On self-affine measures with equal Hausdorff and Lyapunov dimensions

Let $\mu$ be a self-affine measure on ${\mathbb{R}}^d$ associated to a self-affine IFS ${\varphi_{\lambda}(x) = A_{\lambda}x + v_{\lambda}}{\lambda\in\Lambda}$andaprobabilityvector$p=(p{\lambda})_{\lambda}>0$. Assume the strong separation condition holds. Let ${\gamma }_1\ge \mathrm{.}\mathrm{.}\mathrm{.}\ge {\gamma }_d$ and $D$ be the Lyapunov exponents and dimension corresponding to $\{A_{\lambda }{\}}_{\lambda \in \mathrm{\Lambda }}$ and $p^{\mathbb{N}}$, and let $\mathbf{G}$ be the group generated by $\{A_{\lambda }{\}}_{\lambda \in \mathrm{\Lambda }}$. We show that if ${\gamma }_{m+1}>{\gamma }_m=\mathrm{.}\mathrm{.}\mathrm{.}={\gamma }_d$, 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 $\dim \mu =D$.