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 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.
Highlights
The winding problem concerns understanding the regularity of functions which map a line segment onto a spiral
We find that the Assouad spectrum is the only ‘dimension’ which distinguishes between spirals with different polynomial winding rates in the superlinear regime
In the well-studied α-models for fluid turbulence polynomial spirals appear as the evolution of the half-line [0, ∞) ⊂ R2 under the resulting two-dimensional flow and the polynomial winding rate depends on the parameter α, see [FHT]
Summary
Spirals appear naturally across mathematics and wider science, often arising via a dynamical system or geometric constraint. A well-studied and important problem in the dimension theory of fractals is to consider how Hölder maps affect a given notion of fractal dimension, see [F]. We establish precisely how much information can be extracted from dimension theory in the context of the Hölder version of the winding problem, proving that the best information comes from the Assouad spectrum, but even this is not sharp. Given two bounded homeomorphic sets X, Y ⊂ Rd, first consider the Hölder mapping problem which asks for sharp estimates on α and β such that there exists an (α, β)-Hölder map f with f(X) = Y. The polynomial spirals we consider here are examples where sharp information is not provided by dimension theory, but where the Assouad spectrum performs the best
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