Relatively compact Siegel disks with non-locally connected boundaries

TL;DR

This paper constructs holomorphic maps with Siegel disks that have non-locally connected, indecomposable continuum boundaries, which are still contained within the domain of the map, revealing complex boundary structures.

Contribution

It provides explicit examples of injective holomorphic maps with Siegel disks having non-locally connected boundaries, a novel boundary behavior in complex dynamics.

Findings

Existence of holomorphic maps with Siegel disks having non-locally connected boundaries

Boundaries are indecomposable continua

Siegel disks are contained within the domain of the map

Abstract

We construct holomorphic maps with a Siegel disk whose boundary is not locally connected (and is an indecomposable continuum), yet compactly contained in the domain of definition of the map. Our examples are injective and defined on a subset of C.