Quadratic Julia sets with positive area (Q456662)

The authors prove the existence of quadratic polynomials having a Julia set with positive Lebesgue measure. They provide such examples with a Cremer fixed point, with a Siegel disk, or with infinitely many satellite renormalizations.NEWLINENEWLINEThe main part of the article is to show the example of the Cremer case. It is a key point to prove the following proposition. There exists a non empty set \(S\) of bounded type irrationals such that: for all \(\alpha \in S\) and all \(\varepsilon>0\), there exists \(\alpha^{\prime} \in S\) with (1) \(|\alpha^{\prime}-\alpha|<\varepsilon\); (2) the quadratic polynomial \(P_{\alpha^{\prime}}\) has a cycle in the Euclidean disk \(D(0, \varepsilon) \backslash 0\); (3) area \((K_{\alpha^{\prime}})\geq (1-\varepsilon)\) area \((K_{\alpha})\), where \(K_{\alpha^{\prime}}\) and \(K_{\alpha}\) denote the filled-in Julia set of \(P_{\alpha^{\prime}}\) and \(P_{\alpha}\) respectively. As an immediate consequence, there exists a Cauchy sequence \(\{\alpha_n\}\) with a limit \(\alpha\) such that \(P_{\alpha}\) is the Cremer quadratic polynomial with positive Lebesgue measure.NEWLINENEWLINEThe proofs are based on three tools. The first one is McMullen's Lebesgue density near the boundary of a Siegel disk with bounded rotation number. The second one is Chéritat's techniques of parabolic explosion and Yoccoz's renormalization techniques to control the shape of the Siegel disk. The last one is Inou and Shishikura's results of parabolic renormalization to control the post-critical sets of perturbed polynomials having an indifferent fixed point.