A Ruelle-McMullen formula for the volume dimension of skew products in ${\mathbb{C}}^2$

Ruelle gave an explicit second-order expansion at $c=0$ of the Hausdorff dimension of the Julia set of the quadratic family $f_c(z)=z^2+c$. McMullen later extended this result to polynomial perturbations of $z^d$ for arbitrary degree $d\ge 2$. In this paper we study an analogue of this problem for skew products in ${\mathbb{C}}^2$. Since holomorphic dynamical systems in higher dimensions are non-conformal, we replace the Hausdorff dimension by the \emph{volume dimension}, a dynamically defined notion we introduced in our earlier work and characterized as the zero of a natural pressure function. We consider families of holomorphic skew products of the form $$f_t(z,w)=(z^d,w^d+t(c_1(z)w^{d-1}+c_2(z)w^{d-2}+\cdots +c_d(z)))\mathrm{.}$$ Our main result gives an explicit second-order expansion of the volume dimension of the Julia set $J(f_t)$ as $t\to 0$ in terms of the coefficients $c_k(z)$.