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The initial and terminal cluster sets of an analytic curve
2016, Gauthier, Paul
For an analytic curve γ:(a,b)→C, the set of values approaches by γ(t), as t↘a and as t↗b can be any two continuua of C∪{∞}.
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Spherical arc-length as a global conformal parameter for analytic curves in the Riemann sphere
2016, Gauthier, Paul, Nestoridis, Vassili, Papadopoulos, Athanase
We prove that for every analytic curve in the complex plane C, Euclidean and spherical arc-lengths are global conformal parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in Rn and Cn and we discuss the situation of curves in the Riemann sphere C {∞}.