On self-affine measures with equal Hausdorff and Lyapunov dimensions

Abstract

Let $\mu$ be a self-affine measure on ${\mathbb{R}}^d$ associated to a self-affine IFS $\{{\phi }_{\lambda }(x)=A_{\lambda }x+v_{\lambda }{\}}_{\lambda \in \Lambda }$ 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 $\dim \mu =D$.