Non-spectrality of planar self-affine measures with three-elements digit set
Non-spectrality of planar self-affine measures with three-elements digit set
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DOI：
10.1016/j.jfa.2008.12.012
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发表时间：
2009-07
影响因子：
1.7
通讯作者：
Jian-Lin Li
中科院分区：
文献类型：
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作者：
Jian-Lin Li
本文证明了仿射迭代函数系[式：见正文]的自仿射测度μM，D是唯一确定的.测度μM，D的谱性与非谱性的判定问题近年来受到了广泛的关注。关于μM，D的非谱问题之一是估计L2（μM，D）中正交指数的个数并求出它们。本文证明了：对于扩张整数矩阵M∈M2（Z），如果ac−bd <$3Z给出的三元数集D，则L2（μM，D）中至多存在3个相互正交的指数，且最佳指数数为3.这证实了关于平面上自仿射测度的非谱性的三元数集猜想
The self-affine measure μM,Dassociated with an affine iterated function system [Formula: see text] is uniquely determined. The problems of determining the spectrality or non-spectrality of a measure μM,Dhave been received much attention in recent years. One of the non-spectral problem on μM,Dis to estimate the number of orthogonal exponentials in L2(μM,D) and to find them. In the present paper we show that for an expanding integer matrix M∈M2(Z) and the three-elements digit set D given by if ac−bd∉3Z, then there exist at most 3 mutually orthogonal exponentials in L2(μM,D), and the number 3 is the best. This confirms the three-elements digit set conjecture on the non-spectrality of self-affine measures in the plane