Reducible Linear Quasi{periodic Systems with Positive Lyapunov Exponent and Varying Rotation Number Consider a Class of Linear Diierential Equations

A linear system in two dimensions is studied. The coeecients are 2{periodic on three angles, j ; j = 1; 2; 3; and these angles are linear with respect to time, with incommensurable frequencies. The system has positive Lyapunov coeecients and the rotation number changes in a continuous way when some parameter moves. A lift to T 3 R 2 ; however, is only of class L p ; for any p < 2: (1) where M 2 Sl(2; R); A(t) 2 sl(2; R) and _ = d=dt: Corresponding evolutions frequently occur as fundamental solutions of second order equations of Hill (or Schrr odinger) type x + q(t)x = 0: (2) Generically also the converse is true: the equation (1) can be associated to a second order equation of the form (2). 1 Preliminaries Presently we consider an example of this where M(t; t 0) = C(t) e t 0 0 e ?t ! C(t 0) ?1 ; (3) for > 0; where C(t) 2 SO(2; R): First note that this ow is reducible, in the sense that the transformation C renders it into a ow generated by the constant matrix 0 0 ? ! : Let us further specify C(t) = v 1 (t) ?v 2 (t) v 2 (t) v 1 (t) ! with v 2 1 + v 2 2 1: