Inhomogeneous self-affine carpets

TL;DR

This paper studies the dimension theory of inhomogeneous self-affine carpets, revealing new phenomena where expected formulas for box dimensions can fail, thus advancing understanding of non-conformal fractals.

Contribution

It explicitly computes upper and lower box dimensions for broad classes of inhomogeneous self-affine carpets and uncovers phenomena absent in self-similar cases.

Findings

Expected formulas for upper box dimension can fail in self-affine settings.

Explicit dimension calculations for large classes of inhomogeneous self-affine carpets.

Discovery of new phenomena not present in self-similar fractals.

Abstract

We investigate the dimension theory of inhomogeneous self-affine carpets. Through the work of Olsen, Snigireva and Fraser, the dimension theory of inhomogeneous self-similar sets is now relatively well-understood, however, almost no progress has been made concerning more general non-conformal inhomogeneous attractors. If a dimension is countably stable, then the results are immediate and so we focus on the upper and lower box dimensions and compute these explicitly for large classes of inhomogeneous self-affine carpets. Interestingly, we find that the `expected formula' for the upper box dimension can fail in the self-affine setting and we thus reveal new phenomena, not occurring in the simpler self-similar case.