Abstract: 

Around 1980, David Ruelle considered an asymptotic expansion of the Hausdorff dimension of Julia sets of the quadratic family f_c(z) = z^2+c near c=0. In particular, Ruelle’s results states that H.dimJ_c = 1 + |c|^2/(2\log 2) + O(|c|^3) as |c| \to 0. In this talk, we consider an analogue of this question for the family f_t(z,w)=(z^d,w^d+t(c_2(x)w^{d-2} + … + c_d(x))) of polynomial skew products of C^2 near t=0. As holomorphic dynamical systems in higher dimensions are non-conformal, we replace Hausdorff dimension by volume dimension (that we introduced earlier) and consider its asymptotic expansion. This is joint work with Fabrizio Bianchi.