On the structure of uniformly hyperbolic chain control sets

Abstract

The following theorem is proved: Let Q be an isolated compact chain control set of a control-affine system on a smooth manifold M. If Q is uniformly hyperbolic without center bundle, then the lift of Q to the extended state space U×M, where U is the space of control functions, is a graph over U. In other words, for every control u∈U there is a unique x∈Q such that the corresponding state trajectory φ(t,x,u) evolves in Q.