Algebraic Coding of Expansive Groupautomorphisms

Let be an expansive automorphisms of compact connected abelian group X whose dual group ^ X is cyclic w.r.t. (i.e. ^ X is generated by f n : n 2 Zg for some 2 ^ X). Then there exists a canonical group homomorphism from the spacè 1 (Z;Z) of all bounded two-sided sequences of integers onto X such that = , where is the shift on`1 (Z;Z). We prove that there exists a sooc subshift V ` 1 (Z;Z) such that the restriction of to V is surjective and almost one-to-one. In the special case where is a hyperbolic toral automorphism with a single eigenvalue > 1 and all other eigenvalues of absolute value < 1 we show that, under certain technical and possibly unnecessary conditions, the restriction of to the two-sided beta-shift V ` 1 (Z;Z) is surjective and almost one-to-one. The proofs are based on the study of homoclinic points and an associated algebraic construction of symbolic representations in 12] and 7]. Earlier results in this direction were obtained by 1. Introduction The classical constructions of symbolic representations of hyperbolic toral automorphisms are based on their geometrical properties and make no sig-niicant use of algebra (cf. 1], 3], 6], 21]). In 23] a diierent approach is proposed, based on arithmetical ideas, leading to a symbolic representation