This argument is from Ermenko-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 $\mu_f$.

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 $\mu_f$.