Pointwise regularity of parameterized affine zipper fractal curves fractal curves

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 Hölder exponent
for a subinterval of the spectrum. We give an equivalent characterization
for the existence of regular pointwise Hölder exponent for Lebesgue almost every
point. In this case, we extend the multifractal analysis to the full spectrum. In
particular, we apply our results for de Rham’s curve.