On the packing dimension of box-like self-affine sets in the plane

TL;DR

This paper studies the packing dimension of a broad class of planar self-affine sets called 'box-like', including various known fractals, by deriving a pressure formula based on singular values under certain conditions.

Contribution

It introduces a generalized framework for box-like self-affine sets, extending previous models to include rotations and reflections, and provides a formula for their dimensions.

Findings

Derived a pressure formula for dimensions of box-like self-affine sets.

Included sets with rotational and reflectional components in the analysis.

Confirmed the formula under the rectangular open set condition.

Abstract

We consider a class of planar self-affine sets which we call "box-like". A box-like self-affine set is the attractor of an iterated function system (IFS) of affine maps where the image of the unit square, [0,1]^2, under arbitrary compositions of the maps is a rectangle with sides parallel to the axes. This class contains the Bedford-McMullen carpets and the generalisations thereof considered by Lalley-Gatzouras, Bara\'nski and Feng-Wang as well as many other sets. In particular, we allow the mappings in the IFS to have non-trivial rotational and reflectional components. Assuming a rectangular open set condition, we compute the packing and box-counting dimensions by means of a pressure type formula based on the singular values of the maps.