TY - JOUR
AU1 - Beardon, A. F.
AU2 - Minda, D.
AB - In this paper we review the familiar collection of results that concern holomorphic maps of a disc or half-plane into itself that are due to Schwarz, Pick, Julia, Denjoy and Wolff. We give a coherent geometric treatment of these results entirely in terms of the ideas of geodesics, horocycles and G-spaces as introduced by Busemann. In particular, we show that the results of Wolff and Julia hold for all weak contractions of the hyperbolic metric (whether holomorphic or not); holomorphicity plays no role in the arguments. These results apply to holomorphic maps because the Schwarz–Pick lemma implies that holomorphic maps are weak contractions. An important ingredient in the proofs are several projections of the hyperbolic plane onto a geodesic which are weak contractions relative to the hyperbolic distance.
TI - Geometric Julia–Wolff Theorems for Weak Contractions
JF - Computational Methods and Function Theory
DO - 10.1007/s40315-022-00475-z
DA - 2023-12-01
UR - https://www.deepdyve.com/lp/springer-journals/geometric-julia-wolff-theorems-for-weak-contractions-lORIXixEc0
SP - 741
EP - 770
VL - 23
IS - 4
DP - DeepDyve
ER -