Composition operators and Rational Inner Functions on the bidisc

In the present article, composition operators induced by Rational Inner Functions on the bidisc ${\mathbb{D}}^2$ are studied, acting on the weighted Bergman space $A_{\beta }^2({\mathbb{D}}^2)\mathrm{.}$ We prove that under mild conditions that Rational Inner Functions with one singularity on ${\mathbb{T}}^2$ induce unbounded composition operator on $A^2({\mathbb{D}}^2)\mathrm{.}$ We also prove that under the condition of stability of the polynomial inducing the Rational Inner Function, the composition operator is bounded between two different Bergman spaces.