Shift Radix Systems - A Survey

Abstract

Let $d\ge 1$ be an integer and $\mathbf{r}=(r_0,\dots ,r_{d-1})\in {\mathbf{R}}^d$. The {\em shift radix system} ${\tau }_{\mathbf{r}}:{\mathbb{Z}}^d\to {\mathbb{Z}}^d$ is defined by $${\tau }_{\mathbf{r}}(\mathbf{z})=(z_1,\dots ,z_{d-1},-\lfloor \mathbf{r}\mathbf{z}\rfloor {)}^t(\mathbf{z}=(z_0,\dots ,z_{d-1}{)}^t).$$ ${\tau }_{\mathbf{r}}$ has the {\em finiteness property} if each $\mathbf{z}\in {\mathbb{Z}}^d$ is eventually mapped to $0$ under iterations of ${\tau }_{\mathbf{r}}$. In the present survey we summarize results on these nearly linear mappings. We discuss how these mappings are related to well-known numeration systems, to rotations with round-offs, and to a conjecture on periodic expansions w.r.t.\ Salem numbers. Moreover, we review the behavior of the orbits of points under iterations of ${\tau }_{\mathbf{r}}$ with special emphasis on ultimately periodic orbits and on the finiteness property. We also describe a geometric theory related to shift radix systems.

45 pages, 16 figures

References in corpus (1)

• Pentagonal Domain Exchange

Cited by in corpus (1)

• An introduction to $p$-adic systems: A new kind of number systems inspired by the Collatz $3n+1$ conjecture