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!
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2 Answers
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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
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$\begingroup$ I konw that:$Dim_{H}(J(f))<2$ implies $LebJ(f))=0$. How about the inverse? $\endgroup$– RiemannCommented Aug 3, 2013 at 17:12
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$\begingroup$ Zakeri-Petersen type should be $\log a_{n} = O( \sqrt{n} )$. $\endgroup$– RiemannCommented Aug 6, 2013 at 7:39
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$\begingroup$ The inverser statement is false in general, you could have $Leb(K) = 0$ for some compact subset $K$ of $\mathbf C$ and $Dim_H(K) = 2$. For instance, for quadratic polynomials it was well know the existence of Julia sets with $Dim_H(K) = 2$ (work of Shishikura earlier of 90s) and we didn't even know if there exist or not Julia sets with positive Lebesgue measure. It was proved in $2004$ by X.Buff and A.Chéritat. $\endgroup$ Commented Sep 22, 2013 at 8:32
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It seems that the state of the art is this paper of Zhou and Liao and references therein.
answered Aug 2, 2013 at 15:53