On self-affine measures with equal Hausdorff and Lyapunov dimensions

TL;DR

This paper establishes conditions under which self-affine measures have equal Hausdorff and Lyapunov dimensions, linking algebraic properties of the affine transformations with measure-theoretic dimension results.

Contribution

It provides new criteria involving Lyapunov exponents, group actions, and Furstenberg measures for the exact dimensionality and dimension equality of self-affine measures.

Findings

Conditions for measure exact dimensionality established.

Dimension equality between Hausdorff and Lyapunov dimensions proved.

Link between group irreducibility and measure dimension demonstrated.

Abstract

Let $\mu$ be a self-affine measure on $\mathbb{R}^{d}$ associated to a self-affine IFS $\{\varphi_{\lambda}(x) = A_{\lambda}x + v_{\lambda}\}_{\lambda\in\Lambda}$ and a probability vector $p=(p_{\lambda})_{\lambda}>0$. Assume the strong separation condition holds. Let $\gamma_{1}\ge...\ge\gamma_{d}$ and $D$ be the Lyapunov exponents and dimension corresponding to $\{A_{\lambda}\}_{\lambda\in\Lambda}$ and $p^{\mathbb{N}}$, and let $\mathbf{G}$ be the group generated by $\{A_{\lambda}\}_{\lambda\in\Lambda}$. We show that if $\gamma_{m+1}>\gamma_{m}=...=\gamma_{d}$, if $\mathbf{G}$ acts irreducibly on the vector space of alternating $m$-forms, and if the Furstenberg measure $\mu_{F}$ satisfies $\dim_{H}\mu_{F}+D>(m+1)(d-m)$, then $\mu$ is exact dimensional with $\dim\mu=D$.