Metric domains, holomorphic mappings and nonlinear semigroups

Abstract

Abstract. We study nonlinear semigroups of holomorphic mappings on certain domains in complex Banach spaces. We examine, in particular, their differentiability and their representations by exponential and other product formulas. In addition, we also construct holomorphic retractions onto the stationary point sets of such semigroups. Introductionwhere the limit is taken with respect to the topology of D.A subset W of D is said to be the stationary point set of S if it consists of all the points a ∈ D such that F t (a) = a for all t ∈ (0, T).In other words,1991 Mathematics Subject Classification. Primary: 32H15, 34G20, 46G20, 47H17, 47H20.

Another look at the Burns-Krantz theorem

“…Since aut(∆) is a real Banach algebra (see, for example, [23] and [2]), we get that f ∈ aut(∆). Moreover, f generates a group of hyperbolic automorphisms on ∆ if and only if f…”

“…Since G (∆) is a real cone (see, for example, [23]) we get that g must belong to G (∆). Assertion (ii) is proved.…”

The weak lower semicontinuity of the Kobayashi distance and its applications

“…This is no longer true in the infinite dimensional case. Here we have to mention that the locally uniform convergence of iterates is a necessary claim in many applications ( [21], [29], [30], [31], [32], [33], [34]). …”

A flower structure of backward flow invariant domains for semigroups

“…This explains the name "continuous semigroup" in our terminology. Furthermore, it follows by a result of E. Berkson and H. Porta [8] that each continuous semigroup is differentiable in t ∈ R + = [0, ∞), (see also [1] and [30]). So, for each continuous semigroup (semiflow) S = {F t } t≥0 ⊂ Hol(D), the limit lim…”

“…Let now D = ∆ be the open unit disk in C. In this case, G(∆) is a real cone in Hol(∆, C), while aut(∆) ⊂ G(∆) is a real Banach space (see, for example, [30]). Moreover, by the Berkson-Porta representation formula, a function f belongs to G(∆) if and only if there is a point τ ∈ ∆ and a function p ∈ Hol(∆, C) with positive real part (Re p(z) ≥ 0 everywhere) such that…”