Complete Conjugacy Invariants of Nonlinearizable Holomorphic Dynamics

TL;DR

This paper characterizes the conjugacy classes of nonlinearizable holomorphic germs near fixed points using hedgehogs and rotation numbers, establishing a complete invariant classification.

Contribution

It introduces a correspondence between hedgehogs and maximal abelian subgroups, and shows conjugacy is determined by rotation number and hedgehog conformal class.

Findings

Nonlinearizable germs with common hedgehogs commute.

Conjugacy is characterized by equal rotation numbers and conformally equivalent hedgehogs.

Hedgehogs uniquely determine conjugacy classes of nonlinearizable germs.

Abstract

Perez-Marco proved the existence of non-trivial totally invariant connected compacts called hedgehogs near the fixed point of a nonlinearizable germ of holomorphic diffeomorphism. We show that if two nonlinearisable holomorphic germs with a common indifferent fixed point have a common hedgehog then they must commute. This allows us to establish a correspondence between hedgehogs and nonlinearizable maximal abelian subgroups of Diff$(\bf C,0)$. We also show that two nonlinearizable germs are conjugate if and only if their rotation numbers are equal and a hedgehog of one can be mapped conformally onto a hedgehog of the other. Thus the conjugacy class of a nonlinearizable germ is completely determined by its rotation number and the conformal class of its hedgehogs.