Narrow Results

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 nonlinear version of the classical Banach–Mazur distance, called symplectic Banach–Mazur distance and denoted by $d_{\mathrm{\text{SBM}}}$. The relevant persistence modules come from filtered symplectic homology and are stable with respect to $d_{\mathrm{\text{SBM}}}$. 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 $({ℝ}^n,|\cdot {|}_{\infty })$ into this space for all $n\in \mathbb{N}$. On the other hand, in the case of domains in $T^{\ast }{S}^2$, we can show that the corresponding metric space has infinite diameter. Finally, we discuss the existence of closed geodesics whose energies can be controlled.

We use generating function techniques developed by Givental, Théret and ourselves to deduce a proof in $ℂ{\text{P}}^d$ of the homological generalization of Franks theorem due to Shelukhin. This result proves in particular the Hofer–Zehnder conjecture in the nondegenerate case: every Hamiltonian diffeomorphism of $ℂ{\text{P}}^d$ that has at least $d+2$ nondegenerate periodic points has infinitely many periodic points. Our proof does not appeal to Floer homology or the theory of $J$-holomorphic curves. An appendix written by Shelukhin contains a new proof of the Smith-type inequality for barcodes of Hamiltonian diffeomorphisms that arise from Floer theory, which lends itself to adaptation to the setting of generating functions.