= Bowen metric
{c}
{title2=$d_n(x,y)=\max_{0\leq k<n}d(f^kx,f^ky)$}
For a <continuous map> on a <compact metric space>, the Bowen metric compares the first $n$ points of two forward <orbits>. Its radius-$\epsilon$ ball is $B_n(x,\epsilon)=\{y:d_n(x,y)<\epsilon\}$. If $s_n(\epsilon)$ is the largest cardinality of an $\epsilon$-<separated set> in this <metric>, then <topological entropy> is $\lim_{\epsilon\downarrow0}\limsup_{n\to\infty}n^{-1}\log s_n(\epsilon)$. The <uniform continuity> of the identity between two <compatible metrics> gives the comparison of separated-set counts needed for independence of the original <metric>.
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