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The local behavior of principal and chordal principal cluster sets

John T. Gresser
1972 Transactions of the American Mathematical Society  
Let K be the unit circle, and let/ be a function whose domain is the open unit disk and whose range is a subset of the Riemann sphere. We define a set, called the boundary principal cluster set of/at £0 6 K, which characterizes the behavior of the principal cluster sets of fat points £ s K which are near £0 and distinct from £0. It is shown that iff is continuous, then the principal and boundary principal cluster sets off at £o are equal for nearly every point i0 6 Í. A similar result holds for
more » ... ar result holds for chordal principal cluster sets. Examples are provided that indicate directions in which the result cannot be improved. Some results concerning points that are accessible through sets which are unions of arcs are also presented.
doi:10.1090/s0002-9947-1972-0294643-3 fatcat:62pey2cspnbxplsfrpzzgbhjzy