Exact Hausdorff dimension of some sofic self-affine fractals

TL;DR

This paper develops a new technique to explicitly compute the Hausdorff dimension of certain sofic self-affine fractals in three-dimensional space, transforming previously incalculable limits into explicit solutions.

Contribution

It introduces a novel method to calculate the exact Hausdorff dimension of some sofic self-affine fractals, including the first such calculation in D.

Findings

Dimension expressed as solution to an infinite-degree equation

Dimension equals spectral radius of a linear operator

First explicit calculation of Hausdorff dimension in D for sofic sets

Abstract

Previous work has shown that the Hausdorff dimension of sofic affine-invariant sets is expressed as a limit involving intricate matrix products. This limit has typically been regarded as incalculable. However, in several highly non-trivial cases, we demonstrate that the dimension can in fact be calculated explicitly. Specifically, the dimension is expressed as the solution to an infinite-degree equation with explicit coefficients, which also corresponds to the spectral radius of a certain linear operator. Our result provides the first non-trivial calculation of the exact Hausdorff dimension of sofic sets in $\mathbb{R}^3$. This is achieved by developing a new technique inspired by the work of Kenyon and Peres (1998).