# Reference request: Chain recurrent set of suspension semiflow

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?