Positive area and inaccessible fixed points for hedgehogs

Abstract

Let $f$ be a germ of holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in $\mathbb{C}$ (i.e. $f(0)=0,f^{\prime }(0)=e^{2{\it\pi}i{\it\alpha}},{\it\alpha}\in \mathbb{R}-\mathbb{Q}$). Perez-Marco showed the existence of a unique family of non-trivial invariant full continua containing the fixed point called Siegel compacta. When $f$ is non-linearizable (i.e. not holomorphically conjugate to the rigid rotation $R_{{\it\alpha}}(z)=e^{2{\it\pi}i{\it\alpha}}z$) the invariant compacts obtained are called hedgehogs. Perez-Marco developed techniques for the construction of examples of non-linearizable germs [Fixed points and circle maps. Acta Math. 179(2) (1997), 243–294]; these were used by the author to construct hedgehogs of Hausdorff dimension one, and adapted by Cheritat [Relatively compact Siegel disks with non-locally connected boundaries, Preprint, 2009] to construct Siegel disks with pseudo-circle boundaries. We use these techniques to construct hedgehogs of positive area and hedgehogs with inaccessible fixed points.