  This paper studies set-invariance and stabilization of hyperbolic sets over
rate-limited channels for discrete-time control systems. We first investigate
structural and control-theoretic properties of hyperbolic sets, in particular
such that arise by adding small control terms to uncontrolled systems admitting
(classical) hyperbolic sets. Then we derive a lower bound on the invariance
entropy of a hyperbolic set in terms of the difference between the unstable
volume growth rate and the measure-theoretic fiber entropy of associated random
dynamical systems. We also prove that our lower bound is tight in two extreme
cases. Furthermore, we apply our techniques to the problem of local uniform
stabilization to a hyperbolic set. Finally, we discuss an example built on the
H\'enon horseshoe.
