ABSTRACT
For continuous transformations on a compact metric space, one can define a topological analogue of the metric entropy by replacing the measurable partitions in the definition of metric entropy by open covers. We start by giving three equivalent ways of defining topological entropy. A proof of the Misiurewicz and Szlenk Theorem is given that derives an explicit formula for the calculation of the topological entropy of a continuous piecewise monotone map on the unit interval. We then prove the Variational Principle, stating the topological entropy is the supremum of the metric entropy over all invariant measures. We end the chapter with a study of measures of maximal entropy, that is, measures with corresponding metric entropy equal to the topological entropy.
