Hausdorff Dimension of a Class of Self-Affine Sets

TL;DR

This paper derives exact formulas for the Hausdorff dimension of certain self-affine fractals generated by affine iterated function systems, extending to hybrid systems and providing topological classifications.

Contribution

It introduces a unified framework for calculating Hausdorff dimensions of complex self-affine systems, including hybrid and overlapping cases, with a topological analysis in the plane.

Findings

Exact Hausdorff dimension formulas under the Open Set Condition.

Dimension formulas for hybrid systems combining affine maps and similarities.

Topological classification of systems in the plane, identifying a parameter bottleneck.

Abstract

In this paper, exact Hausdorff dimension formulas for a class of self-affine attractors generated by affine Iterated Function Systems are derived. We consider systems containing an affine map whose $n$-th iterate is a similarity contraction, alongside standard similarities whose linear parts commute with the symmetric operator $A^\top A$, where $A$ is the linear part of the affine map. We prove that the attractor of such a system exists uniquely, and, under the Open Set Condition, we compute its exact Hausdorff dimension. We extend this framework to systems where all map compositions of some fixed length are similarities, and to systems where overlaps are exact homothetic copies of the attractor. We unify these approaches to establish dimension formulas for hybrid systems that combine multiple eventually contractive affine maps with universally aligned similarities. Finally, we conclude with a topological classification of these systems in the plane. For a two-map system comprising an affine map whose second iterate is a similarity with contraction ratio $c$, alongside an $f$-aligned similarity with ratio $r$, we prove that the precise parameter balance $c + r = 1$ acts as a strict topological bottleneck uniquely guaranteeing both the open set condition and the connectedness of the attractor.