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 at most subexponential expansion 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.