Pointwise regularity of parameterized affine zipper fractal curves

TL;DR

This paper investigates the local regularity and multifractal spectrum of affine zipper fractal curves, providing dimension calculations and characterizations for typical points, with applications to de Rham's curve.

Contribution

It offers new insights into the pointwise regularity and multifractal analysis of affine zipper fractals under dominated splitting conditions.

Findings

Calculated Hausdorff dimension of level sets of pointwise Hölder exponents.

Characterized conditions for the existence of regular pointwise Hölder exponents.

Extended multifractal spectrum analysis to almost every point for these fractals.

Abstract

We study the pointwise regularity of zipper fractal curves generated by affine mappings. Under the assumption of dominated splitting of index-1, we calculate the Hausdorff dimension of the level sets of the pointwise H\"older exponent for a subinterval of the spectrum. We give an equivalent characterization for the existence of regular pointwise H\"older exponent for Lebesgue almost every point. In this case, we extend the multifractal analysis to the full spectrum. In particular, our results apply for de Rham's curve.