Topological Entropy and the Preimage Structure of Maps

My aim in this article is to provide an accessible introduction to the notion of topological entropy and (for context) its measure theoretic analogue, and then to present some recent work applying related ideas to the structure of iterated preimages for a continuous (in general non-invertible) map of a compact metric space to itself. These ideas will be illustrated by two classes of examples, from circle maps and symbolic dynamics. My focus is on motivating and explaining definitions; most results are stated with at most a sketch of the proof. The informed reader will recognize imagery from Bowen’s exposition of topological entropy [Bow78] which I have freely adopted for motivation. 1 Measure-theoretic entropy How much can we learn from observations using an instrument with finite resolution? A simple model of a single observation on a ”state space” X is a finite partition P = {A1, . . . , AN} of X into atoms, grouping the points (states) in X according to the reading they induce on our instrument. A measure μ on X with total measure μ(X) = 1 defines the probability of a given reading as pi = μ(Ai), i = 1, . . . , N. Shannon [Sha63] (see also [Khi57]) noted that the ”entropy” of the partition