Symplectic dynamics via a tube construction

Abstract

This is the first in a series of papers studying symplectic dynamics on Liouville domains $(W, λ)$ from a geometric approach. We employ Usher's tube construction to build a higher-dimensional Liouville domain from $(W,λ; H)$, where $H$ is a Hamiltonian function on $(W, λ)$, possibly non-autonomous. We establish dynamical stabilities with respect to a new Thompson-type pseudo-metric on Hamiltonians, bounded below by Banach--Mazur type distances. We also explore relations between the topological entropies of the Hamiltonian dynamics and its induced Reeb dynamics, examine symplectic capacities, especially the Gromov width, under the tube construction, and give a categorification of the Thompson-type metric via a generalized tube construction (called the tunnel construction). Finally, we investigate the relation between the filtered symplectic homology of the resulting tube and the Hamiltonian Floer homologies of the input Hamiltonians, as well as their iterates, on the given base $(W, λ)$.