Dimension of diagonal self-affine sets and measures via non-conformal partitions

Abstract

Let $\Phi:=\left\{ (x_{1},...,x_{d})\rightarrow\left(r_{i,1}x_{1}+a_{i,1},...,r_{i,d}x_{d}+a_{i,d}\right)\right\} _{i\in\Lambda}$ be an affine diagonal IFS on $\mathbb{R}^{d}$. Suppose that for each $1\le j_{1}<j_{2}\le d$ there exists $i\in\Lambda$ so that $|r_{i,j_{1}}|\ne|r_{i,j_{2}}|$, and that for each $1\le j\le d$ the IFS $\left\{ t\rightarrow r_{i,j}t+a_{i,j}\right\} _{i\in\Lambda}$ on the real line is exponentially separated. Under these assumptions we show that the Hausdorff dimension of the attractor of $\Phi$ is equal to $\min\left\{ \dim_{A}\Phi,d\right\} $, where $\dim_{A}\Phi$ 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 $\Phi$ are all contained in a $1$-dimensional subgroup, the dimension of an associated self-affine measure $\mu$ is equal to the minimum of its Lyapunov dimension and $d$. Most of the proof is dedicated to an entropy increase result for convolutions of $\mu$ with general measures $\theta$ of non-negligible entropy, where entropy is measured with respect to non-conformal partitions corresponding to the Lyapunov exponents of $\mu$. It turns out that with respect to these partitions, the entropy across scales of repeated self-convolutions of $\theta$ behaves quite differently compared to the conformal case. The analysis of this non-conformal multi-scale entropy is the main ingredient of the proof, and is also the main novelty of this paper.