Let $(X,d)$ be a compact [metric space](/page/Metric%20Space), let $f:X\to X$ be continuous, and let $\Lambda\subset X$ be a nonempty compact set with $f(\Lambda)\subseteq \Lambda$. Let $g:\Lambda\to\Lambda$ denote the restriction $g=f|_\Lambda$. Suppose that for some integer $k\ge 2$, the dynamical system $(\Lambda,g)$ factors onto the one-sided full $k$-shift indexed by $\mathbb{N}_0:=\{0,1,2,\dots\}$ or the two-sided full $k$-shift: that is, there exist a full shift space $\Sigma_k\in\{\{1,\dots,k\}^{\mathbb{N}_0},\{1,\dots,k\}^{\mathbb{Z}}\}$, its shift map $\sigma:\Sigma_k\to\Sigma_k$, and a continuous surjection $\pi:\Lambda\to\Sigma_k$ such that

\begin{align*} \pi\circ g=\sigma\circ\pi. \end{align*}

Then

\begin{align*} h_{\mathrm{top}}(f)\ge h_{\mathrm{top}}(f|_\Lambda)\ge \log k. \end{align*}

In particular, $h_{\mathrm{top}}(f)>0$.