On behaviour of holomorphically contractible systems under non-monotonic sequences of sets

Arkadiusz Lewandowski

doi:10.1007/s00013-015-0850-6

Abstract

The new results concerning the continuity of holomorphically contractible systems treated as set functions with respect to non-monotonic sequences of sets are given. In particular, continuity properties of Kobayashi and Caratheodory pseudodistances, as well as Lempert and Green functions with respect to sequences of domains converging in the Hausdorff metric are delivered.

Highlights

It is known that both Caratheodory and Kobayashi pseudodistances depend continuously on increasing and decreasing sequences of domains
The pseudodistances mentioned above are particular examples of a wider class of holomorphically contractible systems, i.e. systems of functions dD : D × D → [0, +∞), D running through all domains in all Cn’s, such that dD is forced to be p, the hyperbolic distance on D, the unit disc on the plane and all holomorphic mappings are contractions with respect to the system
In the present note, inspired by [1], we shall give a very general result stating the continuity of holomorphically contractible systems under the sequences of domains convergent with respect to the Hausdorff distance (for two nonempty bounded sets A, B it is defined as H(A, B) := inf{δ > 0 : A ⊂ B(δ) and B ⊂ A(δ)}, where for a set S and a positive number ε, the set S(ε) := s∈S B(s, ε) is the ε-envelope of S; B(x, r) denotes the open Euclidean ball of center x and radius r)

Summary

Introduction

It is known that both Caratheodory and Kobayashi pseudodistances depend continuously on increasing and decreasing sequences of domains (in the latter case, adding some regularity assumptions on the limiting domain; cf. [3] and references therein). In the present note, inspired by [1], we shall give a very general result stating the continuity of holomorphically contractible systems under the sequences of domains convergent with respect to the Hausdorff distance (for two nonempty bounded sets A, B it is defined as H(A, B) := inf{δ > 0 : A ⊂ B(δ) and B ⊂ A(δ)}, where for a set S and a positive number ε, the set S(ε) := s∈S B(s, ε) is the ε-envelope of S; B(x, r) denotes the open Euclidean ball of center x and radius r).

Results

Conclusion