Let $(X,d)$ be a compact [metric space](/page/Metric%20Space), and let $\varphi:\mathbb{R}\times X\to X$ be a continuous flow, written $\varphi_t(x)=\varphi(t,x)$, such that $\varphi_0=\operatorname{id}_X$ and $\varphi_{t+s}=\varphi_t\circ\varphi_s$ for all $s,t\in\mathbb{R}$. Define the topological entropy $h_{\mathrm{top}}(\varphi)$ of the flow by the continuous-time separated-set convention, and define $h_{\mathrm{top}}(\varphi_a)$ as the topological entropy of the continuous map $\varphi_a:X\to X$ by the discrete-time separated-set convention. Then for every $a>0$,

\begin{align*} h_{\mathrm{top}}(\varphi_a)=a\,h_{\mathrm{top}}(\varphi). \end{align*}