Infinitely renormalizable parameters for quadratic polynomials

Question

Let $P_{c}(z)=z^2 + c$, where $c\in \mathbb{C}$. Did we know that the set of infinitely renormalizable parameters has Lebesgue measure $0$ in complex plane or has Hausdorff dimension $2$?

Answer 1

It is an open problem whether this set has Lebesgue measure 0. Indeed, the question whether the boundary of the Mandelbrot set has measure zero remains open, and the set of finitely renormalisable parameter in the boundary has measure zero by a result of Shishikura (see the discussion in Avila and Moreira, arXiv:math/0306156).

I am less sure about the Hausdorff dimension, but I believe that the best result may still be Lyubich's theorem that the dimension is at least 1 (How big is the set of infinitely renormalizable quadratics? in: Voronezh Winter Mathematical Schools, 131–143, Amer. Math. Soc. Transl. Ser. 2, 184, Amer. Math. Soc., Providence, RI, 1998.)