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Structural stability of holomorphic functions has been the subject of muchresearch in the last fifty years. Due to various technicalities, however, mostof that work has focused on so-called finite-type functions (functions whoseset of singular values has finite cardinality). Recent developments in thefield go beyond this setting. In this paper we extend Eremenko and Lyubich'sresult on natural families of entire maps to the case where the set of singularvalues is not the entire complex plane, showing under this assumption that theset $M_f$ of entire functions quasiconformally equivalent to $f$ admits thestructure of a complex manifold (of possibly infinite dimension). Moreover, wewill consider functions with wandering domains -- another hot topic of researchin complex dynamics. Given an entire function $f$ with a simply connectedwandering domain $U$, we construct an analogue of the multiplier of a periodicorbit, called a distortion sequence, and show that, under some hypotheses, thedistortion sequence moves analytically as $f$ moves within appropriateparameter families.

Holomorphic motions, natural families of entire maps, and multiplier-like objects for wandering domains