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.)

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    $\begingroup$ Another example comes out of my mind is the Sturmian sequences which are symbolic coding of irrational rotations. And there are also generalizations to some cases in higher dimension, You can find more details in the book 'Substitutions in dynamics, arithmetics and combinatorics' by Pytheas Fogg. $\endgroup$
    – Siming Tu
    Sep 14, 2017 at 12:40
  • $\begingroup$ Thanks, Siming. Zero entropy systems are less preferable for my goals, I'm afraid. $\endgroup$ Sep 15, 2017 at 14:19


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