On primes and period growth for Hamiltonian diffeomorphisms

Previous Article
Équidistribution, comptage et approximation par irrationnels quadratiques
JMD Home
This Issue
Next Article
Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition

January 2012, 6(1): 41-58. doi: 10.3934/jmd.2012.6.41

On primes and period growth for Hamiltonian diffeomorphisms

1.	Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801

Received November 2011 Published May 2012

Abstract
Related Papers

Here we use Vinogradov's prime distribution theorem and a multidimensional generalization due to Harman to strengthen some recent results from [12] and [13] concerning the periodic points of Hamiltonian diffeomorphisms. In particular we establish resonance relations for the mean indices of the fixed points of Hamiltonian diffeomorphisms which do not have periodic points with arbitrarily large periods in $\mathbb{P}^2$, the set of natural numbers greater than one which have at most two prime factors when counted with multiplicity. As an application of these results we extend the methods of [2] to partially recover, using only symplectic tools, a theorem on the periodic points of Hamiltonian diffeomorphisms of the sphere by Franks and Handel from [10].

Keywords: Floer homology., Hamiltonian diffeomorphisms, Periodic points.

Mathematics Subject Classification: 53D40, 37J45, 70H1.

Citation: Ely Kerman. On primes and period growth for Hamiltonian diffeomorphisms. Journal of Modern Dynamics, 2012, 6 (1) : 41-58. doi: 10.3934/jmd.2012.6.41

References:

[1]	V. I. Arnol'd, "Mathematical Methods of Classical Mechanics,", Second edition, 60 (1989).
[2]	B. Collier, E. Kerman, B. Reiniger, B. Turmunkh and A. Zimmer, A symplectic proof of a theorem of Franks,, preprint, ().
[3]	C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations,, Comm. Pure Appl. Math., 37 (1984), 207. doi: 10.1002/cpa.3160370204.
[4]	A. Cotton-Clay, Symplectic Floer homology of area-preserving surface diffeomorphisms,, Geom. Topol., 13 (2009), 2619. doi: 10.2140/gt.2009.13.2619.
[5]	A. Cotton-Clay, A sharp bound on fixed points of area-preserving surface diffeomorphisms,, preprint, ().
[6]	S. Dostoglou and D. A. Salamon, Self-dual instantons and holomorphic curves,, Ann. of Math. (2), 139 (1994), 581.
[7]	A. Floer, Morse theory for Lagrangian intersections,, J. Diff. Geom., 28 (1988), 513.
[8]	J. Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms,, Invent. Math., 108 (1992), 403. doi: 10.1007/BF02100612.
[9]	J. Franks, Area preserving homeomorphisms of open surfaces of genus zero,, New York Jour. of Math., 2 (1996), 1.
[10]	J. Franks and M. Handel, Periodic points of Hamiltonian surface diffeomorphisms,, Geom. Topol., 7 (2003), 713. doi: 10.2140/gt.2003.7.713.
[11]	V. L. Ginzburg, The Conley conjecture,, Ann. of Math. (2), 172 (2010), 1127. doi: 10.4007/annals.2010.172.1129.
[12]	V. L. Ginzburg and B. Z. Gürel, Action and index spectra and periodic orbits in Hamiltonian dynamics,, Geom. Topol., 13 (2009), 2745. doi: 10.2140/gt.2009.13.2745.
[13]	V. L. Ginzburg and E. Kerman, Homological resonances for Hamiltonian diffeomorphisms and Reeb flows,, Internat. Math. Res. Notices, 2010 (): 53. doi: 10.1093/imrn/rnp120.
[14]	G. Harman, Simultaneous Diophantine approximation with primes,, J. London Math. Soc. (2), 39 (1989), 405. doi: 10.1112/jlms/s2-39.3.405.
[15]	N. Hingston, Subharmonic solutions of Hamiltonian equations on tori,, Ann. of Math. (2), 170 (2009), 529. doi: 10.4007/annals.2009.170.529.
[16]	H. V. Lê and K. Ono, Symplectic fixed points, the Calabi invariant and Novikov homology,, Topology, 34 (1995), 155. doi: 10.1016/0040-9383(94)E0015-C.
[17]	Y. Long, "Index Theory for Symplectic Paths with Applications,", Progr. Math., 207 (2002).
[18]	D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index,, Comm. Pure Appl. Math., 45 (1992), 1303. doi: 10.1002/cpa.3160451004.
[19]	P. Seidel, "Floer Homology and the Symplectic Isotopy Problem,", D. Phil thesis, (1997).
[20]	P. Seidel, $\pi_1$ of symplectic automorphism groups and invertibles in quantum homology rings,, Geom. Funct. Anal., 7 (1997), 1046. doi: 10.1007/s000390050037.
[21]	P. Seidel, Symplectic Floer homology and the mapping class group,, Pac. J. Math., 206 (2002), 219. doi: 10.2140/pjm.2002.206.219.
[22]	P. Seidel, Braids and symplectic four-manifolds with abelian fundamental group,, Turkish J. Math., 26 (2002), 93.
[23]	I. M. Vinogradov, "The Method of Trigonometric Sums in the Theory of Numbers,", Translated from the Russian, (1954).

show all references

References:

[1]	V. I. Arnol'd, "Mathematical Methods of Classical Mechanics,", Second edition, 60 (1989).
[2]	B. Collier, E. Kerman, B. Reiniger, B. Turmunkh and A. Zimmer, A symplectic proof of a theorem of Franks,, preprint, ().
[3]	C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations,, Comm. Pure Appl. Math., 37 (1984), 207. doi: 10.1002/cpa.3160370204.
[4]	A. Cotton-Clay, Symplectic Floer homology of area-preserving surface diffeomorphisms,, Geom. Topol., 13 (2009), 2619. doi: 10.2140/gt.2009.13.2619.
[5]	A. Cotton-Clay, A sharp bound on fixed points of area-preserving surface diffeomorphisms,, preprint, ().
[6]	S. Dostoglou and D. A. Salamon, Self-dual instantons and holomorphic curves,, Ann. of Math. (2), 139 (1994), 581.
[7]	A. Floer, Morse theory for Lagrangian intersections,, J. Diff. Geom., 28 (1988), 513.
[8]	J. Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms,, Invent. Math., 108 (1992), 403. doi: 10.1007/BF02100612.
[9]	J. Franks, Area preserving homeomorphisms of open surfaces of genus zero,, New York Jour. of Math., 2 (1996), 1.
[10]	J. Franks and M. Handel, Periodic points of Hamiltonian surface diffeomorphisms,, Geom. Topol., 7 (2003), 713. doi: 10.2140/gt.2003.7.713.
[11]	V. L. Ginzburg, The Conley conjecture,, Ann. of Math. (2), 172 (2010), 1127. doi: 10.4007/annals.2010.172.1129.
[12]	V. L. Ginzburg and B. Z. Gürel, Action and index spectra and periodic orbits in Hamiltonian dynamics,, Geom. Topol., 13 (2009), 2745. doi: 10.2140/gt.2009.13.2745.
[13]	V. L. Ginzburg and E. Kerman, Homological resonances for Hamiltonian diffeomorphisms and Reeb flows,, Internat. Math. Res. Notices, 2010 (): 53. doi: 10.1093/imrn/rnp120.
[14]	G. Harman, Simultaneous Diophantine approximation with primes,, J. London Math. Soc. (2), 39 (1989), 405. doi: 10.1112/jlms/s2-39.3.405.
[15]	N. Hingston, Subharmonic solutions of Hamiltonian equations on tori,, Ann. of Math. (2), 170 (2009), 529. doi: 10.4007/annals.2009.170.529.
[16]	H. V. Lê and K. Ono, Symplectic fixed points, the Calabi invariant and Novikov homology,, Topology, 34 (1995), 155. doi: 10.1016/0040-9383(94)E0015-C.
[17]	Y. Long, "Index Theory for Symplectic Paths with Applications,", Progr. Math., 207 (2002).
[18]	D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index,, Comm. Pure Appl. Math., 45 (1992), 1303. doi: 10.1002/cpa.3160451004.
[19]	P. Seidel, "Floer Homology and the Symplectic Isotopy Problem,", D. Phil thesis, (1997).
[20]	P. Seidel, $\pi_1$ of symplectic automorphism groups and invertibles in quantum homology rings,, Geom. Funct. Anal., 7 (1997), 1046. doi: 10.1007/s000390050037.
[21]	P. Seidel, Symplectic Floer homology and the mapping class group,, Pac. J. Math., 206 (2002), 219. doi: 10.2140/pjm.2002.206.219.
[22]	P. Seidel, Braids and symplectic four-manifolds with abelian fundamental group,, Turkish J. Math., 26 (2002), 93.
[23]	I. M. Vinogradov, "The Method of Trigonometric Sums in the Theory of Numbers,", Translated from the Russian, (1954).

[1]	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
[2]	C. Morales. On spiral periodic points and saddles for surface diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1191-1195. doi: 10.3934/dcds.2011.29.1191
[3]	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
[4]	Morimichi Kawasaki, Ryuma Orita. Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories. Journal of Modern Dynamics, 2017, 11: 313-339. doi: 10.3934/jmd.2017013
[5]	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
[6]	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
[7]	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
[8]	Vadim Yu. Kaloshin and Brian R. Hunt. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II. Electronic Research Announcements, 2001, 7: 28-36.
[9]	Vadim Yu. Kaloshin and Brian R. Hunt. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I. Electronic Research Announcements, 2001, 7: 17-27.
[10]	Carlos Matheus, Jean-Christophe Yoccoz. The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis. Journal of Modern Dynamics, 2010, 4 (3) : 453-486. doi: 10.3934/jmd.2010.4.453
[11]	K. H. Kim, F. W. Roush and J. B. Wagoner. Inert actions on periodic points. Electronic Research Announcements, 1997, 3: 55-62.
[12]	Charles Pugh, Michael Shub. Periodic points on the $2$-sphere. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1171-1182. doi: 10.3934/dcds.2014.34.1171
[13]	John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047
[14]	Richard Miles, Thomas Ward. Directional uniformities, periodic points, and entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3525-3545. doi: 10.3934/dcdsb.2015.20.3525
[15]	Anna Gierzkiewicz, Klaudiusz Wójcik. Lefschetz sequences and detecting periodic points. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 81-100. doi: 10.3934/dcds.2012.32.81
[16]	V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277
[17]	Juan Campos, Rafael Ortega. Location of fixed points and periodic solutions in the plane. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 517-523. doi: 10.3934/dcdsb.2008.9.517
[18]	Wenxiang Sun, Yun Yang. Hyperbolic periodic points for chain hyperbolic homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3911-3925. doi: 10.3934/dcds.2016.36.3911
[19]	Grzegorz Graff, Michał Misiurewicz, Piotr Nowak-Przygodzki. Periodic points of latitudinal maps of the $m$-dimensional sphere. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6187-6199. doi: 10.3934/dcds.2016070
[20]	P. Chiranjeevi, V. Kannan, Sharan Gopal. Periodic points and periods for operators on hilbert space. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4233-4237. doi: 10.3934/dcds.2013.33.4233

2018 Impact Factor: 0.295

American Institute of Mathematical Sciences