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.
"Topological entropy and the preimage structure of maps.." Real Anal. Exchange 29 (1) 9 - 42, 2003-2004.