Non-invertible planar self-affine sets

TL;DR

This paper investigates the dimension differences between non-invertible and invertible planar self-affine sets, revealing that for generic cases, dimensions match when large and differ when small, using thermodynamical formalism.

Contribution

It provides a comprehensive comparison of dimensions for non-invertible and invertible self-affine sets in the plane, with a complete characterization under dominated and irreducible matrices.

Findings

Dimensions coincide for large sets in generic cases.

Dimensions differ for small sets in generic cases.

Complete characterization of pressure behavior for dominated and irreducible matrices.

Abstract

We compare the dimension of a non-invertible self-affine set to the dimension of the respective invertible self-affine set. In particular, for generic planar self-affine sets, we show that the dimensions coincide when they are large and differ when they are small. Our study relies on thermodynamical formalism where, for dominated and irreducible matrices, we completely characterize the behavior of the pressures.