Bijective and General Arithmetic Codings for Pisot Toral Automorphisms

Abstract

Let T be an algebraic automorphism of {\Bbb T}^m having the following property: the characteristic polynomial of its matrix is irreducible over \Bbb Q, and a Pisot number β is one of its roots. We define the mapping pt acting from the two-sided β-compactum onto {\Bbb T}^m as follows: \varphi_t (\bar{\varepsilon}) = \sum_{k\epsilonZ} \varepsilon_kT^{-k}t, where t is a fundamental homoclinic point for T, i.e., a point homoclinic to 0 such that the linear span of its orbit is the whole homoclinic group (provided that such a point exists). We call such a mapping an arithmetic coding of T. This paper aimed to show that under some natural hypothesis on β (which is apparently satisfied for all Pisot units) the mapping pt is bijective a.e. with respect to the Haar measure on the torus. Moreover, we study the case of more general parameters t, not necessarily fundamental, and relate the number of preimages of pt to certain number-theoretic quantities. We also give several full criteria for T to admit a bijective arithmetic coding and consider some examples of arithmetic codings of Cartan actions. This work continues the study begun in l25r for the special case m e 2.