论文标题
在$ sl（2，\ mathbb {r}）$ - 与次级碰撞的非理性旋转上
On $SL(2,\mathbb{R})$-cocycles over irrational rotations with secondary collisions
论文作者
论文摘要
我们考虑一个偏斜的产品$ f_ {a} =（σ_为，a）$在非理性旋转上$σ_Ω（x）= x +ω$的circle $ \ mathbb {t}^{1} $。据认为，转换$ a：\ mathbb {t}^{1} \ to sl（2，\ mathbb {r}）$是a $ c^{1} $ - 地图具有form $ a（x）= r（x）= r（φ（x））z（λ（λ））$ r（λ（λ））$ a $ r（φ）角度$φ$和$ z（λ）= diag \ {λ，λ^{ - 1} \} $是对角矩阵。假设$λ（x）\geλ_{0}> 1 $具有足够大的常数$λ_{0} $，并且函数$φ$使得$ \cosφ（x）$仅具有简单的零，我们研究由$ f_ {a} $生成的同伴生成的双曲线属性。我们采用关键集合方法来表明，在功能$φ$的衍生产品的一些其他要求下，二次碰撞补偿了由于主要碰撞而导致的双曲线的弱化，而$ f_ {a} $产生的合子碰撞与二次碰撞可能被部分消除相反。
We consider a skew product $F_{A} = (σ_ω, A)$ over irrational rotation $σ_ω(x) = x + ω$ of a circle $\mathbb{T}^{1}$. It is supposed that the transformation $A: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ being a $C^{1}$-map has the form $A(x) = R(φ(x)) Z(λ(x))$, where $R(φ)$ is a rotation in $\mathbb{R}^{2}$ over the angle $φ$ and $Z(λ)= diag\{λ, λ^{-1}\}$ is a diagonal matrix. Assuming that $λ(x) \ge λ_{0} > 1$ with a sufficiently large constant $λ_{0}$ and the function $φ$ be such that $\cos φ(x)$ possesses only simple zeroes, we study hyperbolic properties of the cocycle generated by $F_{A}$. We apply the critical set method to show that, under some additional requirements on the derivative of the function $φ$, the secondary collisions compensate weakening of the hyperbolicity due to primary collisions and the cocycle generated by $F_{A}$ becomes hyperbolic in contrary to the case when secondary collisions can be partially eliminated.
