On primes and period growth for Hamiltonian diffeomorphisms

Ely Kerman

doi:10.3934/jmd.2012.6.41

Article Contents

2012, Volume 6, Issue 1: 41-58. Doi: 10.3934/jmd.2012.6.41

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On primes and period growth for Hamiltonian diffeomorphisms

Ely Kerman^1,

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

Received: October 31, 2011

Published: April 2012

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Abstract

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].

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Mathematics Subject Classification: 53D40, 37J45, 70H12.

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