Livsic theorems for connected Lie groups (Q2719052)

This paper is devoted to the extension of the well-known Livšič theorem related to hyperbolic dynamics. Let \(\varphi\) be a hyperbolic diffeomorphism on a basic set \(\Lambda\) and let \(G\) be a connected Lie group. Let \(f:\Lambda\to G\) be Hölder. The authors show that, under some natural assumption on \(f\), the following holds: \(f=u\varphi\cdot u^{-1}\) for some Hölder \(u\). Further, assuming ``some kind of'' hyperbolicity assumption, they show that if \(u:\Lambda\to G\) is a measurable solution to \(f=u \varphi \cdot u^{-1}\) a.e., then \(u\) must be Hölder. These results extend the Livšič theorem when \(G\) is compact or abelian.