Dynamical properties of the negative beta-transformation
Dynamical properties of the negative beta-transformation
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负β-变换的动力学特性
DOI：
10.1017/s0143385711000514
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发表时间：
2011
影响因子：
0.9
通讯作者：
W. Steiner
中科院分区：
文献类型：
--
作者：
Lingmin Liao;W. Steiner
摘要分析了Ito和Sadahiro最近研究的负β变换的动力学性质。与经典的贝塔变换相反，负贝塔变换的绝对连续不变测度的密度在一定区间上可能为零。通过对这一性质的详细研究，我们证明了(−β)-变换对所有的β>；1都是精确的，从而证实了Góra的一个猜想，并完成了对Faller的一个研究。我们还证明了1的(−β)-展开式在β趋于1时的极限行为与图-莫尔斯序列有关。精确度的一个结果是，每个Yrrap数都是β>；1，使得1的(−β)展开最终是周期性的，它是Perron数。这推广了Parry数的一个著名性质。然而，Parry数的集合不同于Yrrap数的集合。
Abstract We analyse dynamical properties of the negative beta-transformation, which has been studied recently by Ito and Sadahiro. Contrary to the classical beta-transformation, the density of the absolutely continuous invariant measure of the negative beta-transformation may be zero on certain intervals. By investigating this property in detail, we prove that the (−β)-transformation is exact for all β>1, confirming a conjecture of Góra, and intrinsic, which completes a study of Faller. We also show that the limit behaviour of the (−β)-expansion of 1 when β tends to 1 is related to the Thue–Morse sequence. A consequence of the exactness is that every Yrrap number, which is a β>1 such that the (−β) -expansion of 1 is eventually periodic, is a Perron number. This extends a well-known property of Parry numbers. However, the set of Parry numbers is different from the set of Yrrap numbers.