Theorems Cantor Structure and Local Product Coordinates for a Two-Strip Horseshoe Attributions

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Let $(R,d)$ be a nonempty compact [metric space](/page/Metric%20Space), let $U$ be an open neighbourhood of $R$, and let $F:U\to F(U)$ be a homeomorphism onto its image. Assume that the iterates and inverse branches appearing below are defined on the relevant subsets of $R$.

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Suppose that $R$ contains two disjoint compact sets $H_0,H_1\subset R$, called Markov strips, with the following two-strip horseshoe property. For every pair of integers $m,n\geq 0$ and every word $(s_{-m},\dots,s_n)\in\{0,1\}^{m+n+1}$, define the cylinder

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\begin{align*} C[s_{-m},\dots,s_n]:=\{x\in R: F^k(x)\in H_{s_k}\text{ for every }-m\leq k\leq n\}, \end{align*}

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where negative iterates are interpreted using the inverse branch determined by the preceding strip itinerary. Each such cylinder is a nonempty compact subset of $R$. If the word is extended by either symbol at the next positive or negative time, the two corresponding descendant cylinders are disjoint compact subsets of the parent cylinder. Finally, the cylinder diameters tend to $0$ uniformly as $m,n\to\infty$.

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Define the maximal invariant horseshoe set

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\begin{align*} \Lambda:=\{x\in R: F^k(x)\in H_0\cup H_1\text{ for every }k\in\mathbb Z\}. \end{align*}

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Assume that for every bi-infinite sequence $s=(s_k)_{k\in\mathbb Z}\in\{0,1\}^{\mathbb Z}$ the intersection

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\begin{align*} \bigcap_{m,n\geq 0} C[s_{-m},\dots,s_n] \end{align*}

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consists of exactly one point, and that the coding map

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\begin{align*} \pi:\{0,1\}^{\mathbb Z}\to\Lambda \end{align*}

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which sends $s$ to this unique point is a homeomorphism.

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Let

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\begin{align*} \Sigma^-:=\{0,1\}^{\mathbb Z_{<0}},\qquad \Sigma^+:=\{0,1\}^{\mathbb N_0} \end{align*}

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with their product topologies, and let

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\begin{align*} \Theta:\Sigma^-\times\Sigma^+\to\{0,1\}^{\mathbb Z} \end{align*}

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be the concatenation map defined by $(\Theta(s^-,s^+))_i=s^-_i$ for $i<0$ and $(\Theta(s^-,s^+))_i=s^+_i$ for $i\geq 0$. Assume that the local unstable plaques in $\Lambda$ are exactly the sets $(\pi\circ\Theta)(\Sigma^-\times\{s^+\})$ with $s^+\in\Sigma^+$ fixed, and that the local stable plaques in $\Lambda$ are exactly the sets $(\pi\circ\Theta)(\{s^-\}\times\Sigma^+)$ with $s^-\in\Sigma^-$ fixed.

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Then $\Lambda$ is compact, perfect, and totally disconnected. Moreover, if $m,n\geq 0$ and $(a_{-m},\dots,a_n)\in\{0,1\}^{m+n+1}$, then the finite cylinder neighbourhood

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\begin{align*} \Lambda[a_{-m},\dots,a_n]:=\pi(\{s\in\{0,1\}^{\mathbb Z}:s_i=a_i\text{ for every }-m\leq i\leq n\}) \end{align*}

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is homeomorphic, under the restriction of $\pi\circ\Theta$, to the product

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\begin{align*} \Sigma^-[a_{-m},\dots,a_{-1}]\times\Sigma^+[a_0,\dots,a_n], \end{align*}

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where

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\begin{align*} \Sigma^-[a_{-m},\dots,a_{-1}]:=\{s^-\in\Sigma^-:s^-_i=a_i\text{ for every }-m\leq i<0\} \end{align*}

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and

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\begin{align*} \Sigma^+[a_0,\dots,a_n]:=\{s^+\in\Sigma^+:s^+_i=a_i\text{ for every }0\leq i\leq n\}. \end{align*}

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These past and future cylinder factors are Cantor sets. Hence $\Lambda$ has local product structure by stable and unstable Cantor sets in the symbolic coordinates supplied by $\pi$.

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