On the Hausdorff dimension of a 2-dimensional Weierstrass curve
Dynamical Systems
2018-08-21 v2 Classical Analysis and ODEs
Probability
Abstract
We compute the Hausdorff dimension of a two-dimensional Weierstrass function, related to lacunary (Hadamard gap) power series, that has no L\'evy area. This is done by interpreting it as a pullback attractor of a dynamical system based on the Baker transformation. A lower bound for the Hausdorff dimension is obtained by investigating the pushforward of the Lebesgue measure on the graph along scaled neighborhoods of stable fibers of the underlying dynamical system following the graph. Scaling ideas are crucial. They become accessible by self similarity properties of a mapping whose increments coincide with vertical distances on the stable fibers.
Keywords
Cite
@article{arxiv.1806.09585,
title = {On the Hausdorff dimension of a 2-dimensional Weierstrass curve},
author = {Peter Imkeller and Goncalo dos Reis},
journal= {arXiv preprint arXiv:1806.09585},
year = {2018}
}
Comments
16 pages, 2 Figures; (V2 updates funding wrt V1)