On primes and period growth for Hamiltonian diffeomorphisms

TL;DR

This paper employs advanced prime distribution theorems to analyze the periodic points of Hamiltonian diffeomorphisms, revealing resonance relations and partially recovering known results through symplectic methods.

Contribution

It introduces new resonance relations for mean indices of fixed points and extends results on periodic points without large period points using prime distribution theorems.

Findings

Resonance relations for mean indices of fixed points.

Partial recovery of a theorem on sphere Hamiltonian diffeomorphisms.

Analysis of periodic points with restricted prime factorization.

Abstract

Here we use Vinogradov's prime distribution theorem and a multi-dimensional generalization due to Harman to strengthen some recent results 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 partially recover, using only symplectic tools, a theorem on the periodic points of Hamiltonian diffeomorphisms of the sphere by Franks and Handel.