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S. Aubry, P. Le Daeron and G. André, Classical ground-states of one-dimensional models for incommensurate structures,, unpublished preprint, (1982).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

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