On totally disconnected generalised Sierpinski carpets

TL;DR

This paper investigates the structure of generalized Sierpinski carpets, establishing conditions for total disconnectedness and providing examples with maximal box-counting dimension.

Contribution

It identifies specific pattern families that ensure total disconnectedness and extends results to a broader setting, including a dimension-maximal example.

Findings

Certain pattern families guarantee total disconnectedness.

Results extend to more general constructions.

An example with box-counting dimension 2 is provided.

Abstract

Generalised Sierpinski carpets are planar sets that generalise the well-known Sierpinski carpet and are defined by means of sequences of patterns. We study the structure of the sets at the kth iteration in the construction of the generalised carpet, for k greater than or equal to 1. Subsequently, we show that certain families of patterns provide total disconnectedness of the resulting generalised carpets. Moreover, analogous results hold even in a more general setting. Finally, we apply the obtained results in order to give an example of a totally disconnected generalised carpet with box-counting dimension 2.