Pluripotential theory, semigroups and boundary behavior of infinitesimal generators in strongly convex domains

Abstract

Abstract. We characterize infinitesimal generators of semigroups of holomorphic selfmaps of strongly convex domains using the pluricomplex Green function and the pluricomplex Poisson kernel. Moreover, we study boundary regular fixed points of semigroups. Among other things, we characterize boundary regular fixed points both in terms of the boundary behavior of infinitesimal generators and in terms of pluripotential theory.

Evolution families and the Loewner equation I: the unit disc

“…On the other hand, by definition of Herglotz vector field, fixed a point t, the map D ∋ z → G(z, t) is the infinitesimal generator of a semigroup of holomorphic self-maps of the unit disc for almost every t ∈ [0, +∞). Therefore, by [5,Thm. 0…”

“…In particular, in the proof of the above theorem we use a result of [5], from which it follows that (dρ D ) (z,w) (G(z, t), G(w, t)) ≤ 0 for all t ≥ 0 and z = w, where ρ D is the hyperbolic distance on D. This estimate allows us to avoid considering displacement of fixed points in order to obtain suitable bounds. In fact, a version of Theorem 1.1 holds more generally on complex complete hyperbolic manifolds whose Kobayashi distance is C 1 (see [6]).…”

A Julia-Wolff-Carathéodory theorem for infinitesimal generators in the unit ball

“…Thanks to [BCD,Theorem 0.4], we have that lim t→1 − d dt (G 1 (te 1 )) = β, and then we are done, because d dt (G 1 (te 1 )) = dG te 1 (e 1 ), e 1 . (iv) Without loss of generality we can assume p = e 1 and v = e 2 , so that the quotient we would like to study is…”

Analytic Semigroups of Holomorphic Mappings and Composition Operators

“…The one-dimesional case. First we consider the one-dimensional case and quote a familiar formula of [BP78] which gave a great push in the development of semigroup theory [Aba92, Gor93, RS96, CD05, CDP06, ES10], Loewner's evolution equation theory [BCD10] and the theory of semigroups of composition operators. Write D for the unit (open) disk in C.…”