Let Mn denote the space of complex n×n matrices, and let Ωn denote the spectral unit ball of Mn, namely the set of matrices in Mn whose eigenvalues lie within the open unit disc. As a step towards the eventual classification of the holomorphic automorphisms of Ωn, we prove that every such automorphism F satisfying F(0) = 0 and F′(0) = I has the property that F(x) is conjugate to x for each x∈Ωn. This result is obtained by combining earlier work of Ransford and White with a general theorem about spectrum‐preserving maps proved in this paper. 1991 Mathematics Subject Classification 15A18.

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Let {/? A } be an analytic family of rational maps and denote by J (A) the Julia set of 7? A . We prove that the upper semicontinuous regularization /*(A) of /(A) (which coincides with /(A) for all A in a dense open set) is a meromorphic multifunction, and give applications that illustrate the instability of Julia sets. In a similar vein, we also consider forward orbits of critical points and limit sets of Kleinian groups.

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A two-generator group 〈f, g〉 of Möbius maps is determined, up to conjugacy, by the numbers β = tr2(f) − 4, β′ = tr2(g) − 4 and γ = tr (fgf−1g−1) − 2, provided that γ ≠ 0. We study the subset D of C3 of those (β, β′, γ) which arise from discrete groups. In particular, we identify precisely D[setmn ]D.