Projections of four corner Cantor set: Total self-similarity, spectrum and unique codings

Derong Kong, Beibei Sun
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Abstract

Given , the four corner Cantor set is a self-similar set generated by the iterated function system For let be the orthogonal projection of onto a line with an angle to the -axis. In principle, is a self-similar set having overlaps. In this paper we give a complete characterization on which the projection is totally self-similar. We also study the spectrum of , which turns out that the spectrum achieves its maximum value if and only if is totally self-similar. Furthermore, when is totally self-similar, we calculate its Hausdorff dimension and study the subset which consists of all having a unique coding. In particular, we show that for Lebesgue almost every . Finally, for we prove that the possibility for to contain an interval is strictly smaller than that for to have an exact overlap.
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四角康托集合的投影:总自相似性、频谱和唯一编码
设 ,四角康托集合是由迭代函数系统生成的自相似集合 设 为 的正投影到与 轴成一定角度的直线上。原则上,它是一个有重叠的自相似集合。本文给出了投影完全自相似的完整特征。我们还研究了 , 的频谱,结果发现只有当且仅当完全自相似时,频谱才会达到最大值。此外,当完全自相似时,我们会计算其豪斯多夫维度,并研究由所有具有唯一编码的子集组成的子集。特别是,我们证明,对于 Lebesgue,几乎每个 。最后,我们证明,包含一个区间的可能性严格小于具有精确重叠的可能性。
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