Gemini 2.5 Flash
Gemini 2.5 Flash 71 tok/s
Gemini 2.5 Pro 48 tok/s Pro
GPT-5 Medium 12 tok/s Pro
GPT-5 High 21 tok/s Pro
GPT-4o 81 tok/s Pro
Kimi K2 231 tok/s Pro
GPT OSS 120B 435 tok/s Pro
Claude Sonnet 4 33 tok/s Pro
2000 character limit reached

Low-action holomorphic curves and invariant sets (2401.14445v2)

Published 25 Jan 2024 in math.SG and math.DS

Abstract: We prove a compactness theorem for sequences of low-action punctured holomorphic curves of controlled topology, in any dimension, without imposing the typical assumption of uniformly bounded Hofer energy. In the limit, we extract a family of closed Reeb-invariant subsets. Then, we prove new structural results for the U-map in ECH and PFH, implying that such sequences exist in abundance in low-dimensional symplectic dynamics. We obtain applications to symplectic dynamics and the geometry of surfaces. First, we prove generalizations to higher genus surfaces and three-manifolds of the celebrated Le Calvez-Yoccoz theorem. Second, we show that for any closed Riemannian or Finsler surface a dense set of points have geodesics passing through them that visit different sections of the surface. Third, we prove a version of Ginzburg-G\"urel's "crossing energy bound" for punctured holomorphic curves, of arbitrary topology, in symplectizations of any dimension.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (51)
  1. S. Allais. On the Hofer-Zehnder conjecture on ℂ⁢PdℂsuperscriptP𝑑\mathbb{C}{\rm P}^{d}blackboard_C roman_P start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT via generating functions. Internat. J. Math., 33(10-11):Paper No. 2250072, 58, 2022.
  2. New examples in smooth ergodic theory. Ergodic diffeomorphisms. Trudy Moskov. Mat. Obšč., 23:3–36, 1970.
  3. M. Asaoka and K. Irie. A C∞superscript𝐶C^{\infty}italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT closing lemma for Hamiltonian diffeomorphisms of closed surfaces. Geom. Funct. Anal., 26(5):1245–1254, 2016.
  4. M. Batoréo. On hyperbolic points and periodic orbits of symplectomorphisms. J. Lond. Math. Soc. (2), 91(1):249–265, 2015.
  5. I. Bendixson. Sur les courbes définies par des équations différentielles. Acta Math., 24(1):1–88, 1901.
  6. B. Bramham. Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves. Ann. of Math. (2), 181(3):1033–1086, 2015.
  7. J. Chaidez and O. Edtmair. 3D convex contact forms and the Ruelle invariant. Invent. Math., 229(1):243–301, 2022.
  8. Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective. arXiv preprint arXiv:2111.03983, 2021.
  9. Invariant sets and hyperbolic closed Reeb orbits. arXiv preprint arXiv:2309.04576, 2023.
  10. On the barcode entropy of Reeb flows. arXiv preprint arXiv:2401.01421, 2024.
  11. C. C. Conley. Some abstract properties of the set of invariant sets of a flow. Illinois J. Math., 16:663–668, 1972.
  12. G. Contreras and M. Mazzucchelli. Proof of the C2superscript𝐶2C^{2}italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-stability conjecture for geodesic flows of closed surfaces. To appear in Duke Math. J., arXiv preprint arXiv:2109.10704.
  13. D. Cristofaro-Gardiner and R. Hind. Boundaries of open symplectic manifolds and the failure of packing stability. arXiv preprint arXiv:2307.01140v2, 2023.
  14. Proof of the simplicity conjecture. Ann. of Math. (2), 199:181–257, 2024.
  15. Proof of Hofer–Wysocki–Zehnder’s two or infinity conjecture. arXiv preprint arXiv:2310.07636, 2023.
  16. Torsion contact forms in three dimensions have two or infinitely many Reeb orbits. Geom. Topol., 23(7):3601–3645, 2019.
  17. The asymptotics of ECH capacities. Invent. Math., 199(1):187–214, 2015.
  18. A note on the existence of U-cyclic elements in periodic Floer homology. arXiv preprint arXiv:2110.13844, 2021.
  19. Periodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms. arXiv preprint arXiv:2110.02925, 2021.
  20. A. Denjoy. Sur les courbes definies par les équations différentielles à la surface du tore. J. Math. Pures Appl. (9), 11:333–375, 1932.
  21. Finsler geodesics, periodic Reeb orbits, and open books. Eur. J. Math., 3(4):1058–1075, 2017.
  22. O. Edtmair and M. Hutchings. PFH spectral invariants and C∞superscript𝐶C^{\infty}italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT closing lemmas. arXiv preprint arXiv:2110.02463, 2021.
  23. J. W. Fish. Target-local Gromov compactness. Geom. Topol., 15(2):765–826, 2011.
  24. J. W. Fish and H. Hofer. Feral curves and minimal sets. Ann. of Math. (2), 197(2):533–738, 2023.
  25. T. Fisher and B. Hasselblatt. Hyperbolic flows. Zurich Lectures in Advanced Mathematics. EMS Publishing House, Berlin, [2019] ©2019.
  26. J. Franks. The Conley index and non-existence of minimal homeomorphisms. In Proceedings of the Conference on Probability, Ergodic Theory, and Analysis (Evanston, IL, 1997), volume 43, pages 457–464, 1999.
  27. Hyperbolic fixed points and periodic orbits of Hamiltonian diffeomorphisms. Duke Math. J., 163(3):565–590, 2014.
  28. Non-contractible periodic orbits in Hamiltonian dynamics on closed symplectic manifolds. Compos. Math., 152(9):1777–1799, 2016.
  29. Hamiltonian pseudo-rotations of projective spaces. Invent. Math., 214(3):1081–1130, 2018.
  30. Barcode entropy of geodesic flows. To appear in J. Eur. Math. Soc., arXiv preprint arXiv:2212.00943.
  31. M. Handel. There are no minimal homeomorphisms of the multipunctured plane. Ergodic Theory Dynam. Systems, 12(1):75–83, 1992.
  32. S. Hozoori. Dynamics and topology of conformally Anosov contact 3-manifolds. Differential Geometry and its Applications, 73:101679, 2020.
  33. M. Hutchings. An index inequality for embedded pseudoholomorphic curves in symplectizations. J. Eur. Math. Soc. (JEMS), 4(4):313–361, 2002.
  34. M. Hutchings. The embedded contact homology index revisited. In New perspectives and challenges in symplectic field theory, volume 49 of CRM Proc. Lecture Notes, pages 263–297. Amer. Math. Soc., Providence, RI, 2009.
  35. M. Hutchings. Lecture notes on embedded contact homology. In Contact and symplectic topology, volume 26 of Bolyai Soc. Math. Stud., pages 389–484. János Bolyai Math. Soc., Budapest, 2014.
  36. M. Hutchings and M. Sullivan. The periodic Floer homology of a Dehn twist. Algebr. Geom. Topol., 5:301–354, 2005.
  37. M. Hutchings and C. H. Taubes. Gluing pseudoholomorphic curves along branched covered cylinders. I. J. Symplectic Geom., 5(1):43–137, 2007.
  38. M. Hutchings and C. H. Taubes. Gluing pseudoholomorphic curves along branched covered cylinders. II. J. Symplectic Geom., 7(1):29–133, 2009.
  39. M. Hutchings and C. H. Taubes. The Weinstein conjecture for stable Hamiltonian structures. Geom. Topol., 13(2):901–941, 2009.
  40. K. Irie. Dense existence of periodic Reeb orbits and ECH spectral invariants. J. Mod. Dyn., 9:357–363, 2015.
  41. P. Kronheimer and T. Mrowka. Monopoles and three-manifolds, volume 10 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2007.
  42. P. Le Calvez and J.-C. Yoccoz. Un théorème d’indice pour les homéomorphismes du plan au voisinage d’un point fixe. Ann. of Math. (2), 146(2):241–293, 1997.
  43. R. D. Mauldin, editor. The Scottish Book. Birkhäuser/Springer, Cham, second edition, 2015. Mathematics from the Scottish Café with selected problems from the new Scottish Book, Including selected papers presented at the Scottish Book Conference held at North Texas University, Denton, TX, May 1979.
  44. C. T. McMullen. Renormalization and 3-manifolds which fiber over the circle, volume 142 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1996.
  45. M. Meiwes. On the barcode entropy of Lagrangian submanifolds. arXiv preprint arXiv:2401.07034, 2024.
  46. R. Prasad. High-dimensional families of holomorphic curves and the dynamics on three-dimensional energy surfaces. In preparation.
  47. R. Prasad. Invariant probability measures from pseudoholomorphic curves I. J. Mod. Dyn., 19:31–74, 2023.
  48. R. Prasad. Invariant probability measures from pseudoholomorphic curves II: Pseudoholomorphic curve constructions. J. Mod. Dyn., 19:75–160, 2023.
  49. J. M. Salazar. Instability property of homeomorphisms on surfaces. Ergodic Theory Dynam. Systems, 26(2):539–549, 2006.
  50. P. Seidel. Symplectic Floer homology and the mapping class group. Pacific J. Math., 206(1):219–229, 2002.
  51. C. H. Taubes. Embedded contact homology and Seiberg-Witten Floer cohomology I. Geom. Topol., 14(5):2497–2581, 2010.
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
Lightbulb On Streamline Icon: https://streamlinehq.com

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now
List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets

This paper has been mentioned in 1 post and received 3 likes.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up to View