map f. Definition of Bowen and Dinaburg This definition uses a metric on X (actually, a uniform structure would suffice). This is a narrower definition than that of Adler, Konheim, and McAndrew, as it requires the additional metric structure on the topological space (but is independent of the choice of metrics generating the given topology). However, in practice, the Bowen-Dinaburg topological entropy is usually much easier to calculate. Let (X, d) be a compact metric space and f: X → X be a continuous map. For each natural number n, a new metric dn is defined on X by the formula Given any ε > 0 and n ≥ 1, two points of X are ε-close with respect to this metric if their first n iterates are ε-close. This metric allows one to distinguish in a neighborhood of an orbit the points that move away from each other during the iteration from the points that travel together. A subset E of X is said to be (n, ε)-separated if each pair of distinct points of E is at least ε apart in the metric dn. Denote by N(n, ε) the maximum cardinality of an (n, ε)-separated set. The topological entropy of the map f is defined by Interpretation Since X is compact, N(n, ε) is finite and represents the number of distinguishable orbit segments of length n, assuming that we cannot distinguish points within ε of one another. A straightforward argument shows that the limit defining h(f) always exists in the extended real line (but could be infinite). This limit may be interpreted as the measure of the average exponential growth of the number of distinguishable orbit segments. In this sense, it measures complexity of the topological dynamical system (X, f). Rufus Bowen extended this definition of topological entropy in a way which permits X to be non-compact under the assumption that the map f is uniformly continuous. Properties Topological entropy is an invariant of topological dynamical systems, meaning that it is preserved by topological conjugacy. Let be an expansive homeomorphism of a compact metric space and let be a topological generator. Then the topological entropy of relative to is equal to the topological entropy of , i.e. Let be a continuous transformation of a compact metric space , let be the measure-theoretic entropy of with respect to and let be the set of all -invariant Borel probability measures on