Dimension of invariant measures for affine iterated function systems

Let {Si}i∈Λ be a finite contracting affine iterated function system (IFS) on Rd. Let (Σ,σ) denote the two-sided full shift over the alphabet Λ, and let π:Σ→Rd be the coding map associated with the IFS. We prove that the projection of an ergodic σ-invariant measure on Σ under π is always exact dimensional, and its Hausdorff dimension satisfies a Ledrappier–Young-type formula. Furthermore, the result extends to average contracting affine IFSs. This completes several previous results and answers a folklore open question in the community of fractals. Some applications are given to the dimension of self-affine sets and measures.