Let \(\varphi \) be a holomorphic self-map of the bidisc that is Lipschitz on the closure. We show that the composition operator \(C_{\varphi }\) is compact on the Bergman space if and only if \(\varphi (\overline{\mathbb {D}^2})\cap \mathbb {T}^2=\emptyset \) and \(\varphi (\overline{\mathbb {D}^2}\setminus \mathbb {T}^2) \cap b\mathbb {D}^2=\emptyset \) . In the last section of the paper, we prove a result on \(C^2\) -smooth bounded pseudoconvex domains in \(\mathbb {C}^{n}\) .
