On affine iterated function systems which robustly admit an invariant affine subspace

TL;DR

This paper provides conditions under which affine iterated function systems consistently have invariant affine subspaces despite translation changes, revealing new examples that challenge existing theorems on self-affine sets.

Contribution

It introduces a simple sufficient condition for invariant affine subspaces in affine IFSs and presents new examples that defy Falconer's dimension results and the open set condition.

Findings

Identified conditions for persistent invariant affine subspaces in affine IFSs.

Constructed examples of similarity IFSs violating the open set condition.

Provided counterexamples to Falconer's dimension theorem in specific cases.

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.