Algebraic Coding of Expansive Group Automorphisms and Two-sided Beta-Shifts

Abstract

Let α be an expansive automorphisms of compact connected abelian group X whose dual group is cyclic w.r.t. α (i.e. is generated by for some ). Then there exists a canonical group homomorphism ξ from the space of all bounded two-sided sequences of integers onto X such that , where σ is the shift on . We prove that there exists a sofic subshift such that the restriction of ξ to V is surjective and almost one-to-one. In the special case where α is a hyperbolic toral automorphism with a single eigenvalue and all other eigenvalues of absolute value we show that, under certain technical and possibly unnecessary conditions, the restriction of ξ to the two-sided beta-shift 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 [13] and [7]. Earlier results in this direction were obtained by Vershik ([24]–[26]), Kenyon and Vershik ([10]), and Sidorov and Vershik ([20]–[21]).