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

TL;DR

This paper establishes conditions under which the dimension of self-affine measures equals the Lyapunov dimension, especially in three dimensions, and explores the impact of Diophantine properties and entropy on the measure's dimension.

Contribution

It provides new criteria for the dimension of self-affine measures to match the Lyapunov dimension in three dimensions and extends results to projections and Diophantine systems.

Findings

Dimension equals Lyapunov dimension in 3D under strong separation conditions.

Dimension equals the ambient space when the system is Diophantine with sufficient entropy.

Results on the dimension of orthogonal projections of the measures.

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 $\mu$.