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 $\{\varphi _{\lambda }(x)=A_{\lambda }x+v_{\lambda }\}_{\lambda \in \Lambda }$ and a probability vector $p=(p_{\lambda })_{\lambda }>0$. Assume the strong separation condition holds. Let $\gamma _{1}\ge \cdots \ge \gamma _{d}$ 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 $\gamma _{m+1}>\gamma _{m}=\cdots =\gamma _{d}$, 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 $\dim \mu =D$.