Solutions to the twisted cocycle equation over hyperbolic systems

Cohomology of fiber-bunched twisted cocycles over hyperbolic systems

“…Moreover, this result generalizes the main result of [22] in the case where G = GL(d, R). Indeed, it was observed in [22,Remark 3.4] that in this case, instead asking for the group to admit a bi-invariant metric (recall that GL(d, R) does not admit such a metric), one can assume some bounded distortion condition in the twisted cocycles. Roughly speaking, this condition asks for each of the terms on the lefthand side of (5) to be uniformly bounded.…”

“…In fact, the main results of those works can be obtained as corollaries of the previous one by taking α = Id. Moreover, this result generalizes the main result of [22] in the case where G = GL(d, R). Indeed, it was observed in [22,Remark 3.4] that in this case, instead asking for the group to admit a bi-invariant metric (recall that GL(d, R) does not admit such a metric), one can assume some bounded distortion condition in the twisted cocycles.…”

“…In order to prove our main result, we follow the approaches of [3], which in turn was inspired by [18,20], and [19,22]. The main idea consists of constructing invariant holonomies, which are a family of linear maps with good properties (see Proposition 3.1), and then using this family to explicitly construct the transfer map on a dense set under the additional assumption that f admits a fixed point.…”

“…The case where α is not the identity, on the other hand, despite its many applications, has received much less attention. To the best of the author's knowledge, the best result in this setting is a theorem by Walkden [22], who obtained an analogous result to the original Livšic's theorem under the assumptions that G is a connected Lie group admitting a bi-invariant metric * and the automorphism α satisfies some 'growth' conditions. The objective of this paper is to extend the results of [22] to the case when G = GL(d, R).…”

“…and the very same Hölder estimate holds for these maps when y, z ∈ W u ε (x). It is worth noting that the main ideas beyond this concept, although not under this name, were present in [22] (see also [18,20] for the case α = Id). On the other hand, the construction of these holonomies in that setting was greatly simplified owing to the existence of a bi-invariant metric.…”