Dimension of the Green measure of skew products with dominant exponent in the fibers

Let $f(z,w)=(p(z),q(z,w))$ be a polynomial skew product of ${\mathbb{C}}^2$ of degree $d⩾2$ and let $\mu$ and ${\mu }_p$ be the Green measures associated to $f$ and $p$ respectively. Let ${\lambda }_1>{\lambda }_2$ and ${\lambda }_p$ be the Lyapunov exponents of $(f,\mu )$ and $(p,{\mu }_p)$ respectively. We show that if ${\lambda }_2={\lambda }_p$, then $\mu$ is exact dimensional and: \begin{equation*}\dim_Hμ=\frac{\log d}{λ_1}+\frac{\log d}{λ_2}.\end{equation*} The proof relies on the continuity of the conditional measures obtained by Jonsson, on the fact that $\mu$ has constant Jacobian $d^2$, and on a fibered version of a normal form theorem for the inverse branches of the iterates of $f$.