Dimension of diagonal self-affine measures with exponentially separated projections

TL;DR

This paper determines the dimension of diagonal self-affine measures with exponentially separated projections, confirming a conjecture by Rapaport and removing previous linearity restrictions.

Contribution

It proves the dimension formula for such measures under exponential separation and distinct Lyapunov exponents, extending prior results by removing linear subgroup assumptions.

Findings

Dimension equals the minimum of ambient dimension and Lyapunov dimension.

Established new results on dimension and entropy increase for random measures.

Confirmed a conjecture of Rapaport without linearity restrictions.

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.