The dimensions of inhomogeneous self-affine sets

TL;DR

This paper establishes bounds on the upper box dimension of inhomogeneous self-affine sets, linking it to affinity and condensation set dimensions, and explores conditions for these bounds to be tight, extending classical results.

Contribution

It provides new bounds and conditions for the dimension of inhomogeneous self-affine sets, unifying and extending previous theoretical results in the field.

Findings

Upper box dimension bounded by maximum of affinity and condensation set dimensions

Conditions identified for the bound to be attained

Extension of Falconer's results to inhomogeneous self-affine sets

Abstract

We prove that the upper box dimension of an inhomogeneous self-affine set is bounded above by the maximum of the affinity dimension and the dimension of the condensation set. In addition, we determine sufficient conditions for this upper bound to be attained, which, in part, constitutes an exploration of the capacity for the condensation set to mitigate dimension drop between the affinity dimension and the corresponding homogeneous attractor. Our work improves and unifies previous results on general inhomogeneous attractors, low-dimensional affine systems, and inhomogeneous self-affine carpets, while providing inhomogeneous analogues of Falconer's seminal results on homogeneous self-affine sets.