Bijective and general arithmetic codings for Pisot automorphisms of the torus

Abstract

Let $T$ be an algebraic automorphism of $\mathbb{T}^{m}$ having the following property: the characteristic polynomial of its matrix is irreducible over $\mathbb{Q}$, and a Pisot number $\beta$ is one of its roots. We define the mapping $\phi_{\mathbf{t}}$ acting from the two-sided $\beta$-compactum onto $\mathbb{T}^{m}$ as follows: \[ \phi_{\mathbf{t}}(\bar{\epsilon})= \sum_{k\in\mathbb{Z}}\epsilon_{k}T^{-k}\mathbf{t}, \] where $\mathbf{t}$ is a fundamental homoclinic point for $T$, i.e., a point homoclinic to $\mathbf{0}$ such that the linear span of its orbit is the whole homoclinic group (provided such a point exists). We call such a mapping an arithmetic coding of $T$. This paper is aimed to show that under some natural hypothesis on $\beta$ (which is apparently satisfied for all Pisot units) the mapping $\phi_{\mathbf{t}}$ is bijective a.e. with respect to the Haar measure on the torus. Besides, we study the case of more general parameters $\mathbf{t}$, not necessarily fundamental, and relate the number of preimages of $\phi_{\mathbf{t}}$ to certain number-theoretic quantities. We also give several full criteria for $T$ to admit a bijective arithmetic coding. This work continues the study begun in [Sidorov & Vershik, Journal Dynam. Control Systems 4 (1998), 365-399] for the special case $m=2$.