Computational Methods and Function Theory 9 (2009), No. 2, 473--484
Copyright Heldermann Verlag 2009
Patrick Ahern
ahern@math.wisc.edu , University of Wisconsin, Mathematics Department, Madison, WI 53706, U.S.A.
Anthony G. O'Farrell
anthonyg.ofarrell@gmail.com , National University of Ireland, Mathematics Department, Maynooth, Co. Kildare, Ireland.
Let G be a group. We say that an element f∈ G is reversible in G if it is conjugate to its inverse, i.e. there exists g∈ G such that g-1fg=f-1. We denote the set of reversible elements by R(G). For f∈ G, we denote by R_f(G) the set (possibly empty) of reversers of f, i.e. the set of g∈ G such that g-1fg=f-1. We characterise the elements of R(G) and describe each Rf(G), where G is the the group of biholomorphic germs in one complex variable. That is, we determine all solutions to the equation fogof = g, in which f and g are holomorphic functions on some neighbourhood of the origin, with f(0)=g(0)=0 and f'(0)≠0≠g'(0).
Keywords: biholomorphic germ, reversible, group.
MSC 2000: Primary 30D05; Secondary 39B32, 37F99, 30C35.
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