Is the Hausdorff dimension $Dim_{H}(J(f))$ of the Julia set less than 2 for quadratic rational map? Let $f(z)$ be a quadratic rational map with two Siegel disks which can be normalized to be $$f(z)=z\frac{z+e^{2\pi i\alpha}}{e^{2\pi i\beta}z+1}.$$ If one of the ratation numbers $\alpha$ and $\beta$ is of bounded type, can we get the result: 
$$Dim_{H}(J(f))<2.$$ I know when both $\alpha$ and $\beta$ are bounded type, the resule is true!
 A: Yes this is true (the first answer). Even if the rotation numbers are Zakeri-Petersen type $\log a_{n+1} = O( \sqrt{n} )$ (this class includes bounded-type). We can prove for instance that the the Julia sets for quadratic polynomials are of measure zero, and the Julia set is locally connected.
I think one can deduce this by mating two quadratic polynomials $P_\alpha (z) = e^{i 2 \pi \alpha} z + z^2$ and $P_\beta (z) = e^{i 2 \pi \beta} z + z^2$. With a suitable normalization you get a quadratic rational maps 
$$ f(z) = z \frac{z+e^{i 2 \pi \alpha}}{1+e^{i 2 \pi \beta} z}$$
and the previous properties are preserved under the operation of mating.
You can refer to:
Petersen-Zakeri paper: http://annals.math.princeton.edu/2004/159-1/p01
Petersen-Yampolsky paper on mating: http://arxiv.org/abs/math/9808009
The answer of the question is no. You can build quadratic rational maps such that its Julia set has Hausdorff dimension $2$. In the moduli space (isomorphic to $\mathbf C^2$) you can approach the infinity in some directions such that, the rational map become quadratic-like in some open subset containing exactly one critical point (the second one being "far"). The you proceed exactly like Shishikura (for quadratic polynomial), by iterating twice the parabolic renormalization.
You can refer to Shishikura's paper: http://arxiv.org/pdf/math/9201282.pdf
A: It seems that the state of the art is this paper of Zhou and Liao and references therein.
