Hofer growth of $C^1$-generic Hamiltonian flows

TL;DR

This paper demonstrates that on specific closed symplectic manifolds, a generic cyclic subgroup of Hamiltonian diffeomorphisms exhibits undistorted behavior in the Hofer metric, revealing new geometric properties of these groups.

Contribution

It establishes the undistorted nature of generic cyclic subgroups in the universal cover of Hamiltonian diffeomorphisms with respect to the Hofer metric on certain symplectic manifolds.

Findings

Generic cyclic subgroups are undistorted in the Hofer metric.

The result applies to certain closed symplectic manifolds.

Provides new insights into the geometry of Hamiltonian diffeomorphism groups.

Abstract

We prove that on certain closed symplectic manifolds a $C^1$-generic cyclic subgroup of the universal cover of the group of Hamiltonian diffeomorphisms is undistorted with respect to the Hofer metric.