Finite and periodic orbits of shift radix systems (Q628842)

For \(r=(r_0,\ldots,r_{d-1})\in {\mathbb R}^d\) define the function \[ \tau_r:{\mathbb Z}^d\to {\mathbb Z}^d, z=(z_0,\ldots,z_{d-1})\mapsto(z_1,\ldots,z_{d-1},-\lfloor rz\rfloor) , \] where \(rz\) is the scalar product of the vectors \(r\) and \(z\). If each orbit of \(\tau_r\) ends up at \(0\), \(\tau_r\) is called a shift radix system. The authors study periodicity properties of the mappings \(\tau_r\) in the case when the roots of the polynomial \(t^d+ r_{d-1}t^{d-1}+\ldots+r_0\) have modulus \(\leq 1\) with at least one equality.