Let $U \subset \mathbb{R}^2$ be open, let $F: U \to U$ be a $C^1$ diffeomorphism, and let $R \subset U$ be a compact rectangle. Suppose that $F$ has a two-strip horseshoe structure on $R$ with horizontal strips $H_0,H_1 \subset R$ and vertical strips $V_0,V_1 \subset R$, in the following sense:
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For each $i \in \{0,1\}$, the restricted map $F|_{H_i}: H_i \to V_i$ is a $C^1$ diffeomorphism onto $V_i$, and
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\begin{align*}
R \cap F^{-1}(R) = H_0 \cup H_1.
\end{align*}
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Assume that the strips $H_0$ and $H_1$ are separated in $R$, that
is the maximal invariant set in $R$, and assume that $H_0 \cap \Lambda$ and $H_1 \cap \Lambda$ are relatively open and relatively closed in $\Lambda$.
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Assume the following explicit Markov-cylinder conclusions hold. Let $\mathcal{B}_{\mathrm{nb}}(\mathbb{R}^2)$ denote the set of all nonempty bounded subsets of $\mathbb{R}^2$. Define the diameter map $\operatorname{diam}:\mathcal{B}_{\mathrm{nb}}(\mathbb{R}^2)\to[0,\infty)$ by
For every integer $N \geq 0$ and every word $w=(w_{-N},\dots,w_N) \in \{0,1\}^{\{-N,\dots,N\}}$, define
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\begin{align*}
C_N(w) := \{x \in R : F^k(x) \in H_{w_k} \text{ for every } k \in \{-N,\dots,N\}\}.
\end{align*}
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Assume that each such set $C_N(w)$ is nonempty and compact. Moreover, for every $s=(s_n)_{n \in \mathbb{Z}} \in \{0,1\}^{\mathbb{Z}}$, if $w_N=(s_{-N},\dots,s_N)$, then $C_{N+1}(w_{N+1}) \subset C_N(w_N)$ for every $N \geq 0$ and
Let $\Sigma_2 := \{0,1\}^{\mathbb{Z}}$ with the [product topology](/page/Product%20Topology), and define the full two-shift $\sigma:\Sigma_2 \to \Sigma_2$ by