Let $R \subset \mathbb{R}^2$ be a compact rectangle with the Euclidean [subspace topology](/page/Subspace%20Topology), let $F: R \to R$ be continuous on every invariant set mentioned below, and let $\Lambda \subset R$ be a nonempty compact set such that $F(\Lambda)=\Lambda$. Define
by $(\sigma a)_k = a_{k+1}$ for every $a = (a_k)_{k \in \mathbb{Z}} \in \Sigma_2$ and every $k \in \mathbb{Z}$. Assume the Smale horseshoe coding data are explicitly supplied: there exists a homeomorphism
\begin{align*}
h \circ F|_{\Lambda} = \sigma \circ h.
\end{align*}
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Then $F|_{\Lambda}: \Lambda \to \Lambda$ is topologically transitive, has dense periodic points, has sensitive dependence on initial conditions with respect to the Euclidean metric restricted to $\Lambda$, and has topological entropy
If a map satisfies only the branch-crossing part of the horseshoe construction, but the construction still supplies a nonempty compact invariant subsystem $\Lambda_0 \subset R$ and a homeomorphism
and $F|_{\Lambda_0}$ is topologically transitive, has dense periodic points, and has sensitive dependence on initial conditions. No conclusion is asserted about whether the larger maximal invariant set in $R$ has exactly the same dynamics; it may contain additional invariant orbits.