某些表面流重标度的谱不相交性
Spectral disjointness of rescalings of some surface flows
摘要：
本文研究了两类流的自相似问题：（1）圆旋转上和具有对称对数奇点的屋顶函数下的特殊流;（2）区间交换变换上和屋顶函数下的特殊流，这两类特殊流有两种类型：带一个非IET间断点的分段常数流和带非零斜率的分段线性流。证明了如果$\{T^f_t\}_{t\in\mathbb R}$与（1）相同，则对于旋转的全测度集，对于每两个不同的自然数$K$和$L$，我们有$\{T^f_{Kt}\}_{t\in\mathbb R}$和$\{T^f_{Lt}\}_{t\in\mathbb R}$是谱不相交的.类似地，如果$\{T^f_t\}_{t\in\mathbb R}$如（2）中所示，则对于IET的完整测量集，a.e.在分段常数的情况下，附加不连续性（$f$的）的位置和每两个不同的自然数$K$和$L$，流$\{T^f_{Kt}\}_{t\in\mathbb R}$和$\{T^f_{Lt}\}_{t\in\mathbb R}$是谱不相交的。
英文摘要：
We study self-similarity problem for two classes of flows: (1) special flows over circle rotations and under roof functions with symmetric logarithmic singularities (2) special flows over interval exchange transformations and under roof functions which are of two types * piecewise constant with one additional discontinuity which is not a discontinuity of the IET; * piecewise linear over exchanged intervals with non-zero slope. We show that if $\{T^f_t\}_{t\in\mathbb R}$ is as in (1) then for a full measure set of rotations, and for every two distinct natural numbers $K$ and $L$, we have that $\{T^f_{Kt}\}_{t\in\mathbb R}$ and $\{T^f_{Lt}\}_{t\in\mathbb R}$ are spectrally disjoint. Similarly, if $\{T^f_t\}_{t\in\mathbb R}$ is as in (2), then for a full measure set of IET's, a.e. position of the additional discontinuity (of $f$, in piecewise constant case) and every two distinct natural numbers $K$ and $L$, the flows $\{T^f_{Kt}\}_{t\in\mathbb R}$ and $\{T^f_{Lt}\}_{t\in\mathbb R}$ are spectrally disjoint.
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