Geometry and dynamics on Liouville domains in $T^*\mathbb T^2$

TL;DR

This paper develops a new framework for Liouville domains in the cotangent bundle of the torus, introducing a systolic ratio-based convexity and analyzing their geometric and dynamical properties using symplectic invariants.

Contribution

It introduces a novel convexity notion for Liouville domains in $T^* ext{T}^2$ and explores their geometric and dynamical relations using advanced symplectic tools.

Findings

Established relations between subclasses of Liouville domains.

Analyzed large-scale geometry with respect to Banach-Mazur distance.

Verified normalized capacities for a broad class of Liouville domains.

Abstract

Parallel to the study of toric domains, symplectically convex, and dynamically convex domains in $(\mathbb R^4, \omega_{\rm std})$, we build an analogous framework and corresponding subclasses for Liouville domains in $(T^*\mathbb T^2,\omega_{\rm can})$. A key feature of this framework is the introduction of a new notion of convexity, based on systolic ratios. Via various machinery in quantitative symplectic geometry, including ECH capacities, shape invariant, dynamical zeta function, etc., we investigate the relations between subclasses of Liouville domains in $T^*\mathbb T^2$, obtain large-scale geometry of Liouville domains in $T^*\mathbb T^2$ with respect to coarse Banach-Mazur distance, provide a non-flat codisc bundle of torus even under the action of exact symplectomorphisms, and verify the agreement of normalized capacities for a wide class of Liouville domains in $T^*\mathbb T^2$.