Scaled Entropy for Dynamical Systems

Abstract

In order to characterize the complexity of a system with zero entropy we introduce the notions of scaled topological and metric entropies. We allow asymptotic rates of the general form \(e^{\alpha a(n)}\) determined by an arbitrary monotonically increasing “scaling” sequence \(a(n)\). This covers the standard case of exponential scale corresponding to \(a(n)=n\) as well as the cases of zero and infinite entropy. We describe some basic properties of the scaled entropy including the inverse variational principle for the scaled metric entropy. Furthermore, we present some examples from symbolic and smooth dynamics that illustrate that systems with zero entropy may still exhibit various levels of complexity.