On self-affine measures associated to strongly irreducible and proximal systems

Abstract

Let $\mu$ be a self-affine measure on $\mathbb{R}^{d}$ associated to an affine IFS $\Phi$ and a positive probability vector $p$. Suppose that the maps in $\Phi$ do not have a common fixed point, and that standard irreducibility and proximality assumptions are satisfied by their linear parts. We show that $\dim\mu$ is equal to the Lyapunov dimension $\dim_{L}(\Phi,p)$ whenever $d=3$ and $\Phi$ satisfies the strong separation condition (or the milder strong open set condition). This follows from a general criteria ensuring $\dim\mu=\min\{d,\dim_{L}(\Phi,p)\}$, from which earlier results in the planar case also follow. Additionally, we prove that $\dim\mu=d$ whenever $\Phi$ is Diophantine (which holds e.g. when $\Phi$ is defined by algebraic parameters) and the entropy of the random walk generated by $\Phi$ and $p$ is at least $(\chi_{1}-\chi_{d})\frac{(d-1)(d-2)}{2}-\sum_{k=1}^{d}\chi_{k}$, where $0>\chi_{1}\ge...\ge\chi_{d}$ are the Lyapunov exponents. We also obtain results regarding the dimension of orthogonal projections of $\mu$.