# A strongly irreducible affine iterated function system with two invariant measures of maximal dimension

## TL;DR

This paper constructs examples of affine iterated function systems with two invariant measures of maximal dimension, even when the linear parts do not preserve any finite union of proper subspaces, challenging previous assumptions.

## Contribution

It provides the first known examples of such systems with multiple maximal dimension measures without invariant subspace preservation.

## Key findings

- Existence of affine IFS with two maximal dimension measures without invariant subspace preservation
- Challenges previous beliefs about the uniqueness of maximal dimension measures in affine IFS
- Expands understanding of measure behavior in affine dynamical systems

## Abstract

A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb{R}^d$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the attractor. In the class of measures on the attractor which arise as the projections of shift-invariant measures on the coding space, this self-similar measure is the unique measure of maximal dimension. In the context of affine iterated function systems it is known that there may be multiple shift-invariant measures of maximal dimension if the linear parts of the affinities share a common invariant subspace, or more generally if they preserve a finite union of proper subspaces of $\mathbb{R}^d$. In this note we construct examples where multiple invariant measures of maximal dimension exist even though the linear parts of the affinities do not preserve a finite union of proper subspaces.