论文标题
椭圆多项式螺旋的分形结构
The fractal structure of elliptical polynomial spirals
论文作者
论文摘要
我们研究了椭圆多项式螺旋的分形方面；也就是说，在两个轴方向上具有不同多项式衰减速率的平面螺旋形。我们对这些螺旋式进行完整的维度分析，并明确计算它们的中间，盒子计数和补充型尺寸。一个令人兴奋的特征是，这些螺旋形在Assouad Spectrum中表现出两个相变，这是已知具有该特性的第一类天然分形类。我们继续使用这些维信息来获得地图的Hölder规律性的界限，这些范围可以将一个螺旋变形为另一个螺旋，从而概括了“绕组问题”，即螺旋形成何时是Bi-lipschitz等同于线段的螺旋。一个新颖的特征是使用分数布朗运动和尺寸轮廓来绑定Hölder指数。
We investigate fractal aspects of elliptical polynomial spirals; that is, planar spirals with differing polynomial rates of decay in the two axis directions. We give a full dimensional analysis of these spirals, computing explicitly their intermediate, box-counting and Assouad-type dimensions. An exciting feature is that these spirals exhibit two phase transitions within the Assouad spectrum, the first natural class of fractals known to have this property. We go on to use this dimensional information to obtain bounds for the Hölder regularity of maps that can deform one spiral into another, generalising the 'winding problem' of when spirals are bi-Lipschitz equivalent to a line segment. A novel feature is the use of fractional Brownian motion and dimension profiles to bound the Hölder exponents.
