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

TL;DR

This paper investigates the continuity properties of holomorphically contractible systems, such as Kobayashi and Carathéodory pseudodistances, under non-monotonic domain sequences converging in the Hausdorff metric.

Contribution

It provides new results on the continuity of these systems as set functions with respect to non-monotonic sequences of domains.

Findings

Continuity properties of Kobayashi and Carathéodory pseudodistances established.

Continuity of Lempert and Green functions under Hausdorff convergence shown.

Results extend understanding of holomorphic invariants under non-monotonic domain changes.

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 Carath\'eodory pseudodistances, as well as Lempert and Green functions with respect to sequences of domains converging in Hausdorff metric are delivered.