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Symmetric shift radix systems and finite expansions. (English) Zbl 1164.11007

The authors study some generalizations of the number systems and expansions of Rényi. Let \({\boldsymbol r}=(r_{1},\ldots,r_{d})\in {\mathbb R}^{d}\), \({\boldsymbol z}=(z_{1},\ldots, z_{d})\in {\mathbb Z}^{d}\) and \(\tilde{\tau}_{{\boldsymbol r}}({\boldsymbol z})=(z_{2},\ldots, z_{d},-\left\lfloor{\boldsymbol r}{\boldsymbol z}\right\rfloor)\), where \({\boldsymbol r}{\boldsymbol z}=r_{1}z_{1}+\ldots+r_{d}z_{d}\) for \(d\geq1\). Then \(({\mathbb Z}^{d},\tilde{\tau}_{{\boldsymbol r}})\) is a shift radix system and it was discussed in other publications. In the present paper they study the so called symmetric shift radix system \(({\mathbb Z}^{d},\tau_{{\boldsymbol r}})\), where \(d\geq 2\) and \(\tau_{{\boldsymbol r}}({\boldsymbol z})=(z_{2},\ldots,z_{d},-\left\lfloor {\boldsymbol r}{\boldsymbol z}+\frac{1}{2}\right\rfloor)\). They introduce two sequences: \(D_{d}:={\boldsymbol r}\in {\mathbb R}^{d}:\forall {\boldsymbol z}\in {\mathbb Z}^{d}\), the sequence \((\tau ^{k}_{{\boldsymbol r}}({\boldsymbol z}))^{\infty}_{k=0}\) is eventualy periodic and \(D^{0}_{d}:={\boldsymbol r}\in {\mathbb R}^{d}:\forall {\boldsymbol z}\in {\mathbb Z}^{d},\;\exists k>0:\tau^{k}_{\boldsymbol r}({\boldsymbol z})=0\). The authors study in details the case \(d=2\). They show that \(D_{2}\subset\Delta :=(\epsilon_{2})^{\omega}=\{(x,y)\in \mathbb R^{2}:x\leq 1, | y| \leq x+1\}\) and \(D^{0}_{2}\subset\frac{\Delta}{2}:=\{(x,y)\in \mathbb R^{2}: x\leq \frac{1}{2}, | y| \leq x+ \frac{1}{2}\}\). The main result consists of the following theorem: Define two segments by \(L_{1}=\{(x,y):| x| \leq \frac{1}{2}, | y| =-x-\frac{1}{2}\}\) and \(L_{2}=\{(\frac{1}{2}, y):\frac{1}{2}<y<1\}\). Then \(D^{0}_{2}=\frac{\Delta}{2}\setminus(L_{1}\cup L_{2})\). Some results are illustrated by figures of graphs.

MSC:

11A63 Radix representation; digital problems
37B10 Symbolic dynamics
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