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

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 $T^*\mathbb T^2$.