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S. Aubry, P. Le Daeron and G. André, Classical ground-states of one-dimensional models for incommensurate structures,, unpublished preprint, (1982).

[2]

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

[3]

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

[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.

[5]

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

[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.

[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.

[8]

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

[9]

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

[10]

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

[11]

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

[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.

[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.

[14]

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

[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.

[16]

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

[17]

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

[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.

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[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.

[35]

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

[36]

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

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).

[2]

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

[3]

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

[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.

[5]

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

[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.

[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.

[8]

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

[9]

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

[10]

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

[11]

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

[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.

[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.

[14]

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

[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.

[16]

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

[17]

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

[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.

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[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.

[35]

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

[36]

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

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