Let $M$ be a smooth manifold equipped with a Riemannian distance $d$, let $f: M \to M$ be a diffeomorphism, and let $\Lambda \subset M$ be a compact $f$-invariant hyperbolic set. Let $\mathcal R = \{R_1,\dots,R_m\}$ be a finite proper Markov partition for $f|_\Lambda$, with each $R_i \subset \Lambda$ a compact Markov rectangle, with $\Lambda = \bigcup_{i=1}^m R_i$, and with the relative interiors of the rectangles pairwise disjoint. Assume explicitly that this proper Markov partition has the following standard coding properties: every finite admissible word is realised by an orbit segment in $\Lambda$; there exist constants $C_{\mathcal R} > 0$ and $\theta_{\mathcal R} \in (0,1)$ such that every two-sided cylinder set $K_N(i)$ defined below has $d$-diameter at most $C_{\mathcal R}\theta_{\mathcal R}^N$; and for every $x \in \Lambda$, the set of compatible future rectangle names and the set of compatible past rectangle names determined by the local product structure at $x$ are finite. Define the transition matrix $A \in \{0,1\}^{m \times m}$ by

\begin{align*} A_{ij} = 1 \quad \text{if and only if} \quad f(R_i) \cap R_j \neq \varnothing. \end{align*}

Let

\begin{align*} \Sigma_A := \{(i_n)_{n \in \mathbb Z} \in \{1,\dots,m\}^{\mathbb Z} : A_{i_n i_{n+1}} = 1 \text{ for every } n \in \mathbb Z\} \end{align*}

be the two-sided subshift of finite type, and let $\sigma: \Sigma_A \to \Sigma_A$ be the left shift, defined by $(\sigma i)_n = i_{n+1}$.

Then there exists a continuous surjective map $\pi: \Sigma_A \to \Lambda$ such that

\begin{align*} f \circ \pi = \pi \circ \sigma. \end{align*}

Moreover, $\pi$ is finite-to-one. If $i = (i_n)_{n \in \mathbb Z} \in \Sigma_A$ and the full orbit of $\pi(i)$ avoids the rectangle boundaries, meaning

\begin{align*} f^n(\pi(i)) \notin \bigcup_{r=1}^m \partial_\Lambda R_r \quad \text{for every } n \in \mathbb Z, \end{align*}

where $\partial_\Lambda R_r$ denotes the boundary of $R_r$ relative to $\Lambda$, then $i$ is the unique element of $\Sigma_A$ coding $\pi(i)$.