Autonomous Hamiltonian flows, Hofer's geometry and persistence modules

TL;DR

This paper introduces new robust obstructions in Hofer's geometry that prevent Hamiltonian diffeomorphisms from being expressed as full powers or embedded into one-parameter subgroups, using persistence modules in Floer homology.

Contribution

It develops a novel application of persistence modules to establish obstructions in Hamiltonian dynamics, advancing the understanding of Hamiltonian diffeomorphisms in symplectic geometry.

Findings

Obstructions to representing Hamiltonian diffeomorphisms as full powers.

Obstructions to embedding into one-parameter subgroups.

Applications to geometry and dynamics of Hamiltonian diffeomorphisms.

Abstract

We find robust obstructions to representing a Hamiltonian diffeomorphism as a full $k$-th power, $k \geq 2,$ and in particular, to including it into a one-parameter subgroup. The robustness is understood in the sense of Hofer's metric. Our approach is based on the theory of persistence modules applied in the context of filtered Floer homology. We present applications to geometry and dynamics of Hamiltonian diffeomorphisms.