Invariance entropy for a class of partially hyperbolic sets

Abstract

Invariance entropy is a measure for the smallest data rate in a noiseless digital channel above which a controller that only receives state information through this channel is able to render a given subset of the state space invariant. In this paper, we derive a lower bound on the invariance entropy for a class of partially hyperbolic sets. More precisely, we assume that $Q$ is a compact controlled invariant set of a control-affine system whose extended tangent bundle decomposes into two invariant subbundles $E^{+}$ and $E^{0-}$ with uniform expansion on $E^{+}$ and weak contraction on $E^{0-}$. Under the additional assumptions that $Q$ is isolated and that the $u$-fibers of $Q$ vary lower semicontinuously with the control $u$, we derive a lower bound on the invariance entropy of $Q$ in terms of relative topological pressure with respect to the unstable determinant. Under the assumption that this bound is tight, our result provides a first quantitative explanation for the fact that the invariance entropy does not only depend on the dynamical complexity on the set of interest.