Rigorous Hausdorff dimension estimates for conformal fractals

TL;DR

This paper introduces a rigorous framework for estimating the Hausdorff dimension of conformal fractals in higher dimensions using finite element methods to approximate eigenfunctions of the Perron-Frobenius operator.

Contribution

It presents a novel, versatile approach combining piecewise linear approximations and mesh schemes to accurately estimate fractal dimensions in $ abla^n$ spaces.

Findings

Provides rigorous bounds for eigenfunction derivatives

Develops a finite element-based approximation scheme

Successfully estimates Hausdorff dimensions of conformal fractals

Abstract

We develop a versatile framework which allows us to rigorously estimate the Hausdorff dimension of maximal conformal graph directed Markov systems in $\mathbb{R}^n$ for $n \geq 2$. Our method is based on piecewise linear approximations of the eigenfunctions of the Perron-Frobenius operator via a finite element framework for discretization and iterative mesh schemes. One key element in our approach is obtaining bounds for the derivatives of these eigenfunctions, which, besides being essential for the implementation of our method, are of independent interest.