Metric Entropy and Topological Entropy

Abstract

This chapter is dedicated to the study of metric entropy, including its relation to topological entropy. After establishing some basic properties of metric entropy, we consider the notion of conditional entropy, and we show how generators can be used to compute metric entropy. We then establish the Shannon–McMillan–Breiman theorem, which can be seen as the fundamental theorem of entropy theory. In particular, it shows that metric entropy can be computed in terms of an invariant local quantity. We also introduce the notion of topological entropy for a continuous transformation of a compact metric space, and we establish the variational principle showing that the topological entropy is the supremum of the metric entropies over all invariant probability measures.