Inhomogeneous attractors and box dimension

TL;DR

This paper explores the dimension theory of inhomogeneous attractors generated by iterated function systems, focusing on upper box dimension estimates, examples, and open questions in the field.

Contribution

It provides a detailed analysis of the upper box dimension of inhomogeneous attractors, highlighting methods to estimate it and discussing cases where estimates are not sharp.

Findings

Derived new bounds for upper box dimension of inhomogeneous attractors

Presented examples illustrating the behavior of box dimension estimates

Identified open problems in the dimension theory of inhomogeneous fractals

Abstract

Iterated function systems (IFSs) are one of the most important tools for building examples of fractal sets exhibiting some kind of `approximate self-similarity'. Examples include self-similar sets, self-affine sets etc. A beautiful variant on the standard IFS model was introduced by Barnsley and Demko in 1985 where one builds an \emph{inhomogeneous} attractor by taking the closure of the orbit of a fixed compact condensation set under a given standard IFS. In this expository article I will discuss the dimension theory of inhomogeneous attractors, giving several examples and some open questions. I will focus on the upper box dimension with emphasis on how to derive good estimates, and when these estimates fail to be sharp.