Inspired by Avila and Lyubich results on measure and Hausdorff dimension of Feigenbaum Julia sets I will present a new explicit criterion to verify that the Julia set of the Feigenbaum fixed point of period-doubling renormalization has Hausdorff dimension less that two (and thus has zero Lebesgue measure). The results of computer experiments show (modulo computer errors) that this criterion is satisfied (joint with Scott Sutherland). I will also discuss applications of similar type criteria to unimodal maps with Fibonacci combinatorics and their Henon-like perturbations (ongoing joint work with Vadim Kaloshin and Mikhail Lyubich).