Hausdorff dimension of Julia set

Question

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.

Answer 1

This argument is from Eremenko-Lyubich survey: Let $f$ be a rational map and $\mu$ an $f$-invariant ergodic measure on $\widehat{\Bbb{C}}$. By 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 $\mu_f$.

Answer 2

Here is a fairly elementary argument, which works in great generality (in particular for rational and transcendental entire functions, and indeed suitably modified for meromorphic functions and classes beyond).

The point is that we can find, near any point of the Julia set, a conformal iterated function system with at least two branches, which are branches of the inverse function of f.

For example, take a repelling periodic point $z_0$. Since such points are dense in the Julia set, we may assume that $z_0$ has simple preimages near every point of the Julia set. (Indeed, the set of points for which this is not the case is finite by the normal-families version of Nevanlinna's theorem, or by an elementary counting argument for polynomials and rational functions.)

Take a small linearising disc neighbourhood $D$ of $z_0$, and a simple preimage $z_1\neq z_0$ of $z_0$ under some $f^n$ that belongs to $D$. For large $k$, there is a univalent branch of $f^{n+k}$ defined on $D$, which first pulls back $D$ into itself $k$ times, and then pulls $D$ back to $z_1$. So we have two branches of $f^{-(n+k)}$ on $D$, one which takes $z_0$ to $z_1$, and one which fixes $z_0$.

Together, these form a conformal iterated function system, whose limit set is contained in the Julia set of $f$. The Hausdorff dimension of such a limit set is always positive, by an elementary argument.