# American Institute of Mathematical Sciences

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

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].
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, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. [2] B. Collier, E. Kerman, B. Reiniger, B. Turmunkh and A. Zimmer, A symplectic proof of a theorem of Franks, preprint, arXiv:1107.1282. [3] C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math., 37 (1984), 207-253. doi: 10.1002/cpa.3160370204. [4] A. Cotton-Clay, Symplectic Floer homology of area-preserving surface diffeomorphisms, Geom. Topol., 13 (2009), 2619-2674. doi: 10.2140/gt.2009.13.2619. [5] A. Cotton-Clay, A sharp bound on fixed points of area-preserving surface diffeomorphisms, preprint, arXiv:1009.0760. [6] S. Dostoglou and D. A. Salamon, Self-dual instantons and holomorphic curves, Ann. of Math. (2), 139 (1994), 581-640. [7] A. Floer, Morse theory for Lagrangian intersections, J. Diff. Geom., 28 (1988), 513-547. [8] J. Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms, Invent. Math., 108 (1992), 403-418. doi: 10.1007/BF02100612. [9] J. Franks, Area preserving homeomorphisms of open surfaces of genus zero, New York Jour. of Math., 2 (1996), 1-19. [10] J. Franks and M. Handel, Periodic points of Hamiltonian surface diffeomorphisms, Geom. Topol., 7 (2003), 713-756. doi: 10.2140/gt.2003.7.713. [11] V. L. Ginzburg, The Conley conjecture, Ann. of Math. (2), 172 (2010), 1127-1180. 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), 2745-2805. 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, 53-68. doi: 10.1093/imrn/rnp120. [14] G. Harman, Simultaneous Diophantine approximation with primes, J. London Math. Soc. (2), 39 (1989), 405-413. doi: 10.1112/jlms/s2-39.3.405. [15] N. Hingston, Subharmonic solutions of Hamiltonian equations on tori, Ann. of Math. (2), 170 (2009), 529-560. 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), 155-176. doi: 10.1016/0040-9383(94)E0015-C. [17] Y. Long, "Index Theory for Symplectic Paths with Applications," Progr. Math., 207, Birkhäuser Verlag, Basel, 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), 1303-1360. doi: 10.1002/cpa.3160451004. [19] P. Seidel, "Floer Homology and the Symplectic Isotopy Problem," D. Phil thesis, University of Oxford, 1997. [20] P. Seidel, $\pi_1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal., 7 (1997), 1046-1095. doi: 10.1007/s000390050037. [21] P. Seidel, Symplectic Floer homology and the mapping class group, Pac. J. Math., 206 (2002), 219-229. doi: 10.2140/pjm.2002.206.219. [22] P. Seidel, Braids and symplectic four-manifolds with abelian fundamental group, Turkish J. Math., 26 (2002), 93-100. [23] I. M. Vinogradov, "The Method of Trigonometric Sums in the Theory of Numbers," Translated from the Russian, revised and annotated by A. Davenport and K. F. Roth, Reprint of the 1954 translation, Dover Publications, Inc., Mineola, NY, 2004.

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##### References:
 [1] V. I. Arnol'd, "Mathematical Methods of Classical Mechanics," Second edition, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. [2] B. Collier, E. Kerman, B. Reiniger, B. Turmunkh and A. Zimmer, A symplectic proof of a theorem of Franks, preprint, arXiv:1107.1282. [3] C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math., 37 (1984), 207-253. doi: 10.1002/cpa.3160370204. [4] A. Cotton-Clay, Symplectic Floer homology of area-preserving surface diffeomorphisms, Geom. Topol., 13 (2009), 2619-2674. doi: 10.2140/gt.2009.13.2619. [5] A. Cotton-Clay, A sharp bound on fixed points of area-preserving surface diffeomorphisms, preprint, arXiv:1009.0760. [6] S. Dostoglou and D. A. Salamon, Self-dual instantons and holomorphic curves, Ann. of Math. (2), 139 (1994), 581-640. [7] A. Floer, Morse theory for Lagrangian intersections, J. Diff. Geom., 28 (1988), 513-547. [8] J. Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms, Invent. Math., 108 (1992), 403-418. doi: 10.1007/BF02100612. [9] J. Franks, Area preserving homeomorphisms of open surfaces of genus zero, New York Jour. of Math., 2 (1996), 1-19. [10] J. Franks and M. Handel, Periodic points of Hamiltonian surface diffeomorphisms, Geom. Topol., 7 (2003), 713-756. doi: 10.2140/gt.2003.7.713. [11] V. L. Ginzburg, The Conley conjecture, Ann. of Math. (2), 172 (2010), 1127-1180. 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), 2745-2805. 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, 53-68. doi: 10.1093/imrn/rnp120. [14] G. Harman, Simultaneous Diophantine approximation with primes, J. London Math. Soc. (2), 39 (1989), 405-413. doi: 10.1112/jlms/s2-39.3.405. [15] N. Hingston, Subharmonic solutions of Hamiltonian equations on tori, Ann. of Math. (2), 170 (2009), 529-560. 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), 155-176. doi: 10.1016/0040-9383(94)E0015-C. [17] Y. Long, "Index Theory for Symplectic Paths with Applications," Progr. Math., 207, Birkhäuser Verlag, Basel, 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), 1303-1360. doi: 10.1002/cpa.3160451004. [19] P. Seidel, "Floer Homology and the Symplectic Isotopy Problem," D. Phil thesis, University of Oxford, 1997. [20] P. Seidel, $\pi_1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal., 7 (1997), 1046-1095. doi: 10.1007/s000390050037. [21] P. Seidel, Symplectic Floer homology and the mapping class group, Pac. J. Math., 206 (2002), 219-229. doi: 10.2140/pjm.2002.206.219. [22] P. Seidel, Braids and symplectic four-manifolds with abelian fundamental group, Turkish J. Math., 26 (2002), 93-100. [23] I. M. Vinogradov, "The Method of Trigonometric Sums in the Theory of Numbers," Translated from the Russian, revised and annotated by A. Davenport and K. F. Roth, Reprint of the 1954 translation, Dover Publications, Inc., Mineola, NY, 2004.
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