May  2011, 30(2): 379-426. doi: 10.3934/dcds.2011.30.379

Attaching maps in the standard geodesics problem on $S^2$

1. 

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

Received  March 2010 Revised  April 2010 Published  February 2011

Unstable manifolds of critical points at infinity in the variational problems relating to periodic orbits of Reeb vector-fields in Contact Form Geometry are viewed in this paper as part of the attaching maps along which these variational problems attach themselves to natural generalizations that they have. The specific periodic orbit problem for the Reeb vector-field $\xi_0$ of the standard contact structure/form of $S^3$ is studied; the extended variational problem is the closed geodesics problem on $S^2$. The attaching maps are studied for low-dimensional (at most $4$) cells. Some circle and ''loop" actions on the loop space of $S^3$, that are lifts (via Hopf-fibration map) of the standard $S^1$-action on the free loop space of $S^2$, are also defined. ''Conjugacy" relations relating these actions are established.
Citation: Abbas Bahri. Attaching maps in the standard geodesics problem on $S^2$. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 379-426. doi: 10.3934/dcds.2011.30.379
References:
[1]

A. Bahri, "Pseudo-Orbits of Contact Forms,", Pitman Research Notes in Mathematics Series No. 173, (1988). Google Scholar

[2]

A. Bahri, "Flow-lines and Algebraic invariants in Contact Form Geometry PNLDE,", "Flow-lines and Algebraic invariants in Contact Form Geometry PNLDE,", 53 (2003). Google Scholar

[3]

A. Bahri, Compactness,, Advanced Nonlinear Stud., 8 (2008), 465. Google Scholar

[4]

A. Bahri, Topological remarks-critical points at infinity and string theory,, Advanced Nonlinear Studies, 9 (2009), 499. Google Scholar

[5]

M. Chas and D. Sullivan, String topology,, preprint, 1 (1999). Google Scholar

[6]

Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet's work,, Ann. Inst. Fourier, 42 (1992), 165. Google Scholar

[7]

H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three,, Inventiones Mathematicae, 114 (1993), 515. doi: doi:10.1007/BF01232679. Google Scholar

[8]

W. Klingenberg, Closed geodesics on surfaces of genus 0,, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 6 (1979), 19. Google Scholar

[9]

L. Menichi, String topology for spheres,, Comment. Math. Helv, 84 (2009), 135. doi: doi:10.4171/CMH/155. Google Scholar

[10]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, Comm. Pure. Appl. Math., 31 (1978), 157. doi: doi:10.1002/cpa.3160310203. Google Scholar

[11]

D. Sullivan, Infinitesimal computations in topology,, I.H.E.S., 47 (1977), 269. Google Scholar

[12]

C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture,, Geom. Topol., 11 (2007), 2117. doi: doi:10.2140/gt.2007.11.2117. Google Scholar

[13]

A. S. Zvarc, Homology of the space of closed curves,, Trudy Moskov, 9 (1960), 3. Google Scholar

show all references

References:
[1]

A. Bahri, "Pseudo-Orbits of Contact Forms,", Pitman Research Notes in Mathematics Series No. 173, (1988). Google Scholar

[2]

A. Bahri, "Flow-lines and Algebraic invariants in Contact Form Geometry PNLDE,", "Flow-lines and Algebraic invariants in Contact Form Geometry PNLDE,", 53 (2003). Google Scholar

[3]

A. Bahri, Compactness,, Advanced Nonlinear Stud., 8 (2008), 465. Google Scholar

[4]

A. Bahri, Topological remarks-critical points at infinity and string theory,, Advanced Nonlinear Studies, 9 (2009), 499. Google Scholar

[5]

M. Chas and D. Sullivan, String topology,, preprint, 1 (1999). Google Scholar

[6]

Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet's work,, Ann. Inst. Fourier, 42 (1992), 165. Google Scholar

[7]

H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three,, Inventiones Mathematicae, 114 (1993), 515. doi: doi:10.1007/BF01232679. Google Scholar

[8]

W. Klingenberg, Closed geodesics on surfaces of genus 0,, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 6 (1979), 19. Google Scholar

[9]

L. Menichi, String topology for spheres,, Comment. Math. Helv, 84 (2009), 135. doi: doi:10.4171/CMH/155. Google Scholar

[10]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, Comm. Pure. Appl. Math., 31 (1978), 157. doi: doi:10.1002/cpa.3160310203. Google Scholar

[11]

D. Sullivan, Infinitesimal computations in topology,, I.H.E.S., 47 (1977), 269. Google Scholar

[12]

C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture,, Geom. Topol., 11 (2007), 2117. doi: doi:10.2140/gt.2007.11.2117. Google Scholar

[13]

A. S. Zvarc, Homology of the space of closed curves,, Trudy Moskov, 9 (1960), 3. Google Scholar

[1]

Abbas Bahri. Recent results in contact form geometry. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 21-30. doi: 10.3934/dcds.2004.10.21

[2]

Fengjie Geng, Junfang Zhao, Deming Zhu, Weipeng Zhang. Bifurcations of a nongeneric heteroclinic loop with nonhyperbolic equilibria. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 133-145. doi: 10.3934/dcdsb.2013.18.133

[3]

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 - A, 2000, 6 (4) : 809-828. doi: 10.3934/dcds.2000.6.809

[4]

K. H. Kim, F. W. Roush and J. B. Wagoner. Inert actions on periodic points. Electronic Research Announcements, 1997, 3: 55-62.

[5]

John Franks, Michael Handel, Kamlesh Parwani. Fixed points of Abelian actions. Journal of Modern Dynamics, 2007, 1 (3) : 443-464. doi: 10.3934/jmd.2007.1.443

[6]

Yulin Zhao, Siming Zhu. Higher order Melnikov function for a quartic hamiltonian with cuspidal loop. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 995-1018. doi: 10.3934/dcds.2002.8.995

[7]

Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153

[8]

Justine Yasappan, Ángela Jiménez-Casas, Mario Castro. Stabilizing interplay between thermodiffusion and viscoelasticity in a closed-loop thermosyphon. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3267-3299. doi: 10.3934/dcdsb.2015.20.3267

[9]

Alain Bensoussan, Shaokuan Chen, Suresh P. Sethi. Linear quadratic differential games with mixed leadership: The open-loop solution. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 95-108. doi: 10.3934/naco.2013.3.95

[10]

Alexander Blokh, Michał Misiurewicz. Dense set of negative Schwarzian maps whose critical points have minimal limit sets. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 141-158. doi: 10.3934/dcds.1998.4.141

[11]

Rafael De La Llave, Michael Shub, Carles Simó. Entropy estimates for a family of expanding maps of the circle. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 597-608. doi: 10.3934/dcdsb.2008.10.597

[12]

Liviana Palmisano. Unbounded regime for circle maps with a flat interval. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2099-2122. doi: 10.3934/dcds.2015.35.2099

[13]

Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098

[14]

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

[15]

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

[16]

Xiaochen Sun, Fei Hu, Yancong Zhou, Cheng-Chew Lim. Optimal acquisition, inventory and production decisions for a closed-loop manufacturing system with legislation constraint. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1355-1373. doi: 10.3934/jimo.2015.11.1355

[17]

Monica Lazzo, Paul G. Schmidt. Convergence versus periodicity in a single-loop positive-feedback system 2. Periodic solutions. Conference Publications, 2011, 2011 (Special) : 941-952. doi: 10.3934/proc.2011.2011.941

[18]

Monica Lazzo, Paul G. Schmidt. Convergence versus periodicity in a single-loop positive-feedback system 1. Convergence to equilibrium. Conference Publications, 2011, 2011 (Special) : 931-940. doi: 10.3934/proc.2011.2011.931

[19]

Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803

[20]

Yi Jing, Wenchuan Li. Integrated recycling-integrated production - distribution planning for decentralized closed-loop supply chain. Journal of Industrial & Management Optimization, 2018, 14 (2) : 511-539. doi: 10.3934/jimo.2017058

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]