Markov partitions and homoclinic points of algebraic $\mathbb Z^d$-actions

We prove that a general class of expansive $\mathbb Z^d$-actions by automorphisms of compact. Abelian groups with completely positive entropy has “symbolic covers” of equal topological entropy. These symbolic covers are constructed by using homoclinic points of these actions. For $d=1$ we adapt a result of Kenyon and Vershik in [7] to prove that these symbolic covers are, in fact, sofic shifts. For $d\ge2$ we are able t o prove the analogous statement only for certain examples, where the existence of such covers yields finitary isomorphisms between topologically nonisomorphic $\mathbb Z^2$-actions.