Sequences of Matrices

In this chapter we study in detail the class of Lyapunov exponents obtained from a linear dynamics with discrete time. More precisely, we consider the Lyapunov exponent obtained from a sequence of matrices as in Chapter 2 . In particular, we obtain lower and upper bounds for the Grobman coefficient, which are quite useful for nonregular sequences. Whereas the lower bound is established without further hypotheses, the upper bound is obtained assuming that the matrices are upper-triangular, which allows for a considerable simplification of the exposition. Nevertheless, we also show that from the point of view of the theory of regularity, one can always reduce the study of an arbitrary sequence of matrices to the study of an upper-triangular sequence. More precisely, there is a coordinate change by orthogonal matrices, which thus keeps the Lyapunov exponent unchanged, bringing the sequence to one that is upper-triangular. In addition, we give several alternative characterizations of regularity, in particular in terms of exponential growth rates of volumes. Moreover, for regular sequences of matrices, we show that the Lyapunov exponent is always a limit and that the angles between the images of two vectors cannot decrease exponentially with time. Finally, we present the stronger notion of regularity for two-sided sequences of matrices. In this case there is a splitting into invariant subspaces on which the convergence is uniform. The notion plays an important role, in particular in connection with ergodic theory.