  Loewner Theory, based on dynamical viewpoint, is a powerful tool in Complex
Analysis, which plays a crucial role in such important achievements as the
proof of famous Bieberbach's conjecture and well-celebrated Schramm's
Stochastic Loewner Evolution (SLE). Recently Bracci et al [Bracci et al, to
appear in J. Reine Angew. Math. Available on ArXiv 0807.1594; Bracci et al,
Math. Ann. 344(2009), 947--962; Contreras et al, Revista Matematica
Iberoamericana 26(2010), 975--1012] have proposed a new approach bringing
together all the variants of the (deterministic) Loewner Evolution in a simply
connected reference domain. We construct an analogue of this theory for the
annulus. In this paper, the first of two articles, we introduce a general
notion of an evolution family over a system of annuli and prove that there is a
1-to-1 correspondence between such families and semicomplete weak holomorphic
vector fields. Moreover, in the non-degenerate case, we establish a
constructive characterization of these vector fields analogous to the
non-autonomous Berkson - Porta representation of Herglotz vector fields in the
unit disk [Bracci et al, to appear in J. Reine Angew. Math. Available on ArXiv
0807.1594].

Complex Variables;Dynamical Systems