Let $(X,d)$ be a compact [metric space](/page/Metric%20Space), let $f: X \to X$ be continuous, and let $N \in \mathbb{N}$. Suppose that $\Lambda \subset X$ is a compact set such that $f^N(\Lambda) \subset \Lambda$. Let $\Sigma_2 := \{0,1\}^{\mathbb{Z}}$ be the full two-sided shift space, and let $\sigma: \Sigma_2 \to \Sigma_2$ be the shift map given by $\sigma((a_k)_{k \in \mathbb{Z}}) = (a_{k+1})_{k \in \mathbb{Z}}$.

Assume that the restricted system $f^N|_\Lambda: \Lambda \to \Lambda$ factors onto the full two-shift: there exists a continuous surjection

\begin{align*} \pi: \Lambda \to \Sigma_2 \end{align*}

such that

\begin{align*} \pi \circ (f^N|_\Lambda) = \sigma \circ \pi. \end{align*}

Then

\begin{align*} h_{\mathrm{top}}(f) \geq \frac{\log 2}{N}. \end{align*}