On the Relation between Topological Entropy and Restoration Entropy

Abstract

In the context of state estimation under communication constraints, several notions of dynamical entropy play a fundamental role, among them: topological entropy and restoration entropy. In this paper, we present a theorem that demonstrates that for most dynamical systems, restoration entropy strictly exceeds topological entropy. This implies that robust estimation policies in general require a higher rate of data transmission than non-robust ones. The proof of our theorem is quite short, but uses sophisticated tools from the theory of smooth dynamical systems.

Keywords

• topological entropy
• restoration entropy
• state estimation under communication constraints
• SRB measures
• Anosov diffeomorphisms