标题： 关于$\mathbb{T}^2$上某些斜移的推广 显示英文标题

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

摘要： 在本文中，我们研究了二环面 $\mathbb{T}^2$ 上的映射，形式为 $T(x,y)=(x+\omega,g(x)+f(y))$，其中 $\omega\in\mathbb{T}$ 是无理数，并且对于一类映射 $f,g:\mathbb{T}\to\mathbb{T}$，其中每个 $g$ 是严格单调的且次数为 2，每个 $f$ 是保持方向的圆同胚。 对于我们的类$f$和$g$，我们证明$T$是最小的，并且恰好有两个不变的和遍历的 Borel 概率测度。 此外，这些测度支持在两个$T$-不变图上。 其中一个图是一个奇异非混沌吸引子，其吸引域由（勒贝格）几乎所有的点组成$\mathbb{T}^2$。 仅对映射$f$和$g$需要一个低正则性假设（Lipschitz），并且结果对于$f$和$g$的 Lipschitz 小扰动是稳健的。 显示英文摘要

摘要： 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 $g$.