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Loewner Theory, based on dynamical viewpoint, is a powerful tool in ComplexAnalysis, which plays a crucial role in such important achievements as theproof of famous Bieberbach's conjecture and well-celebrated Schramm'sStochastic Loewner Evolution (SLE). Recently Bracci et al [Bracci et al, toappear in J. Reine Angew. Math. Available on ArXiv 0807.1594; Bracci et al,Math. Ann. 344(2009), 947--962; Contreras et al, Revista MatematicaIberoamericana 26(2010), 975--1012] have proposed a new approach bringingtogether all the variants of the (deterministic) Loewner Evolution in a simplyconnected reference domain. We construct an analogue of this theory for theannulus. In this paper, the first of two articles, we introduce a generalnotion of an evolution family over a system of annuli and prove that there is a1-to-1 correspondence between such families and semicomplete weak holomorphicvector fields. Moreover, in the non-degenerate case, we establish aconstructive characterization of these vector fields analogous to thenon-autonomous Berkson - Porta representation of Herglotz vector fields in theunit disk [Bracci et al, to appear in J. Reine Angew. Math. Available on ArXiv0807.1594].

Loewner theory in annulus I: Evolution families and differential equations