Vol. 141, No. 2, 1990

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Conformal cluster sets and boundary cluster sets coincide

Raimo Näkki

Vol. 141 (1990), No. 2, 363–382
Abstract

The result stated in the title is proved. No restriction other than the obvious requirement that the cluster sets be taken at a nonisolated boundary point is imposed on the domain where the mapping is defined. The result is then generalized by allowing for certain exceptional sets on the boundary. More refined versions are established in the special case where the domain is the open unit disk. These include the statement that one-sided cluster sets coincide with one-sided radial cluster sets. Again, certain exceptional sets on the boundary are allowed for. Consequences are presented in which the existence of limits along sets on the boundary implies limits inside the domain. Finally, generalizations to the class of homeomorphisms satisfying the Carathéodory Prime End Theorem are indicated.

Mathematical Subject Classification 2000
Primary: 30D40
Secondary: 30C35, 30C85
Milestones
Received: 5 June 1986
Revised: 7 December 1988
Published: 1 February 1990
Authors
Raimo Näkki