Positive area and inaccessible fixed points for hedgehogs

TL;DR

This paper explores the construction of hedgehogs with positive area and inaccessible fixed points for certain holomorphic diffeomorphisms, extending previous work on invariant sets in complex dynamics.

Contribution

It introduces new techniques to construct hedgehogs with positive area and inaccessible fixed points, advancing understanding of invariant sets in complex dynamics.

Findings

Constructed hedgehogs of positive area.

Constructed hedgehogs with inaccessible fixed points.

Extended techniques for non-linearizable germs.

Abstract

Let f be a germ of holomorphic diffeomorphism with an irra- tionally indifferent fixed point at the origin in C (i.e. f(0) = 0, f'(0) = e 2pi i alpha, alpha in R - Q). Perez-Marco showed the existence of a unique family of nontrivial 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_{alpha}(z) = e 2pi i z) the invariant compacts obtained are called hedgehogs. Perez-Marco developed techniques for the construction of examples of non-linearizable germs; these were used by the author to construct hedge- hogs of Hausdorff dimension one, and adapted by Cheritat to construct Siegel disks with pseudo-circle boundaries. We use these techniques to construct hedgehogs of positive area and hedgehogs with inaccessible fixed points.