Compactness of composition operator on weighted Bergman spaces of the polydisc

We study composition operators induced by a smooth symbol between weighted Bergman spaces of the polydisc. We first prove a compactness criterion that only requires knowing what happens on the distinguished boundary. Then we prove simple geometric characterizations of boundedness and compactness on some $A_{\beta }^2({\mathbb{D}}^d)$, particularly for $\beta >d-3$.