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Consider the discrete-time dynamical system defined by a continuous map $f:X\to X$ of a compact metric space. Let $M_f = (X\times [0,1]) / \sim$ be the mapping torus and let $\Phi\colon M_f \times [0,\infty) \to M_f$ be the suspension semiflow. Let $\mathcal{R}(f)$ and $\mathcal{R}(\Phi)$ denote the chain recurrent sets of $f$ and $\Phi$, respectively, and let $i:X\hookrightarrow M_f$ be the embedding $X\hookrightarrow [X\times \{0\}] \subset M_f$.

I suspect that it is well-known that $\mathcal{R}(f) = i^{-1}(\mathcal{R}(\Phi))$, and I'm hoping to cite this in a paper, I but I haven't managed to find a reference in the literature. Could anyone please point me to a suitable reference?

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