一般参数移位基系统的有限性
The finiteness property for shift radix systems with general parameters
摘要：
二维扩展移位基系统（SRS）具有一定的周期轨道。本文的目的是尽可能地描述这种不寻常的点。我们给出了所有包含参数的区域，这些参数对应的SRS产生明显的循环，如$(1), (-1), (1,-1), (1,0), (-1,0)$。证明了如果$\mathbf{r}=(r_0,r_1)\in \mathbb{R}^2$既不属于上述区域也不属于有限区域$1\le r_0\le 4/3, -r_0 \le r_1 <r_0-1$，则$τ_{\mathbf{r}}$只有平凡有界轨道$\mathbf{0}$，这是对非周期轨道SRS的有限性的自然推广.进一步的归约应该是相当复杂的，因为对于所有的$1\le r_0< 4/3$，至少存在一个区间$I$，使得对于点$(r_0,r_1)$，无论何时$r_1\in I$，这都不是真的。
英文摘要：
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 $τ_{\mathbf{r}}$ 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 $r_1\in I$.
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