Dimension of diagonal self-affine measures with exponentially separated projections

TL;DR

This work resolves the dimension of diagonal self-affine measures by proving that, when the Lyapunov exponents are strictly increasing $χ_1<\cdots<χ_d$ and each coordinate IFS $Φ_j$ is exponentially separated, the measure dimension satisfies $\dim μ = \min\{ d, \dim_L(Φ,p)\}$. The authors introduce a disintegration of the self-affine measure into random convolution measures, and develop Ledrappier-Young type formulas for these disintegrations along coordinate projections. A central novelty is the removal of the prior restriction that the linear parts lie in a 1D subgroup, thereby confirming Rapaport's conjecture in greater generality. The approach hinges on a delicate entropy analysis of random measures, using nonconformal partitions and an entropy-increase mechanism for repeated self-convolutions, which may have further implications for projections and typical-parameter behavior in diagonal self-affine systems. Overall, the paper advances the dimension theory for self-affine measures by combining disintegration, LY-type formulas, and entropy techniques to obtain explicit dimension results under mild separation assumptions.

Abstract

Let $ μ$ be a self-affine measure associated with a diagonal affine iterated function system (IFS) $ Φ= \{ (x_{1}, \ldots, x_{d}) \mapsto ( r_{i, 1}x_{1} + t_{i,1}, \ldots, r_{i,d}x_{d} + t_{i,d}) \}_{i\inΛ} $ on $ \mathbb{R}^{d} $ and a probability vector $ p = (p_{i})_{i\inΛ}$. For $ 1 \leq j \leq d $, denote the $ j $-th the Lyapunov exponent by $ χ_{j} := \sum_{i\inΛ} - p_{i} \log | r_{i,j} |$, and define the IFS induced by $ Φ$ on the $j$-th coordinate as $ Φ_{j} := \{ x \mapsto r_{i,j}x + t_{i,j}\}_{i\inΛ}$. We prove that if $ χ_{j_{1}} \neq χ_{j_{2}} $ for $ 1 \leq j_{1} < j_{2} \leq d $, and $ Φ_{j}$ is exponentially separated for $ 1 \leq j \leq d $, then the dimension of $ μ$ 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 $ 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.

Theorems & Definitions (72)

• Definition 1
• Remark 1
• Theorem 1
• Remark 2
• Remark 3
• Corollary 1
• Example 1
• Corollary 2
• proof
• Corollary 3