We consider Hölder continuous cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold $\mathcal {M}$ . We obtain several results for this setting. If a cocycle is bounded in $C^{1+\gamma }$ , we show that it has a continuous invariant family of $\gamma $ -Hölder Riemannian metrics on $\mathcal {M}$ . We establish continuity of a measurable conjugacy between two cocycles assuming bunching or existence of holonomies for both and pre-compactness in $C^0$ for one of them. We give conditions for existence of a continuous conjugacy between two cocycles in terms of their cycle weights. We also study the relation between the conjugacy and holonomies of the cocycles. Our results give arbitrarily small loss of regularity of the conjugacy along the fiber compared to that of the holonomies and of the cocycle.

中文翻译：

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微分同胚在部分双曲系统上共循环

我们考虑一个可访问的部分双曲线系统上的 Hölder 连续余环，其值在紧流形的微分同胚群中$\数学{M}$. 我们为此设置获得了几个结果。如果一个 cocycle 有界$C^{1+\伽玛}$，我们证明它有一个连续不变的族$\伽马$-Hölder 黎曼度量$\数学{M}$. 我们在两个 cocycle 之间建立了可测量共轭的连续性，假设两者都存在聚束或存在 holonomies 和 pre-compactness$C^0$为其中之一。我们根据它们的循环权重给出了两个共循环之间存在连续共轭的条件。我们还研究了共环的共轭性和完全性之间的关系。我们的结果给出了沿着纤维的共轭规律性的任意小的损失与完整的和 cocycle 的相比。