A STRONGLY IRREDUCIBLE AFFINE ITERATED FUNCTION SYSTEM WITH TWO INVARIANT MEASURES OF MAXIMAL DIMENSION

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 Hausdor 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 give an example 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