Let $M$ be a smooth manifold and let $f: M \to M$ be a $C^r$ diffeomorphism, where $r \geq 1$. Suppose $f$ satisfies the hypotheses of the [Smale-Birkhoff Homoclinic Theorem](/theorems/7752) in the following local horseshoe form: there is a hyperbolic periodic point $p \in M$ and a transverse homoclinic point $q \in W^s(p) \cap W^u(p)$ not lying on the orbit of $p$, and for every neighbourhood $U \subset M$ of the homoclinic orbit

\begin{align*} \mathcal{O}(q) := \{f^j(q) : j \in \mathbb{Z}\} \end{align*}

the associated horseshoe construction can be made so that the whole finite return tower lies in $U$. More explicitly, for every such $U$ there are an integer $m\in\mathbb N$, a compact topological rectangle $R\subset M$, and two pairwise disjoint compact vertical strips $R_0,R_1\subset R$ such that, for $F:=f^m$, the union

\begin{align*} \bigcup_{j=0}^{m-1} f^j(R) \end{align*}

is contained in $U$, each $F(R_a)$ crosses $R$ as a horizontal strip for $a\in\{0,1\}$, and the following cylinder property holds. For every sequence $s=(s_n)_{n\in\mathbb Z}\in\{0,1\}^{\mathbb Z}$ and every $N\geq 1$, the finite-itinerary set

\begin{align*} K_N(s):=\{x\in R_0\cup R_1:F^n(x)\in R_{s_n}\text{ for every }-N\leq n\leq N\} \end{align*}

is nonempty and compact, the sets $K_N(s)$ are nested in $N$, and their diameters tend to $0$ as $N\to\infty$ with respect to a metric inducing the topology on $R$.

Then there exist an integer $m \in \mathbb{N}$ and a nonempty compact set $K \subset M$ such that $f^m(K) = K$ and the dynamical system

\begin{align*} f^m|_K: K \to K \end{align*}

is topologically conjugate to the full two-sided shift on two symbols

\begin{align*} \sigma: \{0,1\}^{\mathbb{Z}} \to \{0,1\}^{\mathbb{Z}}. \end{align*}

The set $K$ need not be invariant under $f$, but the finite union

\begin{align*} \widehat K := \bigcup_{j=0}^{m-1} f^j(K) \end{align*}

is compact and $f$-invariant. Moreover, for every neighbourhood $U \subset M$ of $\mathcal{O}(q)$, the construction may be carried out with $\widehat K \subset U$. Consequently, $f$ has infinitely many periodic orbits accumulating on $\mathcal{O}(q)$, in the sense that every neighbourhood of $\mathcal{O}(q)$ contains periodic orbits of $f$. Finally, if $h_{\mathrm{top}}(f)$ denotes topological entropy on a possibly noncompact space as the supremum of $h_{\mathrm{top}}(f|_C)$ over compact $f$-invariant sets $C \subset M$, then $h_{\mathrm{top}}(f)>0$.