Self-duality and codings for expansive automorphisms

Lind and Schmidt have shown that for certain ergodic Z k -actions on a compact abelian group Γ, the homoclinic group H is isomorphic to the Pontryagin dual of Γ. Einsiedler and Schmidt extended these results and showed that Γ is a quotient of a locally compact ring R modulo H. In this paper, we present a dynamical interpretation of R if k = 1 : it is a product of the stable group and the unstable group of Γ, under a suitable topology. As applications, we give a topological interpretation of the Pisot–Vijayaraghavan theorem and we link the results to tessellation theory.