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 80 tok/s
Gemini 2.5 Pro 28 tok/s Pro
GPT-5 Medium 32 tok/s Pro
GPT-5 High 38 tok/s Pro
GPT-4o 125 tok/s Pro
Kimi K2 181 tok/s Pro
GPT OSS 120B 462 tok/s Pro
Claude Sonnet 4.5 35 tok/s Pro
2000 character limit reached

C^0-stability of topological entropy for Reeb flows in dimension 3 (2311.12001v2)

Published 20 Nov 2023 in math.DS and math.SG

Abstract: We study stability properties of the topological entropy of Reeb flows on contact 3-manifolds with respect to the C0-distance on the space of contact forms. Our main results show that a C\infty-generic contact form on a closed co-oriented contact 3-manifold (Y,\xi) is a lower semi-continuity point for the topological entropy, seen as a functional on the space of contact forms of (Y,\xi) endowed with the C0-distance. We also study the stability of the topological entropy of geodesic flows of Riemannian metrics on closed surfaces. In this setting, we show that a non-degenerate Riemannian metric on a closed surface S is a lower semi-continuity point of the topological entropy, seen as a functional on the space of Riemannian metrics on S endowed with the C0-distance.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (48)
  1. Casim Abbas. Finite energy surfaces and the chord problem. Duke Math. J., 96(2):241–316, 1999.
  2. Entropy collapse versus entropy rigidity for Reeb and Finsler flows. Selecta Math. (N.S.), 29(5):Paper No. 67, 2023.
  3. Marcelo R. R. Alves. Cylindrical contact homology and topological entropy. Geom. Topol., 20(6):3519–3569, 2016.
  4. Marcelo R. R. Alves. Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds. J. Mod. Dyn., 10:497–509, 2016.
  5. Marcelo R. R. Alves. Legendrian contact homology and topological entropy. J. Topol. Anal., 11(1):53–108, 2019.
  6. Topological entropy for Reeb vector fields in dimension three via open book decompositions. J. Éc. polytech. Math., 6:119–148, 2019.
  7. C0superscript𝐶0{C}^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT-robustness of topological entropy for geodesic flows. J. Fixed Point Theory Appl., 24(2):42, 2022.
  8. Marcelo R. R. Alves and Matthias Meiwes. Dynamically exotic contact spheres of dimensions ≥7absent7\geq 7≥ 7. Comment. Math. Helv., 94(3):569–622, 2019.
  9. Marcelo R. R. Alves and Matthias Meiwes. Braid stability and the Hofer metric. Ann. Henri Lebesgue, page arXiv:2112.11351 - accepted for publication, 2021.
  10. Marcelo R. R. Alves and Abror Pirnapasov. Reeb orbits that force topological entropy. Ergodic Theory Dynam. Systems, 42(10):3025–3068, 2022.
  11. Frédéric Bourgeois. Contact homology and homotopy groups of the space of contact structures. Math. Res. Lett., 13(1):71–85, 2006.
  12. Compactness results in symplectic field theory. Geom. Topol., 7:799–888, 2003.
  13. Entropy and quasimorphisms. J. Mod. Dyn., 15:143–163, 2019.
  14. Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective. Preprint arXiv:2111.03983, 2021.
  15. Hofer’s geometry and topological entropy. Compos. Math., 159(6):1250–1299, 2023.
  16. Generic properties of 3333-dimensional Reeb flows: Birkhoff sections and entropy. Preprint arXiv:2202.01506, 2022.
  17. Surfaces of section for geodesic flows of closed surfaces. Preprint arXiv:2204.11977, 2022.
  18. Existence of Birkhoff sections for Kupka–Smale Reeb flows of closed contact 3-manifolds. Geom. Funct. Anal., 32(5):951–979, 2022.
  19. Lucas Dahinden. Lower complexity bounds for positive contactomorphisms. Israel J. Math., 224(1):367–383, 2018.
  20. Lucas Dahinden. Positive topological entropy of positive contactomorphisms. J. Symplectic Geom., 18(3):691–732, 2020.
  21. Lucas Dahinden. C0superscript𝐶0{C^{0}}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT-stability of topological entropy for contactomorphisms. Commun. Contemp. Math., 23(06):2150015, 2021.
  22. Dragomir L. Dragnev. Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations. Comm. Pure Appl. Math., 57(6):726–763, 2004.
  23. Barcode entropy for Reeb flows on contact manifolds with Liouville fillings. Preprint arXiv:2305.04770, 2023.
  24. Quantitative conditions for right-handedness of flows. Preprint arXiv:2106.12512, 2021.
  25. Fiberwise volume growth via Lagrangian intersections. J. Symplectic Geom., 4(2):117–148, 2006.
  26. Hansjörg Geiges. An introduction to contact topology, volume 109 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2008.
  27. Étienne Ghys. Right-handed vector fields & the Lorenz attractor. Jpn. J. Math., 4(1):47–61, 2009.
  28. Barcode entropy of geodesic flows. Preprint arXiv:2212.00943, 2022.
  29. Helmut Hofer. Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. Invent. Math., 114(3):515–563, 1993.
  30. Floer homology and Novikov rings. In The Floer memorial volume, volume 133 of Progr. Math., pages 483–524. Birkhäuser, Basel, 1995.
  31. Properties of pseudo-holomorphic curves in symplectisations. II. Embedding controls and algebraic invariants. Geom. Funct. Anal., 5(2):270–328, 1995.
  32. Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics. Ann. Inst. H. Poincaré Anal. Non Linéaire, 13(3):337–379, 1996.
  33. The dynamics on three-dimensional strictly convex energy surfaces. Ann. of Math. (2), 148(1):197–289, 1998.
  34. A Poincaré-Birkhoff theorem for tight Reeb flows on S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. Invent. Math., pages 1–90, 2014.
  35. Michael Hutchings. Braid stability for periodic orbits of area-preserving surface diffeomorphisms. Preprint arXiv:2303.07133, 2023.
  36. Introduction to the modern theory of dynamical systems, volume 54. Cambridge University Press, 1997.
  37. Michael Khanevsky. Non-autonomous curves on surfaces. J. Mod. Dyn., 17:305–317, 2021.
  38. Ricardo Mañé. On the topological entropy of geodesic flows. J. Differential Geom., 45(1):74–93, 1997.
  39. Positive topological entropy of Reeb flows on spherizations. Math. Proc. Cambridge Philos. Soc., 151(1):103–128, 2011.
  40. Matthias Meiwes. Rabinowitz Floer homology, leafwise intersections, and topological entropy. PhD thesis, 2018.
  41. Matthias Meiwes. Topological entropy and orbit growth in link complements. Preprint, arXiv:2308.06047, 2023.
  42. Al Momin. Contact homology of orbit complements and implied existence. J. Mod. Dyn., 5:409–472, 2011.
  43. Sheldon E. Newhouse. Continuity properties of entropy. Ann. of Math. (2), 129(2):215–235, 1989.
  44. Richard Siefring. Relative asymptotic behavior of pseudoholomorphic half-cylinders. Comm. Pure Appl. Math., 61(12):1631–1684, 2008.
  45. Richard Siefring. Intersection theory of punctured pseudoholomorphic curves. Geom. Topol., 15(4):2351–2457, 2011.
  46. Persistence modules, symplectic Banach-Mazur distance and Riemannian metrics. Internat. J. Math., 32(07):2150040, 2021.
  47. Michael Usher. Symplectic Banach-Mazur distances between subsets of ℂnsuperscriptℂ𝑛\mathbb{C}^{n}blackboard_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. J. Topol. Anal., 14(01):231–286, 2022.
  48. Chris Wendl. Lectures on symplectic field theory. arXiv:1612.01009, 2016.
Citations (3)
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 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