Dimension of diagonal self-affine measures with exponentially separated projections

Abstract

Let μ be a self-affine measure associated with a diagonal affine iterated function system (IFS) $\Phi ={\left\{\right(x_1,\dots ,x_d)\mapsto (r_{i,1}{x}_1+t_{i,1},\dots ,r_{i,d}{x}_d+t_{i,d}\left)\right\}}_{i\in \Lambda }$ on $R^d$ and a probability vector $p={\left(p_i\right)}_{i\in \Lambda }$. For $1\le j\le d$, denote the j-th Lyapunov exponent by ${\chi }_j:={\sum }_{i\in \Lambda }-p_i\log \left|r_{i,j}\right|$, and define the IFS induced by Φ on the j-th coordinate as ${\Phi }_j:={\{x\mapsto r_{i,j}x+t_{i,j}\}}_{i\in \Lambda }$. We prove that if ${\chi }_{j_1}\ne {\chi }_{j_2}$ for $1\le j_1<j_2\le d$, and ${\Phi }_j$ is exponentially separated for $1\le j\le d$, then the dimension of μ is the minimum of d and its Lyapunov dimension. This confirms a conjecture of Rapaport [52] by removing the additional assumption that the linear parts of the maps in Φ are contained in a 1-dimensional subgroup. One of the main ingredients of the proof involves disintegrating μ into random measures with convolution structure. In the course of the proof, we establish new results on dimension and entropy increase for these random measures.