关于动力Möbius-Sarnak过程的谱分析与Möbius函数的拓扑熵
On the spectral analysis of dynamical Möbius-Sarnak process and topological entropy of Möbius fonction
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中文总结 AI 辅助
该研究扩展Rokhlin-Sinai机制与谱测度概念，证明动力Möbius-Sarnak过程有可数Lebesgue分量，得到Möbius流拓扑熵的精确值，并验证了相关谱性质与Sarnak猜想的一类情形。
中文摘要 AI 辅助
通过扩展与熵及谱中可数Lebesgue分量相关的Rokhlin-Sinai机制，我们证明了动力Möbius-Sarnak过程具有可数Lebesgue分量。受M. Lin与作者近期工作的启发，我们将谱测度的概念推广到Banach空间上的所有算子。这一推广还进一步受到Bellow-Losert对Wiener的序列谱测度概念的扩展的推动。此外，我们无条件证明了Möbius流的拓扑熵为$\frac{6}{π^2}\log 3$。在其他推论中，我们重现了el Abdalaoui-Nerurkar的近期结果，即对于Möbius函数的任意拟通用测度，配备该测度的Möbius流的谱中具有可数Lebesgue分量。由此可得，Sarnak Möbius正交性猜想对任意具有奇异谱的拓扑动力系统成立。我们进一步证明，Möbius函数的所有可能谱测度相对于Lebesgue测度是绝对连续的。
英文摘要
By extending the Rokhlin-Sinai machinery relating to the entropy and countable Lebesgue component in the spectrum, we establish that the dynamical Möbius-Sarnak process has a countable Lebesgue component. Inspired by recent work of M. Lin and the author, we extend the notion of spectral measure to all operators on Banach spaces. This generalization is further motivated by the Bellow-Losert extension of Wiener's notion of the spectral measure of sequences. Furthermore, we establish unconditionally that the topological entropy of the Möbius flow is given by $\frac{6}{π^2}\log 3$. Among other consequences, we recover a recent result by el Abdalaoui-Nerurkar which asserts that for any quasi-generic measure for the Möbius function, the Möbius flow equipped with this measure has a countable Lebesgue component in its spectrum. It follows that the Sarnak Möbius orthogonality conjecture holds for any topological dynamical system with singular spectrum. We further show that all the potential spectral measures of he Möbius function are absolutely continuous with respect to Lebesgue measure.
