Finite and periodic orbits of shift radix systems

Kirschenhofer, Peter, et al. "Finite and periodic orbits of shift radix systems." Journal de Théorie des Nombres de Bordeaux 22.2 (2010): 421-448. <http://eudml.org/doc/116413>.

@article{Kirschenhofer2010,
abstract = {For $\mathbf\{r\}=(r_0,\ldots ,r_\{d-1\}) \in \mathbb\{R\}^d$ define the function\[ \tau \_\mathbf\{r\}: \, \{\mathbb\{Z\}\}^d \rightarrow \{\mathbb\{Z\}\}^d,\quad \mathbf\{z\}=(z\_0,\ldots ,z\_\{d-1\}) \mapsto (z\_1,\ldots ,z\_\{d-1\},-\left\lfloor \mathbf\{rz\} \right\rfloor ), \]where $\mathbf\{rz\}$ is the scalar product of the vectors $\mathbf\{r\}$ and $\mathbf\{z\}$. If each orbit of $\tau _\mathbf\{r\}$ ends up at $\mathbf\{0\}$, we call $\tau _\mathbf\{r\}$ a shift radix system. It is a well-known fact that each orbit of $\tau _\mathbf\{r\}$ ends up periodically if the polynomial $t^d+r_\{d-1\}t^\{d-1\}+\cdots +r_0$ associated to $\mathbf\{r\}$ is contractive. On the other hand, whenever this polynomial has at least one root outside the unit disc, there exist starting vectors that give rise to unbounded orbits. The present paper deals with the remaining situations of periodicity properties of the mappings $\tau _\mathbf\{r\}$ for vectors $\mathbf\{r\}$ associated to polynomials whose roots have modulus less than or equal to one with equality in at least one case. We show that for a large class of vectors $\{\bf r\}$ belonging to the above class the ultimate periodicity of the orbits of $\tau _\mathbf\{r\}$ is equivalent to the fact that $\tau _\mathbf\{s\}$ is a shift radix system or has another prescribed orbit structure for a certain parameter $\mathbf\{s\}$ related to $\mathbf\{r\}$. These results are combined with new algorithmic results in order to characterize vectors $\mathbf\{r\}$ of the above class that give rise to ultimately periodic orbits of $\tau _\mathbf\{r\}$ for each starting value. In particular, we work out the description of these vectors $\mathbf\{r\}$ for the case $d=3$. This leads to sets which seem to have a very intricate structure.},
affiliation = {Chair of Mathematics and Statistics University of Leoben Franz-Josef-Str. 18 A-8700 Leoben, AUSTRIA; Faculty of Informatics Number Theory Research Group Hungarian Academy of Sciences and University of Debrecen P.O. Box 12 H-4010 Debrecen, HUNGARY; Departamento de Matemática IBILCE - UNESP Rua Cristóvão Colombo, 2265 - Jardim Nazareth 15.054-000 São José do Rio Preto - SP, BRAZIL; Chair of Mathematics and Statistics University of Leoben Franz-Josef-Str. 18 A-8700 Leoben, AUSTRIA},
author = {Kirschenhofer, Peter, Pethő, Attila, Surer, Paul, Thuswaldner, Jörg},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Contractive polynomial; shift radix system; periodic orbit; periodic orbit shift radix system; periodic orbit shift radix system; periodic orbit shift radix system},
language = {eng},
number = {2},
pages = {421-448},
publisher = {Université Bordeaux 1},
title = {Finite and periodic orbits of shift radix systems},
url = {http://eudml.org/doc/116413},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Kirschenhofer, Peter
AU - Pethő, Attila
AU - Surer, Paul
AU - Thuswaldner, Jörg
TI - Finite and periodic orbits of shift radix systems
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 2
SP - 421
EP - 448
AB - For $\mathbf{r}=(r_0,\ldots ,r_{d-1}) \in \mathbb{R}^d$ define the function\[ \tau _\mathbf{r}: \, {\mathbb{Z}}^d \rightarrow {\mathbb{Z}}^d,\quad \mathbf{z}=(z_0,\ldots ,z_{d-1}) \mapsto (z_1,\ldots ,z_{d-1},-\left\lfloor \mathbf{rz} \right\rfloor ), \]where $\mathbf{rz}$ is the scalar product of the vectors $\mathbf{r}$ and $\mathbf{z}$. If each orbit of $\tau _\mathbf{r}$ ends up at $\mathbf{0}$, we call $\tau _\mathbf{r}$ a shift radix system. It is a well-known fact that each orbit of $\tau _\mathbf{r}$ ends up periodically if the polynomial $t^d+r_{d-1}t^{d-1}+\cdots +r_0$ associated to $\mathbf{r}$ is contractive. On the other hand, whenever this polynomial has at least one root outside the unit disc, there exist starting vectors that give rise to unbounded orbits. The present paper deals with the remaining situations of periodicity properties of the mappings $\tau _\mathbf{r}$ for vectors $\mathbf{r}$ associated to polynomials whose roots have modulus less than or equal to one with equality in at least one case. We show that for a large class of vectors ${\bf r}$ belonging to the above class the ultimate periodicity of the orbits of $\tau _\mathbf{r}$ is equivalent to the fact that $\tau _\mathbf{s}$ is a shift radix system or has another prescribed orbit structure for a certain parameter $\mathbf{s}$ related to $\mathbf{r}$. These results are combined with new algorithmic results in order to characterize vectors $\mathbf{r}$ of the above class that give rise to ultimately periodic orbits of $\tau _\mathbf{r}$ for each starting value. In particular, we work out the description of these vectors $\mathbf{r}$ for the case $d=3$. This leads to sets which seem to have a very intricate structure.
LA - eng
KW - Contractive polynomial; shift radix system; periodic orbit; periodic orbit shift radix system; periodic orbit shift radix system; periodic orbit shift radix system
UR - http://eudml.org/doc/116413
ER -