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arXiv 1209.2293math.DS
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具有自由碱和纤维动力学的SL（2，R）上环的简单谱

Trivial and simple spectrum for SL(2,R) cocycles with free base and fiber dynamics

Mario Bessa, Paulo Varandas

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8 pages, revised version
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摘要：

令$AC_D(M,SL(2,\mathbb R))$表示对$(f,A)$，使得$f\in \mathcal A\subset \text{Diff}^{1}(M)$是$C^{1}$ -Anosov传递同构，$A$是关于$f$占优的$SL(2,\mathbb R)$上圈.我们证明了在$AC_D(M,SL(2,\mathbb R))$中（在适当的拓扑中）开稠密的对$(f,A)$关于唯一的最大熵测度$μ_f$具有简单谱.另一方面，相对于单独的拓扑，存在剩余子集$\mathcal{R}\subset \text{Aut}_{Leb}(M)\times L^\infty(M,SL(2,\mathbb R))$，使得$\mathcal{R}$中的任何元素$(f,A)$具有平凡谱或者是双曲谱。然后，我们证明了平凡谱的流行附近的动力学上循环的面积保持映射，也为一般的上循环$\text{Aut}_{Leb}(M)\times L^p(M,SL(2,\mathbb R))$。

英文摘要：

Let $AC_D(M,SL(2,\mathbb R))$ denote the pairs $(f,A)$ so that $f\in \mathcal A\subset \text{Diff}^{1}(M)$ is a $C^{1}$-Anosov transitive diffeomorphisms and $A$ is an $SL(2,\mathbb R)$ cocycle dominated with respect to $f$. We prove that open and densely in $AC_D(M,SL(2,\mathbb R))$ (in appropriate topologies) the pair $(f,A)$ has simple spectrum with respect to the unique maximal entropy measure $μ_f$. On the other hand, there exists a residual subset $\mathcal{R}\subset \text{Aut}_{Leb}(M)\times L^\infty(M,SL(2,\mathbb R))$, with respect to the separate topology, such that any element $(f,A)$ in $\mathcal{R}$ has trivial spectrum or it is hyperbolic. Then, we prove prevalence of trivial spectrum near the dynamical cocycle of an area-preserving map and also for generic cocycles in $\text{Aut}_{Leb}(M)\times L^p(M,SL(2,\mathbb R))$.

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