October  2012, 32(10): 3587-3620. doi: 10.3934/dcds.2012.32.3587

Transport, flux and growth of homoclinic Floer homology

1. 

Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, United States

Received  February 2011 Revised  April 2012 Published  May 2012

We point out an interesting relation between transport in Hamiltonian dynamics and Floer homology. We generalize homoclinic Floer homology from $\mathbb{R}^2$ and closed surfaces to two-dimensional cylinders. The relative symplectic action of two homoclinic points is identified with the flux through a turnstile (as defined in MacKay & Meiss & Percival [19]) and Mather's [20] difference in action $\Delta W$. The Floer boundary operator is shown to annihilate turnstiles and we prove that the rank of certain filtered homology groups and the flux grow linearly with the number of iterations of the underlying symplectomorphism.
Citation: Sonja Hohloch. Transport, flux and growth of homoclinic Floer homology. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3587-3620. doi: 10.3934/dcds.2012.32.3587
References:
[1]

S. Aubry, P. Le Daeron and G. André, Classical ground-states of one-dimensional models for incommensurate structures,, unpublished preprint, (1982). Google Scholar

[2]

G. D. Birkhoff, Nouvelles recherches sur les systèmes dynamiques,, Mem. Pont. Acad. Sci. Nov. Lyncaei, 53 (1935), 85. Google Scholar

[3]

Y. Chekanov, Differential algebra of Legendrian links,, Invent. Math., 150 (2002), 441. doi: 10.1007/s002220200212. Google Scholar

[4]

C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol'd,, Invent. Math., 73 (1983), 33. doi: 10.1007/BF01393824. Google Scholar

[5]

S. de Silva, "Products in the Symplectic Floer Homology of Lagrangian Intersections,", Thesis, (1998). Google Scholar

[6]

A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls,, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649. doi: 10.1016/S0764-4442(97)84777-5. Google Scholar

[7]

A. Fathi, Orbites hétéroclines et ensemble de Peierls,, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1213. doi: 10.1016/S0764-4442(98)80230-9. Google Scholar

[8]

A. Floer, A relative Morse index for the symplectic action,, Comm. Pure Appl. Math., 41 (1988), 393. doi: 10.1002/cpa.3160410402. Google Scholar

[9]

A. Floer, The unregularized gradient flow of the symplectic action,, Comm. Pure Appl. Math., 41 (1988), 775. doi: 10.1002/cpa.3160410603. Google Scholar

[10]

A. Floer, Morse theory for Lagrangian intersections,, J. Diff. Geom., 28 (1988), 513. Google Scholar

[11]

R. Gautschi, J. Robbin and D. Salamon, Heegard splittings and Morse-Smale flows,, Int. J. Math. Math. Sci., 2003 (2003), 3539. Google Scholar

[12]

V. Gelfreich, A proof of the exponentially small transversality of the separatrices for the standard map,, Comm. Math. Phys., 201 (1999), 155. doi: 10.1007/s002200050553. Google Scholar

[13]

V. Gelfreich and C. Simó, High-precision computations of divergent asymptotic series and homoclinic phenomena,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 511. doi: 10.3934/dcdsb.2008.10.511. Google Scholar

[14]

V. Ginzburg, The Conley conjecture,, Ann. of Math. (2), 172 (2010), 1127. doi: 10.4007/annals.2010.172.1129. Google Scholar

[15]

V. Ginzburg and B. Gürel, Action and index spectra and periodic orbits in Hamiltonian dynamics,, Geometry & Topology, 13 (2009), 2745. doi: 10.2140/gt.2009.13.2745. Google Scholar

[16]

S. Hohloch, Homoclinic points and Floer homology,, preprint., (). Google Scholar

[17]

S. Hohloch, Floer homology and homoclinic dynamics,, preprint., (). Google Scholar

[18]

V. Lazutkin, Splitting of separatrices for the Chirikov standard map,, Translated from the Russian and with a preface by V. Gelfreich. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 300 (2003), 25. Google Scholar

[19]

R. MacKay, J. Meiss and I. Percival, Transport in Hamiltonian systems,, Physica D, 13 (1984), 55. doi: 10.1016/0167-2789(84)90270-7. Google Scholar

[20]

J. Mather, A criterion for the nonexistence of invariant circles,, Inst. Hautes Études Sci. Publ. Math., 63 (1986), 153. doi: 10.1007/BF02831625. Google Scholar

[21]

J. Mather, Modulus of continuity for Peierls's barrier,, in, 209 (1987), 177. Google Scholar

[22]

D. McDuff and D. Salamon, "Introduction to Symplectic Topology,", Second edition, (1998). Google Scholar

[23]

J. Palis, On Morse-Smale dynamical systems,, Topology, 8 (1969), 385. Google Scholar

[24]

H. Poincaré, Sur le problème des trois corps et les équations de la dynamique,, Acta Mathematica, 13 (1890), 1. Google Scholar

[25]

H. Poincaré, Les méthodes nouvelles de la méchanique céleste,, Gauthier-Villars et fils, (1899). Google Scholar

[26]

L. Polterovich, On transport in dynamical systems,, (Russian), 43 (1988), 207. Google Scholar

[27]

L. Polterovich, "The Geometry of the Group of Symplectic Diffeomorphism,", Lectures in Mathematics ETH Zürich, (2001). Google Scholar

[28]

L. Polterovich, Growth of maps, distortion of groups and symplectic geometry,, Inv. Math., 150 (2002), 655. doi: 10.1007/s00222-002-0251-x. Google Scholar

[29]

L. Polterovich, Floer homology, dynamics and groups,, in, 217 (2006), 417. Google Scholar

[30]

J. Robbin, Heegard splittings and Floer homology,, preprint, (2000). Google Scholar

[31]

V. Rom-Kedar, Homoclinic tangles-classification and applications,, Nonlinearity, 7 (1994), 441. Google Scholar

[32]

V. Rom-Kedar, Secondary homoclinic bifurcation theorems,, Chaos, 5 (1995), 385. Google Scholar

[33]

D. Salamon, Lectures on Floer homology,, in, 7 (1999), 143. Google Scholar

[34]

M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds,, Pacific J. of Math., 193 (2000), 419. doi: 10.2140/pjm.2000.193.419. Google Scholar

[35]

S. Smale, A structurally stable differentiable homeomorphism with an infinite number of periodic points,, in, (1963), 365. Google Scholar

[36]

S. Smale, Diffeomorphisms with many periodic points,, in, (1965), 63. Google Scholar

show all references

References:
[1]

S. Aubry, P. Le Daeron and G. André, Classical ground-states of one-dimensional models for incommensurate structures,, unpublished preprint, (1982). Google Scholar

[2]

G. D. Birkhoff, Nouvelles recherches sur les systèmes dynamiques,, Mem. Pont. Acad. Sci. Nov. Lyncaei, 53 (1935), 85. Google Scholar

[3]

Y. Chekanov, Differential algebra of Legendrian links,, Invent. Math., 150 (2002), 441. doi: 10.1007/s002220200212. Google Scholar

[4]

C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol'd,, Invent. Math., 73 (1983), 33. doi: 10.1007/BF01393824. Google Scholar

[5]

S. de Silva, "Products in the Symplectic Floer Homology of Lagrangian Intersections,", Thesis, (1998). Google Scholar

[6]

A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls,, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649. doi: 10.1016/S0764-4442(97)84777-5. Google Scholar

[7]

A. Fathi, Orbites hétéroclines et ensemble de Peierls,, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1213. doi: 10.1016/S0764-4442(98)80230-9. Google Scholar

[8]

A. Floer, A relative Morse index for the symplectic action,, Comm. Pure Appl. Math., 41 (1988), 393. doi: 10.1002/cpa.3160410402. Google Scholar

[9]

A. Floer, The unregularized gradient flow of the symplectic action,, Comm. Pure Appl. Math., 41 (1988), 775. doi: 10.1002/cpa.3160410603. Google Scholar

[10]

A. Floer, Morse theory for Lagrangian intersections,, J. Diff. Geom., 28 (1988), 513. Google Scholar

[11]

R. Gautschi, J. Robbin and D. Salamon, Heegard splittings and Morse-Smale flows,, Int. J. Math. Math. Sci., 2003 (2003), 3539. Google Scholar

[12]

V. Gelfreich, A proof of the exponentially small transversality of the separatrices for the standard map,, Comm. Math. Phys., 201 (1999), 155. doi: 10.1007/s002200050553. Google Scholar

[13]

V. Gelfreich and C. Simó, High-precision computations of divergent asymptotic series and homoclinic phenomena,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 511. doi: 10.3934/dcdsb.2008.10.511. Google Scholar

[14]

V. Ginzburg, The Conley conjecture,, Ann. of Math. (2), 172 (2010), 1127. doi: 10.4007/annals.2010.172.1129. Google Scholar

[15]

V. Ginzburg and B. Gürel, Action and index spectra and periodic orbits in Hamiltonian dynamics,, Geometry & Topology, 13 (2009), 2745. doi: 10.2140/gt.2009.13.2745. Google Scholar

[16]

S. Hohloch, Homoclinic points and Floer homology,, preprint., (). Google Scholar

[17]

S. Hohloch, Floer homology and homoclinic dynamics,, preprint., (). Google Scholar

[18]

V. Lazutkin, Splitting of separatrices for the Chirikov standard map,, Translated from the Russian and with a preface by V. Gelfreich. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 300 (2003), 25. Google Scholar

[19]

R. MacKay, J. Meiss and I. Percival, Transport in Hamiltonian systems,, Physica D, 13 (1984), 55. doi: 10.1016/0167-2789(84)90270-7. Google Scholar

[20]

J. Mather, A criterion for the nonexistence of invariant circles,, Inst. Hautes Études Sci. Publ. Math., 63 (1986), 153. doi: 10.1007/BF02831625. Google Scholar

[21]

J. Mather, Modulus of continuity for Peierls's barrier,, in, 209 (1987), 177. Google Scholar

[22]

D. McDuff and D. Salamon, "Introduction to Symplectic Topology,", Second edition, (1998). Google Scholar

[23]

J. Palis, On Morse-Smale dynamical systems,, Topology, 8 (1969), 385. Google Scholar

[24]

H. Poincaré, Sur le problème des trois corps et les équations de la dynamique,, Acta Mathematica, 13 (1890), 1. Google Scholar

[25]

H. Poincaré, Les méthodes nouvelles de la méchanique céleste,, Gauthier-Villars et fils, (1899). Google Scholar

[26]

L. Polterovich, On transport in dynamical systems,, (Russian), 43 (1988), 207. Google Scholar

[27]

L. Polterovich, "The Geometry of the Group of Symplectic Diffeomorphism,", Lectures in Mathematics ETH Zürich, (2001). Google Scholar

[28]

L. Polterovich, Growth of maps, distortion of groups and symplectic geometry,, Inv. Math., 150 (2002), 655. doi: 10.1007/s00222-002-0251-x. Google Scholar

[29]

L. Polterovich, Floer homology, dynamics and groups,, in, 217 (2006), 417. Google Scholar

[30]

J. Robbin, Heegard splittings and Floer homology,, preprint, (2000). Google Scholar

[31]

V. Rom-Kedar, Homoclinic tangles-classification and applications,, Nonlinearity, 7 (1994), 441. Google Scholar

[32]

V. Rom-Kedar, Secondary homoclinic bifurcation theorems,, Chaos, 5 (1995), 385. Google Scholar

[33]

D. Salamon, Lectures on Floer homology,, in, 7 (1999), 143. Google Scholar

[34]

M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds,, Pacific J. of Math., 193 (2000), 419. doi: 10.2140/pjm.2000.193.419. Google Scholar

[35]

S. Smale, A structurally stable differentiable homeomorphism with an infinite number of periodic points,, in, (1963), 365. Google Scholar

[36]

S. Smale, Diffeomorphisms with many periodic points,, in, (1965), 63. Google Scholar

[1]

Chen-Chang Peng, Kuan-Ju Chen. Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1181-1197. doi: 10.3934/dcdsb.2010.14.1181

[2]

Peter Albers, Urs Frauenfelder. Spectral invariants in Rabinowitz-Floer homology and global Hamiltonian perturbations. Journal of Modern Dynamics, 2010, 4 (2) : 329-357. doi: 10.3934/jmd.2010.4.329

[3]

Michael Usher. Floer homology in disk bundles and symplectically twisted geodesic flows. Journal of Modern Dynamics, 2009, 3 (1) : 61-101. doi: 10.3934/jmd.2009.3.61

[4]

Peter Albers, Urs Frauenfelder. Floer homology for negative line bundles and Reeb chords in prequantization spaces. Journal of Modern Dynamics, 2009, 3 (3) : 407-456. doi: 10.3934/jmd.2009.3.407

[5]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785

[6]

Jacobo Pejsachowicz, Robert Skiba. Topology and homoclinic trajectories of discrete dynamical systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1077-1094. doi: 10.3934/dcdss.2013.6.1077

[7]

C. M. Evans, G. L. Findley. Analytic solutions to a class of two-dimensional Lotka-Volterra dynamical systems. Conference Publications, 2001, 2001 (Special) : 137-142. doi: 10.3934/proc.2001.2001.137

[8]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351

[9]

Tiantian Wu, Xiao-Song Yang. A new class of 3-dimensional piecewise affine systems with homoclinic orbits. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5119-5129. doi: 10.3934/dcds.2016022

[10]

Zhihong Xia. Homoclinic points and intersections of Lagrangian submanifold. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 243-253. doi: 10.3934/dcds.2000.6.243

[11]

Paulo Rabelo. Elliptic systems involving critical growth in dimension two. Communications on Pure & Applied Analysis, 2009, 8 (6) : 2013-2035. doi: 10.3934/cpaa.2009.8.2013

[12]

Joachim von Below, Gaëlle Pincet Mailly, Jean-François Rault. Growth order and blow up points for the parabolic Burgers' equation under dynamical boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 825-836. doi: 10.3934/dcdss.2013.6.825

[13]

Boris Paneah. Noncommutative dynamical systems with two generators and their applications in analysis. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1411-1422. doi: 10.3934/dcds.2003.9.1411

[14]

Sergey A. Denisov. Infinite superlinear growth of the gradient for the two-dimensional Euler equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 755-764. doi: 10.3934/dcds.2009.23.755

[15]

Jun Wang, Junxiang Xu, Fubao Zhang. Homoclinic orbits for superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1241-1257. doi: 10.3934/dcds.2010.27.1241

[16]

Fang-Di Dong, Wan-Tong Li, Li Zhang. Entire solutions in a two-dimensional nonlocal lattice dynamical system. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2517-2545. doi: 10.3934/cpaa.2018120

[17]

Jong-Shenq Guo, Chang-Hong Wu. Front propagation for a two-dimensional periodic monostable lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 197-223. doi: 10.3934/dcds.2010.26.197

[18]

Tibye Saumtally, Jean-Patrick Lebacque, Habib Haj-Salem. A dynamical two-dimensional traffic model in an anisotropic network. Networks & Heterogeneous Media, 2013, 8 (3) : 663-684. doi: 10.3934/nhm.2013.8.663

[19]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1

[20]

Piotr Biler, Elio E. Espejo, Ignacio Guerra. Blowup in higher dimensional two species chemotactic systems. Communications on Pure & Applied Analysis, 2013, 12 (1) : 89-98. doi: 10.3934/cpaa.2013.12.89

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]