Quasi-conformal deformations of nonlinearizable germs

Let $f(z) = e^{2πi α}z + O(z^2), α\in \mathbb{R}$ be a germ of holomorphic diffeomorphism in $\mathbb{C}$. For $α$ 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 $α$ 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.

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