Abstract
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 −1 fg = 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 −1 fg = f −1. We characterise the elements of R(G) and describe each R f(G), where G is the the group of biholomorphic germs in one complex variable. That is, we determine all solutions to the equation f ◦ g ◦ f = 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).
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The second author was supported by Grant SFI RFP05/MAT0003 and the ESF Network HCAA.
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Ahern, P., O’Farrell, A.G. Reversible Biholomorphic Germs. Comput. Methods Funct. Theory 9, 473–484 (2009). https://doi.org/10.1007/BF03321741
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DOI: https://doi.org/10.1007/BF03321741