H\"older spiral arcs

TL;DR

This paper provides a precise criterion for when spiral arcs are H"older continuous, unifies classifications for polynomial and elliptical spirals, and improves existing bounds on their H"older properties.

Contribution

It establishes a necessary and sufficient condition for H"older arcs in spiral curves and refines the classification exponents for polynomial and elliptical spirals.

Findings

Sharp condition for H"older arcs established

Unified classification for polynomial and elliptical spirals

Improved bounds on H"older exponents for spiral classes

Abstract

We establish a quantitative necessary and sufficient condition for a spiral arc to be a H\"older arc. The class of spiral arcs contains the polynomial spirals studied by Fraser, and the elliptical spirals studied by Burrell-Falconer-Fraser. As an application, we recover the sharp result on the H\"older winding problem for polynomial spirals. Moreover, we provide a sharp exponent estimate for the H\"older classification of polynomial spirals, which coincides with the corresponding quasiconformal classification estimate, and improve certain exponent bounds of Burrell-Falconer-Fraser on the H\"older classification of elliptical spirals.