Theorems Symbolic Suspension Horseshoe for a Return Map Attributions

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Let $M$ be a smooth manifold, let $\Phi: \mathbb{R} \times M \to M$ be a continuous flow, let $\Sigma \subset M$ be a topological subspace, and let $P: D \to \Sigma$ be a homeomorphism from an open subset $D \subset \Sigma$ onto its image $P(D) \subset \Sigma$. Let $\tau: D \to (0,\infty)$ be a return-time function such that $\Phi_{\tau(x)}(x)=P(x)$ for every $x \in D$.

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Let $U \subset M$ be open. Assume that there exist an integer $m \in \mathbb{N}$, a compact set $R \subset D$ homeomorphic to $[0,1]^2$, and disjoint compact sets $R_0,R_1 \subset R$ with the following properties. Define $F:=P^m$ on its natural domain. The map $F$ is defined on an open neighbourhood of $R_0 \cup R_1$, and for every $N \in \mathbb{N}$ and every word $(a_{-N},\dots,a_N) \in \{0,1\}^{2N+1}$ the set

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\begin{align*} C(a_{-N},\dots,a_N):=\{x \in R : F^j(x) \text{ is defined and } F^j(x) \in R_{a_j} \text{ for every } -N \leq j \leq N\} \end{align*}

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is nonempty and compact. Assume further that, for any metric on $R$ inducing its topology,

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\begin{align*} \sup_{(a_{-N},\dots,a_N)\in\{0,1\}^{2N+1}}\operatorname{diam} C(a_{-N},\dots,a_N) \to 0 \end{align*}

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as $N \to \infty$.

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Define

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\begin{align*} K:=\{x \in R_0 \cup R_1 : F^n(x) \text{ is defined and } F^n(x) \in R_0 \cup R_1 \text{ for every } n \in \mathbb{Z}\}. \end{align*}

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Assume that $P^j$ is defined on $R_0 \cup R_1$ for every integer $j$ with $0 \leq j \leq m-1$, that $\tau$ is continuous at every point of $P^j(R_0 \cup R_1)$ for every such $j$, and define $\tau_m: R_0 \cup R_1 \to (0,\infty)$ by

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\begin{align*} \tau_m(x):=\sum_{j=0}^{m-1}\tau(P^j(x)). \end{align*}

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Assume that

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\begin{align*} \{\Phi_t(x):x \in R_0 \cup R_1,\ 0 \leq t \leq \tau_m(x)\}\subset U. \end{align*}

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Let $K_{\tau_m}$ be the suspension space over $F|_K$ with roof function $\tau_m|_K$, equivalently the quotient of $K\times\mathbb{R}$ by the [equivalence relation](/page/Equivalence%20Relation) generated by $(x,t+\tau_m(x))\sim(F(x),t)$ for $x\in K$ and $t\in\mathbb{R}$. Assume that the evaluation map $E:K_{\tau_m}\to M$ defined by

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\begin{align*} E([(x,t)]):=\Phi_t(x) \end{align*}

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is well-defined and injective.

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Then $K$ is nonempty and compact, $F(K)=K$, and $F^n(x)$ is defined for every $x\in K$ and every $n\in\mathbb{Z}$. There exists a homeomorphism $h:K\to\{0,1\}^{\mathbb{Z}}$ such that $h\circ F=\sigma\circ h$, where $\sigma:\{0,1\}^{\mathbb{Z}}\to\{0,1\}^{\mathbb{Z}}$ is the left shift. The roof function $\tau_m|_K$ is continuous and positive. The set

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\begin{align*} \Lambda:=\{\Phi_t(x):x\in K,\ 0\leq t\leq \tau_m(x)\} \end{align*}

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is a nonempty compact $\Phi$-invariant subset of $U$, and $E$ is a topological conjugacy from the suspension flow on $K_{\tau_m}$ to the restricted flow on $\Lambda$. Equivalently, the restricted flow on $\Lambda$ is topologically conjugate to the suspension flow over $(\{0,1\}^{\mathbb{Z}},\sigma)$ with roof function $\tau_m\circ h^{-1}$.

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