0-1 sequences of the Thue-Morse type and Sarnak's conjecture
0-1 sequences of the Thue-Morse type and Sarnak's conjecture
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DOI：
10.1090/proc/12683
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发表时间：
2013-04
期刊：
影响因子：
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通讯作者：
E. Abdalaoui;S. Kasjan;M. Lema'nczyk
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文献类型：
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作者：
E. Abdalaoui;S. Kasjan;M. Lema'nczyk
我们表明，图像通过$z\mapsto z^m$的连续部分的动力系统的谱措施所产生的0-1序列的Thue-Morse型是成对相互奇异的不同奇数$m\在\N$。Sarnak关于M“obius函数正交性的猜想对这样的动力系统是成立的。同样的猜想，以持有所有系统引起的定期Toeplitz序列。构造了一个Sarnak猜想失败的非正则Toeplitz序列。
We show that the images via $z\mapsto z^m$ of the continuous part of the spectral measures of the dynamical systems generated by the 0-1 sequences of the Thue-Morse type are pairwise mutually singular for different odd numbers $m\in\N$. Sarnak's conjecture on orthogonality with the M\"obius function is shown to hold for such dynamical systems. The same conjecture is shown to hold for all systems induced by regular Toeplitz sequences. A non-regular Toeplitz sequence for which Sarnak's conjecture fails is constructed.