Semigroups in Geometrical Function Theory

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Pluripotential theory, semigroups and boundary behavior of infinitesimal generators in strongly convex domains (2006, background): “For D = D, the unit disc of C, there is a very nice representation formula, due to Berkson and Porta [6] (see also [1] and [34]).”
Loewner chains in the unit disk (2009, background): “Given a semigroup Φ = (φt), it is well-known (see [33], [4]) that there exists a unique holomorphic function G : D → C such that, ∂φt(z) ∂t = G (φt(z)) = G (z) ∂φt(z) ∂z for all z ∈ D and t ≥ 0.”
Evolution families and the Loewner equation I: the unit disc (2008, unknown): “l generators of semigroups of holomorphic self-maps of the unit disc. It is well-known that Gen(D) is closed in Hol(D,C) and a real convex cone in Hol(D,C) with vertex at 0 (see, for example, [1] and [26]). A useful example of inﬁnitesimal generator is given by G= ϕ− id for ϕ∈ Hol(D,D) [26, Corollar”
Analytic flows on the unit disk : Angular derivatives and boundary fixed points (2005, background): “253 Given a semigroup 8= (ϕt), it can be proved (see [Shoikhet 2001; Berkson and Porta 1978]) that there exists a unique analytic function G : D → C such that, for each z ∈ D, the trajectory γz : [0, +∞) → R2, t 7→ γz(t) := ϕt(z) is the solution of the Cauchy problem{ ẋ = f (x, y), ẏ = g(x, y),…”