Lyapunov exponents and partial hyperbolicity of chain control sets on flag manifolds

Abstract

For a right-invariant control system on a flag manifold ${\mathbb{F}}_{\Theta }$ of a real semisimple Lie group, we relate the $\mathfrak{a}$-Lyapunov exponents to the Lyapunov exponents of the system over regular points. Moreover, we adapt the concept of partial hyperbolicity from the theory of smooth dynamical systems to control-affine systems, and we completely characterize the partially hyperbolic chain control sets on ${\mathbb{F}}_{\Theta }$.