A classical theorem of Hutchinson asserts that if an iterated function system
acts on Rd 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 Rd. 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.Comment: This new version has a much more powerful version of the main theorem
and a less direct, more general approach to the proo