Rigorous Hausdorff dimension estimates for conformal fractals

This project explores a range of computational problems that naturally emerge in the dimension theory of conformal dynamical systems. Conformal fractals are intricate geometric objects generated via iterated schemes of conformal (angle-preserving) transformations, and they have numerous interdisciplinary applications in mathematical physics, computer graphics, and data science. Measuring the size of conformal fractal attractors has been one of the central themes in the evolution of modern dynamical systems. One of the most well-known ways for measuring such complex geometric objects is the concept of Hausdorff dimension, which provides a robust way of determining the roughness of a set, extending the idea of dimension beyond integer values. The Hausdorff dimension of conformal fractals cannot be derived via simple analytic closed formulas, and obtaining effective and rigorous estimates becomes a challenging computational problem. The scope of this project is to introduce new methods from numerical partial differential equations with the scope of developing versatile, rigorous, and efficient methods for computing the Hausdorff dimensions of various conformal attractors. The project's topic is naturally interdisciplinary, encompassing a wide range of problems across Real and Complex Analysis, Dynamical Systems, Numerical Analysis, and Large-Scale Computations. The goal is to derive accurate and rigorous Hausdorff dimension estimates for a broad class of conformal fractals by integrating techniques from finite element methods, dynamical systems, and fractal geometry. Finite element analysis is a well-established approach for approximating solutions to a wide range of partial differential equations, with numerous refined methods developed over the years to ensure accurate and reliable numerical results. In contrast, the field of rigorous computation of Hausdorff dimensions for conformal limit sets is still in its infancy. The primary innovation of this project lies in adapting numerical methods typically used for solving PDEs to the estimation of Hausdorff dimensions. This new methodology demands deep analytical foundations and the development of novel theoretical results, presenting significant challenges, especially within the broader context of conformal graph-directed Markov systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.