Persistence modules, symplectic Banach-Mazur distance and Riemannian metrics

We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance between two fiberwise star-shaped domains is measured by a non-linear version of the classical Banach-Mazur distance, called symplectic Banach-Mazur distance and denoted by $d_{SBM}.$ The relevant persistence modules come from filtered symplectic homology and are stable with respect to $d_{SBM}.$ Our main focus is on the space of unit codisc bundles of orientable surfaces of positive genus, equipped with Riemannian metrics. We consider some questions about large scale-geometry of this space and in particular we give a construction of a quasi-isometric embedding of $(\mathbb{R}^n,|\cdot |_\infty)$ into this space for all $n\in \mathbb{N}.$ On the other hand, in the case of domains in $T^*S^2$, we can show that the corresponding metric space has infinite diameter. Finally, we discuss the existence of closed geodesics whose energies can be controlled.