Let $(X,d)$ be a compact [metric space](/page/Metric%20Space) and let $T:X \to X$ be continuous. For $n\ge 1$, define the map $d_n:X\times X\to [0,\infty)$ by
Let $s_n(\varepsilon)$ be the largest cardinality of an $(n,\varepsilon)$-separated subset of $X$ and let $r_n(\varepsilon)$ be the least cardinality of an $(n,\varepsilon)$-spanning subset of $X$. Then