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Aim this article to provide an accessible introduction to the notion of topological entropy and (for context) its measure theoretic analogue, and then to present some 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 tow classes of examples, form 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 which I have freely adopted for motivation.