$\mathcal{MID}$ 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 ($\mathcal {MID}$) 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.