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

Abstract

We examine iteration of certain skew-products on the bidisk whose components are rational inner functions, with emphasis on simple maps of the form Φ(z 1 , z 2 ) = (φ(z 1 , z 2 ), z 2 ). If φ 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 T 2 -symmetric polynomials. We describe the dynamical behavior of such mappings Φ and give criteria for different configurations of fixed point curves and rotation belts in terms of zeros of a related one-variable polynomial.