A reducibility theorem for smooth quasiperiodic linear systems

AbstractWe construct an iterative procedure for finding a change of variables to reduce the linear system. x′ = Ax + P(ϑ)x ϑ′ = ω where P(ϑ) is l times differentiable, to a system with constant coefficients. Under certain conditions on ω and the eigenvalues of A we use the technique of accelerated convergence to overcome the difficulty of small divisors and show that this sequence of transformations converges to a quasiperiodic transformation. As is always the case in such problems, there is an inevitable loss of derivatives. The best previous result, due to Mitropol'skǐi and Samoǐlenko required l>k(k − 1)(2 − k)[k(m + τ) + 2m + 2], where κ is the exponent of the accelerated convergence (1 < κ < 2), and τ is a constant occurring in the relationship between the eigenvalues of A and ω. Our result requires only that l>τ