Dimension of diagonal self-affine measures with exponentially separated projections

Abstract

Let $ \mu $ be a self-affine measure associated with a diagonal affine iterated function system (IFS) $ \Phi = \{ (x_{1}, \ldots, x_{d}) \mapsto ( r_{i, 1}x_{1} + t_{i,1}, \ldots, r_{i,d}x_{d} + t_{i,d}) \}_{i\in\Lambda} $ on $ \mathbb{R}^{d} $ and a probability vector $ p = (p_{i})_{i\in\Lambda}$. For $ 1 \leq j \leq d $, denote the $ j $-th the Lyapunov exponent by $ \chi_{j} := \sum_{i\in\Lambda} - p_{i} \log | r_{i,j} |$, and define the IFS induced by $ \Phi $ 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}} \neq \chi_{j_{2}} $ for $ 1 \leq j_{1} < j_{2} \leq d $, and $ \Phi_{j}$ is exponentially separated for $ 1 \leq j \leq d $, then the dimension of $ \mu $ is the minimum of $ d $ and its Lyapunov dimension. This confirms a conjecture of Rapaport by removing the additional assumption that the linear parts of the maps in $ \Phi $ are contained in a 1-dimensional subgroup. One of the main ingredients of the proof involves disintegrating $ \mu $ 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.