Boundaries of Siegel disks - numerical studies of their dynamics and regularity

TL;DR

This paper numerically investigates the boundary dynamics of Siegel disks in holomorphic maps, revealing universal regularity properties and scaling exponents that depend on critical point order and rotation number.

Contribution

It introduces a numerical method to compute the boundary parameterization of Siegel disks and analyzes their regularity and scaling properties, highlighting universality in these features.

Findings

Boundary regularity is universal across maps with similar critical points.

Scaling exponents depend only on critical point order and rotation number tail.

Numerical methods effectively characterize boundary dynamics of Siegel disks.

Abstract

Siegel disks are domains around fixed points of holomorphic maps in which the maps are locally linearizable (i.e., become a rotation under an appropriate change of coordinates which is analytic in a neighborhood of the origin). The dynamical behavior of the iterates of the map on the boundary of the Siegel disk exhibits strong scaling properties which have been intensively studied in the physical and mathematical literature. In the cases we study, the boundary of the Siegel disk is a Jordan curve containing a critical point of the map (we consider critical maps of different orders), and there exists a natural parameterization which transforms the dynamics on the boundary into a rotation. We compute numerically this parameterization and use methods of harmonic analysis to compute the global Holder regularity of the parameterization for different maps and rotation numbers. We obtain that the regularity of the boundaries and the scaling exponents are universal numbers in the sense of renormalization theory (i.e., they do not depend on the map when the map ranges in an open set), and only depend on the order of the critical point of the map in the boundary of the Siegel disk and the tail of the continued function expansion of the rotation number. We also discuss some possible relations between the regularity of the parameterization of the boundaries and the corresponding scaling exponents.