On self-affine measures with equal Hausdorff and Lyapunov dimensions

Let $μ$ be a self-affine measure on $\mathbb{R}^{d}$ associated to a self-affine IFS $\{φ_λ(x) = A_λx + v_λ\}_{λ\inΛ}$ and a probability vector $p=(p_λ)_λ>0$. Assume the strong separation condition holds. Let $γ_{1}\ge...\geγ_{d}$ and $D$ be the Lyapunov exponents and dimension corresponding to $\{A_λ\}_{λ\inΛ}$ and $p^{\mathbb{N}}$, and let $\mathbf{G}$ be the group generated by $\{A_λ\}_{λ\inΛ}$. We show that if $γ_{m+1}>γ_{m}=...=γ_{d}$, if $\mathbf{G}$ acts irreducibly on the vector space of alternating $m$-forms, and if the Furstenberg measure $μ_{F}$ satisfies $\dim_{H}μ_{F}+D>(m+1)(d-m)$, then $μ$ is exact dimensional with $\dimμ=D$.

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