Measuring Complexity in Dynamical Systems
Benjamin Wilson · Carolina Digital Repository (University of North Carolina at Chapel Hill) · 2019
Measuring the complexity of dynamical systems is important in order to classify them and better understand them. In 1958 Kolmogorov introduced to ergodic theory an analogue of Shannon's information-theoretic entropy as a measure of disorder or uncertainty in a system. Based on this concept and ideas from neuroscience and information theory, we define the intricacy and average sample complexity of a topological dynamical system and a measure-preserving dynamical system. We examine these new complexity measurements in both the topological and measure-theoretic settings, including analysis of symbolic dynamical systems and Markov shifts. We compare these measurements to the usual measure-theoretic and topological entropies, give some properties of these quantities, and look at some questions that they raise.