Preservation of Lyapunov Exponents

This chapter gives a complete characterization of when a dynamics can be decomposed into invariant blocks, up to a coordinate change that leaves the Lyapunov exponent unchanged. In particular, we show that a sequence of invertible matrices can be reduced to a sequence of block matrices with upper-triangular blocks if and only if the space can be decomposed into an invariant splitting such that the angles between complementary invariant subspaces form a tempered sequence. As a nontrivial consequence, we show that any regular sequence of invertible matrices can be reduced to a constant sequence of diagonal matrices. This certainly reminds us of the behavior of the angles between vectors for a regular two-sided sequence of matrices, but here we are only considering one-sided sequences and thus, the results are stronger. On the other hand, the regularity of a two-sided sequence requires a certain compatibility between the forward and backward asymptotic behavior, which has no correspondence for a one-sided sequence. Finally, as an application of the reducibility of a regular one-sided sequence to a constant sequence of matrices, we determine all coordinate changes preserving simultaneously the Lyapunov exponents of all sequences of invertible matrices. These turn out to be the Lyapunov coordinate changes. We purposely avoid duplicating the theory for discrete and continuous time. Instead, we develop in detail the theory for discrete time and we give appropriate references for continuous time.