Quasi-conformal deformations of nonlinearizable germs

Abstract:

Let $f(z) = e^{2\pi i \alpha }z + O(z^2), \alpha \in \mathbb {R}$, be a germ of a 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.