Inhomogeneous IFSs

Iterated function systems (IFSs) are one of the most important and well-studied mechanisms for generating interesting fractal sets. An IFS is a (usually finite) collection of contractions on a compact metric space (e.g. the unit square) and for such a collection there is a unique non-empty compact set which is invariant under the action of the IFS. This set is known as the attractor. Barnsley and Demko introduced a variant on this construction, which I like very much. Instead of requiring that the attractor is invariant under the action of the IFS, one looks for a set which is invariant under the action of the IFS and taking the union with a fixed compact set (the condensation set). There is again a unique non-empty compact set with this property (the inhomogeneous attractor). This idea has applications in image compression, see the easily generated (but quite complicated) images below. The effect is that the attractor is the closure of the orbit of the condensation set under the IFS.

There are many interesting questions about inhomogeneous attractors. I am interested in their dimension in the case when the dimension in question is not countably stable, e.g. the upper box dimension. A naive guess is that the dimension should be the maximum of the dimension of the homogeneous counterpart and the dimension of the condensation set. I verified this for a large class of inhomogeneous self-similar sets here:

► Inhomogeneous self-similar sets and box dimensions, arXiv
        Studia Mathematica, 213, (2012), 133-156.

In the above paper I also studied the lower box dimension, which behaves quite badly in this setting. It turns out that the naive guess does not always hold even for upper box dimension. The upper box dimension can 'jump up' in the presence of carefully chosen overlaps or non-conformal scaling in the IFS (e.g. self-affine sets). Some results in this direction include:

► The dimensions of inhomogeneous self-affine sets (with S. A. Burrell), arXiv
        Annales Academiæ Scientiarum Fennicæ Mathematica, 45, (2020), 313-324.

► Inhomogeneous self-similar sets with overlaps (with S. Baker & Á. Máthé), arXiv
        Ergodic Theory and Dynamical Systems, 39, (2019), 1-18.

► Inhomogeneous self-affine carpets, arXiv
        Indiana University Mathematics Journal, 65, (2016), 1547-1566.

Several interesting questions remain, such as, does the naive formula always hold for inhomogeneous self-similar sets contained in the line?

Together with Tom Bartlett I considered a variant on inhomogeneous attractors in the context of actions of Kleinian groups:

► Dimensions of Kleinian orbital sets (with T. Bartlett), arXiv
        Journal of Fractal Geometry, 10, (2023), 267-278.

I gave a talk about inhomogeneous IFSs at Fractal Geometry and Stochastics 5, Tabartz 2014. Here are the slides.



On the left is an inhomogeneous attractor designed to represent a flock of birds seen from below. The condensation set is the large central bird. In the centre is the associated homogeneous attractor (for comparison). On the right is an inhomogeneous self-affine set designed to look like a forest. The condensation set is the large tree in the foreground. Can you see what the underlying IFS is?