Transport, flux and growth of homoclinic Floer homology

Sonja Hohloch

doi:10.3934/dcds.2012.32.3587

Article Contents

2012, Volume 32, Issue 10: 3587-3620. Doi: 10.3934/dcds.2012.32.3587

This issue Previous Article Perturbed elliptic equations with oscillatory nonlinearities Next Article Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity

Transport, flux and growth of homoclinic Floer homology

Sonja Hohloch^1,

1.
Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540

Received: January 31, 2011

Revised: March 31, 2012

Published: September 2012

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Floer homology,
homoclinic points,
flux,
growth,
two dimensional symplectic dynamical systems.

Mathematics Subject Classification: Primary: 37J05, 37J10, 37J45, 53D40.

Citation:

Related Papers

Cited by

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S. Aubry, P. Le Daeron and G. André, Classical ground-states of one-dimensional models for incommensurate structures, unpublished preprint, 1982.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), 85-216.
[3]	Y. Chekanov, Differential algebra of Legendrian links, Invent. Math., 150 (2002), 441-483.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), 33-49.doi: 10.1007/BF01393824.
[5]	S. de Silva, "Products in the Symplectic Floer Homology of Lagrangian Intersections," Thesis, Merton College, Oxford, 1998.
[6]	A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649-652.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), 1213-1216.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), 393-407.doi: 10.1002/cpa.3160410402.
[9]	A. Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math., 41 (1988), 775-813.doi: 10.1002/cpa.3160410603.
[10]	A. Floer, Morse theory for Lagrangian intersections, J. Diff. Geom., 28 (1988), 513-547.
[11]	R. Gautschi, J. Robbin and D. Salamon, Heegard splittings and Morse-Smale flows, Int. J. Math. Math. Sci., 2003, 3539-3572.
[12]	V. Gelfreich, A proof of the exponentially small transversality of the separatrices for the standard map, Comm. Math. Phys., 201 (1999), 155-216.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), 511-536.doi: 10.3934/dcdsb.2008.10.511.
[14]	V. Ginzburg, The Conley conjecture, Ann. of Math. (2), 172 (2010), 1127-1180.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), 2745-2805.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), 25-55, 285; translation in J. Math. Sci. (N. Y.), 128 (2005), 2687-2705.
[19]	R. MacKay, J. Meiss and I. Percival, Transport in Hamiltonian systems, Physica D, 13 (1984), 55-81.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), 153-204.doi: 10.1007/BF02831625.
[21]	J. Mather, Modulus of continuity for Peierls's barrier, in "Periodic Solutions of Hamiltonian Systems and Related Topics" (eds. P. H. Rabinowitz, et al.) (Il Ciocco, 1986), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 209, Reidel, Dordrecht, (1987), 177-202.
[22]	D. McDuff and D. Salamon, "Introduction to Symplectic Topology," Second edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.
[23]	J. Palis, On Morse-Smale dynamical systems, Topology, 8 (1969), 385-405.
[24]	H. Poincaré, Sur le problème des trois corps et les équations de la dynamique, Acta Mathematica, 13 (1890), 1-271.
[25]	H. Poincaré, Les méthodes nouvelles de la méchanique céleste, Gauthier-Villars et fils, Paris, 1899.
[26]	L. Polterovich, On transport in dynamical systems, (Russian), Uspekhi Mat. Nauk, 43, (1988), 207-208; translation in Russian Math. Surveys, 43 (1988), 251-252.
[27]	L. Polterovich, "The Geometry of the Group of Symplectic Diffeomorphism," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001.
[28]	L. Polterovich, Growth of maps, distortion of groups and symplectic geometry, Inv. Math., 150 (2002), 655-686.doi: 10.1007/s00222-002-0251-x.
[29]	L. Polterovich, Floer homology, dynamics and groups, in "Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology" (ed. P. Biran, et al.), NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, (2006), 417-438.
[30]	J. Robbin, Heegard splittings and Floer homology, preprint, 2000.
[31]	V. Rom-Kedar, Homoclinic tangles-classification and applications, Nonlinearity, 7 (1994), 441-473.
[32]	V. Rom-Kedar, Secondary homoclinic bifurcation theorems, Chaos, 5 (1995), 385-401.
[33]	D. Salamon, Lectures on Floer homology, in "Symplectic Geometry and Topology" (Park City, UT, 1997), IAS/Park City Math. Ser., 7, Amer. Math. Soc., Providence, RI, (1999), 143-229.
[34]	M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. of Math., 193 (2000), 419-461.doi: 10.2140/pjm.2000.193.419.
[35]	S. Smale, A structurally stable differentiable homeomorphism with an infinite number of periodic points, in "Qualitative Methods in the Theory of Non-Linear Vibrations" (Proceedings of the International Symposium on Non-Linear Vibrations, Vol. II, 1961), Izdat. Akad. Nauk Ukrain. SSR, Kiev, (1963), 365-366.
[36]	S. Smale, Diffeomorphisms with many periodic points, in "Differential and Combinatorial Topology" (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton N.J., (1965), 63-80.