Barcode growth for toric-integrable Hamiltonian systems
Erol Barut, Viktor L. Ginzburg
TL;DR
This work connects Hamiltonian dynamics with the growth of barcodes in Floer/symplectic persistence modules, showing that completely integrable, toric-type dynamics yield slow, polynomial barcode growth with degree at most $n$ (half the dimension). It develops a persistence-module framework for symplectic homology, establishes sharp polynomial bounds in toric settings (including non-smooth domains and closed toric manifolds), and proves invariance of these growth rates under interior exact symplectomorphisms. The results contrast with exponential barcode growth in chaotic systems and provide tools to distinguish domains via interior dynamics; they also pose open questions about extending to broader integrable systems and relaxing smoothness assumptions. Overall, the paper deepens the link between dynamical complexity and persistent symplectic invariants with concrete, computable bounds.
Abstract
We continue investigating the connection between the dynamics of a Hamiltonian system and the barcode growth of the associated Floer or symplectic homology persistence module, focusing now on completely integrable systems. We show that for convex/concave or real analytic toric domains and convex/concave or real analytic completely integrable Hamiltonians on closed toric manifolds the barcode has polynomial growth with degree (i.e., slow barcode entropy) not exceeding half of the dimension. This slow polynomial growth contrasts with exponential growth for many systems with sufficiently non-trivial dynamics. We also touch upon the barcode growth function as an invariant of the interior of the domain and use it to distinguish some open domains.