Inhomogeneous attractors and box dimension (2407.08404v1)

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.