An open set of torus-maps conjugate to skew products

Suddhasattwa Das, James Yorke

Published 2015-12-10Version 1

We investigate conditions under which a map of the torus $\Torus$ is conjugate to a skew-product dynamical system of the form $$(x_{n+1},y_{n+1})=(mx_n, g(x_n,y_n))\mod 1,$$ where $m$ is an integer and $g:\Torus\to S^1$ is a $C^1$ map. Skew-product maps are relatively easy to analyze and include a variety of interesting dynamical systems. Notice that the dynamics in the X coordinate is $x_{n+1}=mx_n\mod 1$. We present sufficient conditions for a torus map to be conjugate to a skew-product map. The set of maps which satisfy these conditions is open in the $C^1$ topology.