Almost reducibility for families of sequences of matrices

Abstract

We consider the almost reducibility property of a nonautonomous dynamics with discrete time defined by a sequence of matrices. This corresponds to the reduction of the original nonautonomous dynamics to an autonomous dynamics via a coordinate change that preserves the Lyapunov exponents. In particular, we give a characterization of the almost reducibility of a sequence to a diagonal matrix and we use this result to characterize the class of matrices to which a given sequence is almost reducible. We also consider continuous 1 1 -parameter families of sequences of matrices and we show that the almost reducibility set of such a family is always an F σ δ F_{\sigma \delta } -set. In addition, we show that for any F σ δ F_{\sigma \delta } -set containing zero there exists a family with this set as its almost reducibility set.