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

TL;DR

This paper studies the dynamics of specific rational inner skew-products on the 2-torus, revealing complex behaviors through polynomial analysis and fixed point structures.

Contribution

It provides a detailed description of the dynamics of rational inner skew-products with degree-one components, including criteria for fixed points and rotation behaviors.

Findings

Dynamics characterized by $ ext{T}^2$-symmetric polynomials

Criteria for fixed point curves and rotation belts

Complex global behavior on the 2-torus

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 $\Phi(z_1,z_2) = (\phi(z_1,z_2), z_2)$. If $\phi$ has degree $1$ in the first variable, the dynamics on each horizontal fiber can be described in terms of M\"obius 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.