Contractivity of three-dimensional shift radix systems with finiteness property

Abstract

Let d ≥ 1 be an integer and . We define the shift radix system τ r : ℤ d → ℤ d by The shift radix system τ r has the finiteness property if each a ∈ ℤ d is eventually mapped to 0 under iterations of τ r . The mapping τ r can be written as , where R(r) is a d × d matrix and v is a correction term. It has been conjectured that the fact that τ r has the finiteness property implies that all eigenvalues of R(r) are strictly smaller than one in modulus. The aim of the present paper is to prove this conjecture for the case d = 3.