Quasi-conformal deformations of nonlinearizable germs

Abstract

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