The short version of my question is that I need examples of explicit continuous symbolic codings of invertible dynamical systems.

Here's a longer version. Suppose $(\Omega,\mu,T)$ is an invertible ergodic dynamical system with a probability measure $\mu$. Is there a two-sided subshift on a finite alphabet $(X,\nu,\sigma)$ and a continuous, surjective, finite-to-one map $\phi:X\to\Omega$ such that $\phi\sigma=T\phi$ and $\phi(\nu)=\mu$?

One **example** of such a coding is as follows. Let $d\ge2$ and $\Omega=\mathbb T^d=\mathbb R^d/\mathbb Z^d$. Now let $\mu$ be the Haar (Lebesgue) measure on $\mathbb T^d$ and $T$ be a *Pisot automorphism*. That is, $T$ is given by a $GL(d,\mathbb Z)$-matrix whose characteristic polynomial is irreducible over $\mathbb Q$ and has a root $\beta>1$, and the remaining roots are $<1$ in modulus. Now, let $\mathbf t$ be a homoclinic point for $T$, i.e., $T^n\mathbf t\to\mathbf 0$ as $n\to\pm\infty$.

Define $\phi=\phi_{\mathbf t}$ as follows (here $\mathbf a=(a_n)_{n\in\mathbb Z}$):

$$
\phi_{\mathbf t}(\mathbf a)=\sum_{n\in\mathbb Z} a_nT^{-n}\mathbf t\bmod\mathbb Z^d=\sum_{n\in\mathbb Z} a_n\beta^{-n}\mathbf t\bmod\mathbb Z^d.
$$
Then $\phi_{\mathbf t}$ satisfies the required conditions (K. Schmidt, *Algebraic codings of expansive group automorphisms and two-sided beta-shifts*, Monatsh. Math. **129** (2000), 37-61). Here $\nu=\nu_\beta$ is the Parry measure for the two-sided $\beta$-shift (a unique measure of maximal entropy).

Another trivial example is the baker's map and the full dyadic shift, of course.

More examples?

**UPDATE.** There is a similar, but more general construction which works for all hyperbolic toral automorphisms, due to S. Le Borgne (*Un codage sofique des automorphismes hyperboliques du tore*, Bol. Soc. Bras. Mat. **30** (1999), 61–93.)