On the dimension of planar self-affine sets with non-invertible maps

TL;DR

This paper investigates the dimension of planar self-affine sets generated by non-invertible affine maps, demonstrating conditions under which the dimension matches or is less than the affinity dimension.

Contribution

It establishes the dimension equality under separation conditions and identifies parameter regimes where the dimension is strictly smaller than the affinity dimension.

Findings

Dimension equals affinity dimension under certain separation conditions.

Dimension can be strictly smaller than affinity dimension for specific parameters.

Results apply to typical linear parts of non-invertible mappings.

Abstract

In this paper, we study the dimension of planar self-affine sets, of which generating iterated function system (IFS) contains non-invertible affine mappings. We show that under a certain separation condition, the dimension equals to the affinity dimension for a typical choice of the linear parts of the non-invertible mappings, furthermore, we show that the dimension is strictly smaller than the affinity dimension for certain choices of parameters.