Variational principles of invariance entropy dimension

Abstract

We introduce two types of invariance entropy dimensions (α-invariance entropies), measure-theoretic invariance entropy dimensions (measure-theoretic α-invariance entropies), and measure-theoretic local invariance entropy dimensions (measure-theoretic local α-invariance entropy) to investigate the complexity of a control system with zero invariance entropy, where $\alpha \in R_{+}$. Under some reasonable assumptions, we respectively present two variational principles and inverse variational principles for α-invariance entropies and invariance entropy dimensions, and show that Bowen α-invariance entropy (Bowen invariance entropy dimension) and packing α-invariance entropy (packing invariance entropy dimension) can be determined via the corresponding local entropies of measures.