Cohomology Of Cocycles Over Lie Group And Banach Ring On Anosov System

In this paper, we mainly prove two theorems.Firstly, let M be a compact Riemannian manifold, f : M â†' M be Anosov diffeomorphism, G be a connected Lie group, A : M â†' G be a β- H ¨older continuous function,A : M ×Z â†' G be a cocycle generated by A, and An p= eG, where p : fnp = p. Then there exists a β- H ¨older continuous function C : M â†' G, such that: A(x) = C( f x) ? B(x) ? C(x)-1.We get this conclusion by improving the theorem of M. Pollcott and C. P. Walkden, for Anosov system, center bunching property may be left out.Secondly, we consider that G be a Banach ring,the cocycles A and B have the same periodic data, we explore the conjugacy between A and B: cocycles A and B are β-H ¨older continuous and fiber bunched, in addition, A and B have the same periodic data,i.e. A(n, p) = B(n, p), where p : fnp = p, then there exists a β- H ¨older continuous function C : M â†' G, such that A(x) = C( f x) ? B(x) ? C(x)-1, x ∈ M.