Another look at the Hofer-Zehnder conjecture

TL;DR

This paper presents a simpler proof of a modified version of Shelukhin's theorem, extending the 'two-or-infinitely-many' result to higher-dimensional Hamiltonian diffeomorphisms and advancing the Hofer-Zehnder conjecture.

Contribution

It offers a new, more straightforward proof of a weaker variant of Shelukhin's theorem, emphasizing different aspects of Hamiltonian dynamics and partially confirming the Hofer-Zehnder conjecture.

Findings

Extended the 'two-or-infinitely-many' theorem to higher dimensions.

Provided a simpler proof of a weaker variant of the Hofer-Zehnder conjecture.

Highlighted a different perspective on periodic orbit dynamics.

Abstract

We give a different and simpler proof of a slightly modified (and weaker) variant of a recent theorem of Shelukhin extending Franks' "two-or-infinitely-many" theorem to Hamiltonian diffeomorphisms in higher dimensions and establishing a sufficiently general case of the Hofer-Zehnder conjecture. A few ingredients of our proof are common with Shelukhin's original argument, the key of which is Seidel's equivariant pair-of-pants product, but the new proof highlights a different aspect of the periodic orbit dynamics of Hamiltonian diffeomorphisms.