February  2004, 10(1&2): 21-30. doi: 10.3934/dcds.2004.10.21

Recent results in contact form geometry


Hill Center for the Mathematical Sciences, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, United States

Received  February 2002 Revised  December 2002 Published  October 2003

After briefly recalling the pseudo-holomorphic approach in contact form geometry and after sketching the ways with which this approach defines invariants, we introduce another approach, of more technical type, which starts with a variational problem on Legendrian curves. We show how this approach leads also to the definition of a homology.
Ideally, this homology would be generated by a part of the Morse complex of the variational problem which would involve only periodic orbits. Because of the lack of compactness, it has some additional part which we had characterized in an earlier work [5].
Taking a variant of this approach, we give here a much more restrictive characterization of this additional part which should allow to compute it precisely.
This should indicate that the lack of compactness, seen as creation of additional punctures in the pseudo-holomorphic approach, is much more limited than what would be theoretically allowed and leaves hope that it can be completely computed. The proof of all our claims will be published in [6].
Citation: Abbas Bahri. Recent results in contact form geometry. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 21-30. doi: 10.3934/dcds.2004.10.21

Jaume Llibre, Jesús S. Pérez del Río, J. Angel Rodríguez. Structural stability of planar semi-homogeneous polynomial vector fields applications to critical points and to infinity. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 809-828. doi: 10.3934/dcds.2000.6.809


Alexander Krasnosel'skii, Jean Mawhin. The index at infinity for some vector fields with oscillating nonlinearities. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 165-174. doi: 10.3934/dcds.2000.6.165


Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247


A.M. Krasnosel'skii, Jean Mawhin. The index at infinity of some twice degenerate compact vector fields. Discrete & Continuous Dynamical Systems, 1995, 1 (2) : 207-216. doi: 10.3934/dcds.1995.1.207


Marc Briane. Isotropic realizability of electric fields around critical points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 353-372. doi: 10.3934/dcdsb.2014.19.353


Isaac A. García, Jaume Giné. Non-algebraic invariant curves for polynomial planar vector fields. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 755-768. doi: 10.3934/dcds.2004.10.755


Xiao-Song Yang. Index sums of isolated singular points of positive vector fields. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 1033-1039. doi: 10.3934/dcds.2009.25.1033


Eszter Fehér, Gábor Domokos, Bernd Krauskopf. Tracking the critical points of curves evolving under planar curvature flows. Journal of Computational Dynamics, 2021, 8 (4) : 447-494. doi: 10.3934/jcd.2021017


Kensuke Yoshizawa. The critical points of the elastic energy among curves pinned at endpoints. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 403-423. doi: 10.3934/dcds.2021122


Ferruh Özbudak, Burcu Gülmez Temür, Oǧuz Yayla. Further results on fibre products of Kummer covers and curves with many points over finite fields. Advances in Mathematics of Communications, 2016, 10 (1) : 151-162. doi: 10.3934/amc.2016.10.151


Isaac A. García, Jaume Giné, Susanna Maza. Linearization of smooth planar vector fields around singular points via commuting flows. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1415-1428. doi: 10.3934/cpaa.2008.7.1415


Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481


Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623


Valentina Casarino, Paolo Ciatti, Silvia Secco. Product structures and fractional integration along curves in the space. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 619-635. doi: 10.3934/dcdss.2013.6.619


Enrique R. Pujals, Federico Rodriguez Hertz. Critical points for surface diffeomorphisms. Journal of Modern Dynamics, 2007, 1 (4) : 615-648. doi: 10.3934/jmd.2007.1.615


Keith Promislow, Hang Zhang. Critical points of functionalized Lagrangians. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1231-1246. doi: 10.3934/dcds.2013.33.1231


Marcelo R. R. Alves. Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds. Journal of Modern Dynamics, 2016, 10: 497-509. doi: 10.3934/jmd.2016.10.497


BronisŁaw Jakubczyk, Wojciech Kryński. Vector fields with distributions and invariants of ODEs. Journal of Geometric Mechanics, 2013, 5 (1) : 85-129. doi: 10.3934/jgm.2013.5.85


Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4943-4957. doi: 10.3934/dcds.2021063


Nazar Arakelian, Saeed Tafazolian, Fernando Torres. On the spectrum for the genera of maximal curves over small fields. Advances in Mathematics of Communications, 2018, 12 (1) : 143-149. doi: 10.3934/amc.2018009

2020 Impact Factor: 1.392


  • PDF downloads (73)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]