Planar self-affine sets with equal Hausdorff, box and affinity dimensions

TL;DR

This paper investigates conditions under which certain planar self-affine sets have equal Hausdorff, box-counting, and affinity dimensions, introducing new classes where these dimensions coincide.

Contribution

It provides new conditions and specific classes of self-affine sets with equal Hausdorff, box, and affinity dimensions using ergodic theory and Furstenberg measures.

Findings

Identifies conditions for dimension equality in self-affine sets

Introduces new classes of self-affine sets with matching dimensions

Uses ergodic theory and Furstenberg measures in analysis

Abstract

Using methods from ergodic theory along with properties of the Furstenberg measure we obtain conditions under which certain classes of plane self-affine sets have Hausdorff or box-counting dimensions equal to their affinity dimension. We exhibit some new specific classes of self-affine sets for which these dimensions are equal.