The finiteness property for shift radix systems with general parameters

TL;DR

This paper characterizes parameters for two-dimensional shift radix systems that generate periodic orbits, extending the finiteness property and identifying regions with trivial or non-trivial dynamics.

Contribution

It provides a comprehensive description of parameter regions leading to obvious cycles and generalizes the finiteness property for shift radix systems.

Findings

Identifies all parameter regions with obvious cycles.

Proves that outside these regions, only the trivial orbit exists.

Shows the complexity of the parameter space for certain ranges.

Abstract

There are two-dimensional expanding shift radix systems (SRS) which have some periodic orbits. The aim of the present paper is to describe such unusual points as well as possible. We give all regions that contain parameters the corresponding SRS of which generate obvious cycles like $(1), (-1), (1,-1), (1,0), (-1,0)$. We prove that if $\mathbf{r}=(r_0,r_1)\in \mathbb{R}^2$ neither belongs to the aforementioned regions nor to the finite region $1\le r_0\le 4/3, -r_0 \le r_1 <r_0-1$, then $\tau_{\mathbf{r}}$ only has the trivial bounded orbit $\mathbf{0}$, which is a natural generalization of the established finiteness property for SRS with non-periodic orbits. The further reduction should be quite involving, because for all $1\le r_0< 4/3$ there exists at least one interval $I$ such that for the point $(r_0,r_1)$ this is not true whenever $r_1\in I$.