On some generalizations of skew-shifts on $\mathbb{T}^2$

TL;DR

This paper studies a class of skew-shift maps on the two-torus, showing they are minimal, possess exactly two ergodic measures supported on invariant graphs, including a strange nonchaotic attractor, with robustness under small perturbations.

Contribution

It introduces a broad class of skew-shift maps with degree 2 monotone maps and proves minimality, ergodic measure classification, and existence of a strange nonchaotic attractor.

Findings

Map $T$ is minimal with exactly two invariant ergodic measures.

One invariant measure is supported on a strange nonchaotic attractor.

Results are robust under Lipschitz-small perturbations.

Abstract

In this paper we investigate maps of the two-torus $\mathbb{T}^2$ of the form $T(x,y)=(x+\omega,g(x)+f(y))$ for Diophantine $\omega\in\mathbb{T}$ and for a class of maps $f,g:\mathbb{T}\to\mathbb{T}$, where each $g$ is strictly monotone and of degree 2, and each $f$ is an orientation preserving circle homeomorphism. For our class of $f$ and $g$ we show that $T$ is minimal and has exactly two invariant and ergodic Borel probability measures. Moreover, these measures are supported on two $T$-invariant graphs. One of the graphs is a Strange Nonchaotic Attractor whose basin of attraction consists of (Lebesgue) almost all points in $\mathbb{T}^2$. Only a low regularity assumption (Lipschitz) is needed on the maps $f$ and $g$, and the results are robust with respect to Lipschitz-small perturbations of $f$ and $g$.