Genericity of dimension drop on self-affine sets

TL;DR

This paper demonstrates that, in a generic sense, removing an affine map from a self-affine set in Euclidean space reduces its Hausdorff dimension, addressing an open question in fractal geometry.

Contribution

It proves that generically, the Hausdorff dimension drops when one affine map is removed from a self-affine set, providing partial evidence for a longstanding open problem.

Findings

Removing one affine map reduces Hausdorff dimension in generic cases

Addresses a folklore open question in fractal geometry

Provides partial positive answer to dimension drop conjecture

Abstract

We prove that generically, for a self-affine set in $\mathbb{R}^d$, removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension. This gives a partial positive answer to a folklore open question.