On H\"{o}lder solutions to the spiral winding problem

Abstract

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 H\"{o}lder exponents and establish sharp results for spirals with polynomial winding rates, observing that the sharp H\"{o}lder 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 H\"{o}lder exponents in the winding problem coming directly from knowledge of dimension (and how dimension distorts under H\"{o}lder 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.