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

TL;DR

This paper explores the relationship between Lyapunov exponents and partial hyperbolicity in control systems on flag manifolds, providing a complete characterization of partially hyperbolic chain control sets.

Contribution

It introduces a novel connection between $rak{a}$-Lyapunov exponents and system exponents, and adapts partial hyperbolicity concepts to control-affine systems on flag manifolds.

Findings

Relation between $rak{a}$-Lyapunov exponents and system exponents established

Complete characterization of partially hyperbolic chain control sets achieved

Framework for analyzing control systems on flag manifolds developed

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 $\mathbb{F}_{\Theta}$.