This preprint was accepted February, 1998
Contact: N. Sidorov and A. Vershik
ABSTRACT: We study arithmetic codings of hyperbolic automorphisms of the 2-torus, i.e. mappings acting from the symbolic space of sequences with a finite alphabet endowed with an appropriate structure of additive group onto the torus which preserves this structure and turns the two-sided shift into an automorphism of the torus. The necessary and sufficient condition of the existence of a bijective arithmetic coding is obtained; it is formulated in terms of a certain binary quadratic form constructed by means of a given automorphism. Furthermore, we describe all bijective arithmetic codings in terms the Dirichlet group of the corresponding qudratic field. The arithmetic minimum of that binary quadratic form in the general case equals the minimal possible number of preimages for a.e. point of the torus under such an arithmetical coding.