On the structure of uniformly hyperbolic chain control sets

TL;DR

This paper proves that for uniformly hyperbolic chain control sets without center bundle, the extended state space lift forms a graph over control functions, ensuring unique state trajectories for each control.

Contribution

It establishes a novel structural property of uniformly hyperbolic chain control sets, showing the lift is a graph over control functions, which was previously unknown.

Findings

Lift of Q is a graph over control space U

Unique state trajectory for each control in U

Structural insight into hyperbolic control sets

Abstract

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