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.

翻译：我们建立了一个通用框架，可用于严格估计$\mathbb{R}^n$（$n \geq 2$）空间中极大共形图定向马尔可夫系统的豪斯多夫维数。该方法基于通过有限元框架进行离散化及迭代网格方案，对Perron-Frobenius算子特征函数进行分段线性逼近。我们方法的一个关键要素在于获得这些特征函数导数的界，这除了是实现我们方法所必需的之外，本身也具有独立的研究价值。