AI 中文总结
本研究构造了一类正则性为C^{1+ω}、具无理旋转数且含游荡区间的圆周微分同胚，证明了Denjoy定理在正则性上的锐度，还解决了ω(t)=tlog(1/t)情形下自Herman 1979年工作遗留的公开问题。
AI 中文摘要
设$\omega$为弱于Lipschitz的凹连续模，即当$t$趋于0时$\omega(t)/t$发散。我们构造了一个具有无理旋转数、正则性类为$C^{1+\omega}$的圆周微分同胚，它存在游荡区间。该构造表明，除非对旋转数施加额外限制，否则Denjoy 1932年的定理在正则性上是锐利的。在$\omega(t) = t\log(1/t)$的特殊情形下，该构造解决了可追溯至Herman 1979年关于圆周微分同胚的工作中的一个公开问题，Herman的工作对任意$\varepsilon > 0$给出了$\omega(t) = t\log(1/t)^{1+\varepsilon}$情形的构造。我们的例子是由旋转数快速收敛的周期圆周微分同胚取极限得到的。
英文摘要
Let $ω$ be a concave modulus of continuity that is weaker than Lipschitz, meaning $ω(t)/t$ diverges as $t$ approaches $0$. We construct a diffeomorphism of the circle with irrational rotation number, in the regularity class $C^{1+ω}$, with a wandering interval. This construction implies that Denjoy's 1932 theorem is sharp in regularity, unless additional restrictions are imposed on the rotation number. The construction in the special case $ω(t) = t\log(1/t)$ settles an open problem dating back to Herman's 1979 work on circle diffeomorphisms, which gave constructions for $ω(t) = t\log(1/t)^{1+\varepsilon}$ for every $\varepsilon > 0$. Our examples arise as limits of periodic circle diffeomorphisms with rapidly converging rotation numbers.
Comments50 pages. See Section 1.5 for a statement on AI use
