Livšic's theorem for semisimple Lie groups (Q2757016)

The authors show a Livšic theorem for cocycles of transitive Anosov diffeomorphisms taking values in a connected non-compact linear semisimple Lie group \(G\). Namely, let \(T:M\to M\) be such a diffeomorphism and let \(g:M\to G\) be a map of class \(C^k(k\geq 1)\) with the additional property that \(0<{1 \over\lambda} <\|Ad g(X) \|<\lambda\) for some \(\lambda> 0\) and all \(x\in M\). If \(\mu\) is a \(T\)-invariant probability measure in the Lebesgue measure class and if \(h:M\to G\) is measurable and such that \(h(Tx)= g(x)h(x)\mu\) -- a.e. then \(h\) concides \(\mu\) -- a.e. with a map of class \(C^{k-1}\).NEWLINENEWLINENEWLINEThe proof is very short and elegant and uses old results of \textit{M. I. Brin} [Ergodic Theory Dyn. Syst. 2, 163-183 (1982; Zbl 0511.58032)] and \textit{M. I. Brin} and \textit{J. B. Pesin} [Math. USSR, Izv. 8, 177-219 (1974); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38, 170-212 (1974; Zbl 0304.58017)] on partially hyperbolic maps, applied to the skew-product \(\widehat T:M\times G/ \Gamma\to M\times G/ \Gamma\), \((x,g)\to (Tx,g(x)g)\) for where \(\Gamma\) is a cocompact lattice \(\Gamma\) in \(G\).