Livšic theorems for non-commutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems
Livšic theorems for non-commutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems
复制标题
包括微分同胚群在内的非交换群的利夫希克定理以及阿诺索夫系统共形结构存在性的结果
DOI：
10.1017/s014338570900039x
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发表时间：
2007
影响因子：
0.9
通讯作者：
A. Windsor
中科院分区：
文献类型：
--
作者：
R. Llave;A. Windsor
著名的Livšic定理[A. N. Livšic，Y-系统同调的某些性质，Mat。Zametki 10（1971），555-564; A. N. Livšic，动力系统的上同调，Izv。阿卡德Nauk SSSR序列号36（1972），1296-1320]指出，给定一个流形M，一个李群G，一个M上的传递Anosov同构f和一个Hölder函数η：M <$G，其范围足够接近恒等式，满足η（x）=<$（f（x））<$（x）−1的<$：M <$G的存在性是足够的，满足一个条件-显然是必要的-关于由η生成的限制于周期轨道的上圈。本文给出了主要结果的一个新的证明。这些方法使我们能够处理取值于紧致流形的复同态群中的余圈。这在刚性理论中也有应用。我们开发的本地化程序可以应用到Anosov系统的共形结构的存在性获得一些新的结果。
Abstract The celebrated Livšic theorem [A. N. Livšic, Certain properties of the homology of Y-systems, Mat. Zametki 10 (1971), 555–564; A. N. Livšic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1296–1320] states that given a manifold M, a Lie group G, a transitive Anosov diffeomorphism f on M and a Hölder function η:M↦G whose range is sufficiently close to the identity, it is sufficient for the existence of ϕ:M↦G satisfying η(x)=ϕ(f(x))ϕ(x)−1 that a condition—obviously necessary—on the cocycle generated by η restricted to periodic orbits is satisfied. In this paper we present a new proof of the main result. These methods allow us to treat cocycles taking values in the group of diffeomorphisms of a compact manifold. This has applications to rigidity theory. The localization procedure we develop can be applied to obtain some new results on the existence of conformal structures for Anosov systems.