Hofer's length spectrum of symplectic surfaces

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.