Universal natural families for entire functions
speaker: Gustavo Ferreira (Centre de Recerca Matemàtica (Barcelona))abstract: Given an entire function f, the number of possible parameter spaces containing it is virtually infinite. However, for many applications, it is understood that an "adequate" parameter space is the quasiconformal equivalence class of f, i.e. the set of all entire functions g such that g(h2(z)) = h1(f(z)) for all z, where h1 and h2 are quasiconformal homeomorphisms of the complex plane. A slice of this set depending holomorphically on one or more complex parameters is called a natural family, and it was shown by Eremenko and Lyubich that, if f has finitely many singular values, then its equivalence class is in itself a finite-dimensional complex manifold, and thus a natural family. In this talk, we will study equivalence classes of more general entire functions. We will see that there exists a complex manifold M and a well-defined map F from M to the equivalence class of f that turns the latter into a natural family, and that any other natural family containing f can be naturally "lifted" to M, meaning that M defines a "universal natural family" for f. We will explore the properties of F and M, obtaining several interesting consequences – such as, for instance, the fact that M is finite-dimensional if and only if f has finitely many singular values, in which case we recover Eremenko and Lyubich's result.
timetable:
Thu 30 May, 16:45 - 17:25, Aula Dini
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