关于螺旋缠绕问题的Hölder解
On Hölder solutions to the spiral winding problem
摘要：
缠绕问题涉及理解将线段映射到螺旋线上的函数的规律性。这个问题与流体动力学和保形焊接理论有关，其中螺旋自然出现。在这里，我们解释“规律性”的霍德尔指数和建立尖锐的结果与多项式缠绕率的螺旋，观察到尖锐的霍德尔指数的正向映射和逆满足一个公式，让人想起Sobolev共轭。我们还研究了这些螺旋的维数论，特别是Assouad维数，Assouad谱和盒维数。这里的目的是将直接来自维度知识（以及维度在Hölder图像下如何扭曲）的缠绕问题中Hölder指数的界限与尖锐的结果进行比较。我们发现Assouad光谱提供了最好的信息，但即使这样也不尖锐。我们还发现，Assouad频谱是唯一的“尺寸”，区分不同的多项式缠绕率的超线性制度的螺旋。
英文摘要：
The winding problem concerns understanding the regularity of functions which map a line segment onto a spiral. This problem has relevance in fluid dynamics and conformal welding theory, where spirals arise naturally. Here we interpret `regularity' in terms of Hölder exponents and establish sharp results for spirals with polynomial winding rates, observing that the sharp Hölder exponent of the forward map and its inverse satisfy a formula reminiscent of Sobolev conjugates. We also investigate the dimension theory of these spirals, in particular, the Assouad dimension, Assouad spectrum and box dimensions. The aim here is to compare the bounds on the Hölder exponents in the winding problem coming directly from knowledge of dimension (and how dimension distorts under Hölder image) with the sharp results. We find that the Assouad spectrum provides the best information, but that even this is not sharp. We also find that the Assouad spectrum is the only `dimension' which distinguishes between spirals with different polynomial winding rates in the superlinear regime.
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