$p$-cyclic persistent homology and Hofer distance

In this paper, we prove that the Hofer distance from time-dependent Hamiltonian diffeomorphisms to the set of $p$-th power Hamiltonian diffeomorphisms can be arbitrarily large for symplectic manifold $\Sigma_g \times M$, where $M$ is any closed symplectic manifold, $p$ is sufficiently large and $g \geq 4$. This implies that, on this product, the Hofer distance can be arbitrarily large between time-dependent Hamiltonian diffeomorphisms and autonomous Hamiltonian diffeomorphisms. This generalizes the main result from L. Polterovich and E. Shelukhin’s paper [PS16]. The basic tools we will use are barcode and singular value decomposition which are developed in the paper [UZ16], from which we borrow many proofs and modify them so that they can be adapted to a Floer-type complex equipped with a group action.

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Received 20 July 2016

Accepted 25 July 2018

Published 9 September 2019