Later, Dinaburg and Rufus Bowen gave a different, weaker definition reminiscent of the Hausdorff dimension.
The second definition clarified the meaning of the topological entropy: for a system given by an iterated function, the topological entropy represents the exponential growth rate of the number of distinguishable orbits of the iterates.
An important variational principle relates the notions of topological and measure-theoretic entropy.
Its topological entropy is a nonnegative extended real number that can be defined in various ways, which are known to be equivalent.
For any continuous map f: X → X, the following limit exists: Then the topological entropy of f, denoted h(f), is defined to be the supremum of H(f,C) over all possible finite covers C of X.
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.
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.