Hofer's length spectrum of symplectic surfaces

TL;DR

This paper introduces a new set of symplectic invariants called Hofer's length spectrum for surfaces, which measure minimal energy for translating disks along homology classes, extending classical length spectrum concepts.

Contribution

It defines and analyzes Hofer's length spectrum for symplectic surfaces and constructs continuous quasimorphisms related to periodic non-displaceable disks on genus zero surfaces.

Findings

Defined invariants $l_A$ for symplectic surfaces

Connected invariants to minimal Hofer energy for disk translation

Constructed continuous quasimorphisms for genus zero surfaces

Abstract

Following a question of F. Le Roux, we consider a system of invariants $l_A : H_1(M; \mathbb{Z})\to\mathbb{R}$ of a symplectic surface $M$. These invariants compute the minimal Hofer energy needed to translate a disk of area $A$ along a given homology class and can be seen as a symplectic analogue of the Riemannian length spectrum. When $M$ has genus zero we also construct Hofer- and $C_0$-continuous quasimorphisms $Ham(M) \to H_1(M;\mathbb{R})$ that compute trajectories of periodic non-displaceable disks.