Bowen metric ($d_n(x,y)={\max }_{0\le k<n}d(f^{k}x,f^{k}y)$)

For a continuous map on a compact metric space, the Bowen metric compares the first $n$ points of two forward orbits. Its radius-$ϵ$ ball is $B_n(x,ϵ)=\{y:d_n(x,y)<ϵ\}$. If $s_n(ϵ)$ is the largest cardinality of an $ϵ$-separated set in this metric, then topological entropy is ${\lim }_{ϵ\downarrow 0}{\mathrm{limsup}}_{n\to \infty }{n}^{-1}\log s_n(ϵ)$. The uniform continuity of the identity between two compatible metrics gives the comparison of separated-set counts needed for independence of the original metric.

A radius-$ϵ$ ball for a Bowen metric consists of the points whose first $n$ iterates stay within $ϵ$ of the corresponding iterates of the centre. Balls at a fixed small radius distinguish forward orbit segments at that observational resolution.