A criterion for ''rough'' diagonalizability of systems of linear extensions of compact fluxes (Q1090830)

Consider a linear extension (1) \(dx/dt=P(\phi)x\), \(x\in {\mathbb{R}}^ n\), of a flow \(\phi_ t\) on a compact K such that \(\pi_ 1(K)=0\). The system (1) is said to be rough diagonalizable (r.d.) if for matrices Q near P there exist matrices S(Q) such that (i) \(S^{-1}(Q)QS(Q)-S^{- 1}(Q)\frac{dS(Q)}{dt}\) is diagonal; (ii) S(Q) is continuous with respect to Q. Necessary and sufficient conditions for the system (1) to be r.d. in terms of exponential dichotomy for some perturbed systems is obtained.