Invariance entropy for a class of partially hyperbolic sets

TL;DR

This paper establishes a lower bound on invariance entropy for certain partially hyperbolic control sets, linking it to topological pressure and dynamical complexity, advancing understanding of information requirements for control systems.

Contribution

It introduces a novel lower bound on invariance entropy for partially hyperbolic sets with specific invariant subbundle structures, connecting it to topological pressure.

Findings

Provides a lower bound on invariance entropy in terms of topological pressure.

Shows the bound is tight under certain conditions, explaining entropy's dependence on dynamical complexity.

Offers a quantitative insight into the information rate needed for control of partially hyperbolic systems.

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.