Dynamics of low-degree rational inner skew-products on $\mathbb{T}^2$

We examine iteration of certain skew-products on the bidisk whose components are rational inner functions, with emphasis on simple maps of the form $\Phi (z_1,z_2) = (\phi (z_1,z_2), z_2)$. If $\phi $ has degree $1$ in the first variable, the dynamics on each horizontal fiber can be described in terms of Möbius transformations but the global dynamics on the $2$-torus exhibit some complexity, encoded in terms of certain $\mathbb {T}^2$-symmetric polynomials. We describe the dynamical behavior of such mappings $\Phi $ and give criteria for different configurations of fixed point curves and rotation belts in terms of zeros of a related one-variable polynomial.