Can anyone show me the proof "Hausdorff dimension of Julia set is strictly positive"?
For purpose to prove this we might have to prove the green function of basin of attraction to infinity is Holder continuous.



Hausdorff dimension of Julia set
This argument is from [Eremenko-Lyubich survey](https://www.math.purdue.edu/%7Eeremenko/dvi/AAsurveyp2.pdf): Let $f$ be a rational map and $\mu$ an $f$-invariant ergodic measure on $\widehat{\Bbb{C}}$. By [Ledrappier-Young entropy formula](http://www.scholarpedia.org/article/Pesin_entropy_formula#The_Ledrappier-Young_Entropy_Formula), $h\_\mu(f)$ is the product of the Lyapunov exponent $\chi\_\mu$ by the dimension of the measure. Now pick an invariant ergodic measure with positive entropy supported in the Julia set; e.g. the measure of maximal entropy $\mu\_f$ whose entropy is $\log(\deg f)$. We deduce that
$$
\dim\mu\_f=\frac{\log(\deg f)}{\chi\_{\mu\_f}}
$$  

is non-zero. Now notice that the Hausdorff dimension of $\mathcal{J}(f)={\rm{supp}}\mu\_f$ cannot be smaller than the dimension of $\mu\_f$.


