The Hausdorff dimension of the boundary of the L\'evy dragon

TL;DR

This paper develops a theoretical method to compute the Hausdorff dimension of the boundary of the Le9vy Dragon attractor, involving complex combinatorial topology and eigenvalue analysis of a large matrix.

Contribution

It introduces a new theoretical framework for calculating Hausdorff dimensions of boundaries of iterated function system attractors, demonstrated on the Le9vy Dragon.

Findings

Successfully computed the Hausdorff dimension of the Le9vy Dragon boundary.

Developed a complex combinatorial and eigenvalue analysis method.

Highlighted the role of Perron-Frobenius theory in the computation.

Abstract

A theoretical approach to computing the Hausdorff dimension of the topological boundary of attractors of iterated function systems is developed. The curve known as the L\'evy Dragon is then studied in detail and the Hausdorff dimension of its boundary is computed using the theory developed. The actual computation is a complicated procedure. It involves a great deal of combinatorial topology as well as determining the structure and certain eigenvalues of a $752 \times 752$ matrix. Perron-Frobenius theory plays an important role in analyzing this matrix.