Maximal linear transformation groups preserving the asymptotic properties of linear differential systems: II

We consider linear piecewise differentiable transformations of linear differential systems. We show that the set of such transformations preserving the Lyapunov exponents of each system with finite Lyapunov exponents coincides with the group of generalized Lyapunov transformations. In addition, we show that the set of transformations bringing each linear system with bounded coefficients on the half-line to a system of the same class and preserving its Lyapunov exponents is a group containing the group of Lyapunov transformations as a proper subgroup, and we describe that group. We obtain descriptions of other groups and semigroups of linear piecewise differentiable transformations considered in the theory of Lyapunov exponents and the theory of stability.