  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.
