Quasi-conformal deformations of nonlinearizable germs

TL;DR

This paper investigates the conditions under which nonlinearizable holomorphic germs can be deformed via quasi-conformal conjugacies, revealing new infinite-dimensional families of such deformations when certain dynamical features are present.

Contribution

It demonstrates that nonlinearizable germs with a sequence of repelling periodic orbits can be embedded into infinite-dimensional families of quasi-conformal conjugates, expanding understanding of their deformation space.

Findings

Existence of infinite-dimensional quasi-conformal deformation families

Deformation depends on the presence of converging repelling periodic orbits

No two germs in the family are conformally conjugate

Abstract

Let $f(z) = e^{2\pi i \alpha}z + O(z^2), \alpha \in \mathbb{R}$ be a germ of holomorphic diffeomorphism in $\mathbb{C}$. For $\alpha$ rational and $f$ of infinite order, the space of conformal conjugacy classes of germs topologically conjugate to $f$ is parametrized by the Ecalle-Voronin invariants (and in particular is infinite-dimensional). When $\alpha$ is irrational and $f$ is nonlinearizable it is not known whether $f$ admits quasi-conformal deformations. We show that if $f$ has a sequence of repelling periodic orbits converging to the fixed point then $f$ embeds into an infinite-dimensional family of quasi-conformally conjugate germs no two of which are conformally conjugate.