On affine iterated function systems which robustly admit an invariant affine subspace
Abstract
In this note we give a simple sufficient condition for an affine iterated function system to admit an invariant affine subspace persistently with respect to changes in the translation parameters. This yields further examples of tuples of contracting linear maps which do not satisfy the conclusions of Falconer's theorem on the Hausdorff dimension of almost every self-affine set. We also obtain new examples of iterated function systems of similarity transformations which cannot satisfy the open set condition for any choice of translation parameters, and resolve a related question of Peres and Solomyak.
Cite
@article{arxiv.2111.02324,
title = {On affine iterated function systems which robustly admit an invariant affine subspace},
author = {Ian D. Morris},
journal= {arXiv preprint arXiv:2111.02324},
year = {2022}
}
Comments
This paper has a new title (the previous title was "When does an affine iterated function system preserve an affine subspace for all choices of translation vectors?") and has various minor modifications in respect of referee comments. An improved counterexample to Peres and Solomyak's question is also included. To appear in Proceedings of the AMS