Improved dimension theory of sofic self-affine fractals

TL;DR

This paper provides a combinatorial formula for the Hausdorff dimension of self-affine fractals, extending previous work on sofic sets and establishing conditions for dimension equivalence in higher-dimensional Euclidean spaces.

Contribution

It introduces a new combinatorial expression for the Hausdorff dimension of self-affine fractals, including extensions to higher dimensions and conditions for Minkowski and Hausdorff dimension equality.

Findings

Derived a combinatorial formula for Hausdorff dimension of self-affine fractals.

Extended dimension results to certain sofic sets in higher-dimensional spaces.

Established conditions under which Minkowski and Hausdorff dimensions coincide for planar sofic sets.

Abstract

Follow-up comment by the author: Theorem 2.2 in this paper is a special case of Theorems 1.1 and 4.1 in the article "Weighted thermodynamic formalism on subshifts and applications", Asian J. Math. 16 (2012), by J. Barral and D. J. Feng. In addition, Zhou Feng studied the conditions under which general self-affine fractals, including sofic sets, have the same Hausdorff dimension and box dimension in the paper "On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets", arXiv:2405.03213. I would like to thank Dr. Zhou Feng for pointing out these works. The calculation of the exact Hausdorff dimension of sofic sets presented in this article is refined in my subsequent work "Exact Hausdorff dimension of some sofic self-affine fractals", arXiv:2412.05805. Original abstract: We establish a combinatorial expression for the Hausdorff dimension of a given self-affine fractal in any Euclidean space. This formula includes the extension of the work by Kenyon and Peres (1996) on planar sofic sets and yields an exact value for the dimension of certain sofic sets in $\mathbb{R}^3$ or higher. We also calculate the Minkowski dimension of sofic sets and establish a sufficient and presumably necessary condition for planar sofic sets to have the same Minkowski and Hausdorff dimension. The condition can be regarded as a generalization of the classical result for Bedford-McMullen carpets.