Cohomology of fiber bunched cocycles over hyperbolic systems
Cohomology of fiber bunched cocycles over hyperbolic systems
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双曲系统上纤维束余循环的上同调
DOI：
10.1017/etds.2014.43
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发表时间：
2013
影响因子：
0.9
通讯作者：
V. Sadovskaya
中科院分区：
文献类型：
--
作者：
V. Sadovskaya
考虑Anosov同态上的Hölder连续纤维束$\text{GL}（d，\mathbb{R}）$-值上圈.我们证明了两个这样的上循环是Hölder连续上同调的，如果他们有相等的周期数据，并证明了一个结果的共轭周期数据的上循环。得到了任意常数上圈与其小扰动之间的上同调的一个推论。纤维聚束条件意味着上循环的非共形性由基底的膨胀和收缩控制。我们表明，这个条件可以建立在周期性的数据。上圈的一些重要例子来自于一个自同态的微分及其对不变子丛的限制。我们讨论了应用我们的结果的问题是否Anosov同构光滑共轭的$C^{1}$-小扰动。我们还建立了一个可测的共轭之间的纤维束上圈和一致拟共形的Hölder连续性。我们的主要结果也适用于闭子群$\text{GL}（d，\mathbb{R}）$中的值的上循环，双曲集上的上循环和有限型移位，以及非平凡向量丛上的线性上循环。
We consider Hölder continuous fiber bunched $\text{GL}(d,\mathbb{R})$-valued cocycles over an Anosov diffeomorphism. We show that two such cocycles are Hölder continuously cohomologous if they have equal periodic data, and prove a result for cocycles with conjugate periodic data. We obtain a corollary for cohomology between any constant cocycle and its small perturbation. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base. We show that this condition can be established based on the periodic data. Some important examples of cocycles come from the differential of a diffeomorphism and its restrictions to invariant sub-bundles. We discuss an application of our results to the question of whether an Anosov diffeomorphism is smoothly conjugate to a $C^{1}$-small perturbation. We also establish Hölder continuity of a measurable conjugacy between a fiber bunched cocycle and a uniformly quasiconformal one. Our main results also hold for cocycles with values in a closed subgroup of $\text{GL}(d,\mathbb{R})$, for cocycles over hyperbolic sets and shifts of finite type, and for linear cocycles on a non-trivial vector bundle.