Another look at the Hofer-Zehnder conjecture

Abstract

We give a different and simpler proof of a slightly modified (and weaker) variant of a recent theorem of Shelukhin extending Franks' "two-orinfinitely-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.

On the Hofer-Zehnder conjecture on weighted projective spaces

“…Collier et al gave a proof of Franks theorem in the case of Ham(CP 1 ) using symplectic tools [5]. The higher achievement in proving this conjecture is Shelukhin's theorem showing a homology version of this conjecture in a class of symplectic manifolds including CP d [29] (see also [4,2]). In this article, we prove an extension of this theorem to the weighted projective spaces.…”