Dimension of diagonal self‐affine sets and measures via nonconformal partitions

Abstract Let be an affine diagonal iterated function system (IFS) on . Suppose that for each , there exists so that , and that for each , the IFS on the real line is exponentially separated. Under these assumptions, we show that the Hausdorff dimension of the attractor of is equal to , where is the affinity dimension. This follows from a result regarding self‐affine measures, which says that, under the additional assumption that the linear parts of the maps in are all contained in a one‐dimensional subgroup, the dimension of an associated self‐affine measure is equal to the minimum of its Lyapunov dimension and . Most of the proof is dedicated to an entropy increase result for convolutions of with general measures of nonnegligible entropy, where entropy is measured with respect to nonconformal partitions corresponding to the Lyapunov exponents of . It turns out that with respect to these partitions, the entropy across scales of repeated self‐convolutions of behaves quite differently compared to the conformal case. The analysis of this nonconformal multiscale entropy is the main ingredient of the proof, and is also the main novelty of this paper.