Regions of exponential stability for LTI systems on nonuniform discrete domains

For LTI systems on a class of nonuniform discrete domains, we establish a region in the complex plane for which pole placement is a necessary and sufficient condition for exponential stability of solutions of the system. We study the interesting geometry of this region, comparing and contrasting it with the standard geometry of the regions of exponential stability for ODE systems on R and finite difference/recursive equations on Z. This work connects other results in the literature on the topic and explains the connection geometrically using time scales theory.