$\mathcal{M}\mathcal{I}\mathcal{D}$ and safe quotients for GRWS

Geometrically regular weighted shifts (in short, GRWS) are those with weights $\alpha (N,D)$ given by ${\alpha }_n(N,D)=\sqrt{\frac{p^n+N}{p^n+D}}$, where $p>1$ and $(N,D)$ is fixed in the open unit square $(-1,1)\times (-1,1)$. We study here the zone of pairs $(M,P)$ for which the weight $\frac{\alpha (N,D)}{\alpha (M,P)}$ gives rise to a moment infinitely divisible ($\mathcal{M}\mathcal{I}\mathcal{D}$) or a subnormal weighted shift, and deduce immediately the analogous results for product weights $\alpha (N,D)\alpha (M,P)$, instead of quotients. Useful tools introduced for this study are a pair of partial orders on the GRWS.