Rigorous Hausdorff dimension estimates for conformal fractals

TL;DR

This work introduces a versatile, rigorous pipeline for estimating the Hausdorff dimension of maximal conformal graph directed Markov systems in $\mathbb{R}^n$ by discretizing the Perron-Frobenius operator with finite elements and obtaining robust derivative bounds on the eigenfunction $\rho_t$. By constructing computable upper and lower bounds through matrices that bound the operator’s spectral radius, the authors obtain rigorous interval estimates for the Bowen parameter (dimension) and implement a tail-aware approach for infinite alphabets. The method is demonstrated across a broad spectrum of conformal fractals, including complex and higher-dimensional continued fractions, Schottky groups, and the Apollonian gasket, achieving precise dimension estimates and showcasing the framework’s flexibility and applicability. The results provide both practical computational tools and theoretical bounds that contribute to the dimension spectra of diverse fractal systems, with potential impact on related problems in dynamics and geometric group theory.

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.

Figures (18)

• Figure 1: An example of a triangular mesh of $\mathbb D$.
• Figure 2: A nodal basis function for $\mathbb{P}_1(X^h)$.
• Figure 3: The first (green) and second (red) iterations of a system consisting of quadratic perturbations of linear maps.
• Figure 4: First iteration maps of 3 circle Schottky
• Figure 5: First iteration maps of 3 circle Schottky with mesh

Theorems & Definitions (45)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Theorem 2.7
• Definition 2.8
• Theorem 2.9
• Proposition 2.10