Geodesic
On a question of T. Sheil-Small regarding valency of harmonic maps
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 66 (2012) no. 2.

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The aim of this work is to answer positively a more general question than the following which is due to T. Sheil-Small: Does the harmonic extension in the open unit disc of a mapping f from the unit circle into itself of the form f(e^it) = e^iϕ(t), 0≤ t ≤ 2π where ϕ is a continuously non-decreasing function that satisfies ϕ(2π)-ϕ(0) = 2Nπ, assume every value finitely many times in the disc?
Keywords: Harmonic mapping, cluster set 
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Bshouty, Daoud; Lyzzaik, Abdallah. On a question of T. Sheil-Small regarding valency of harmonic maps. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 66 (2012) no. 2. http://geodesic.mathdoc.fr/item/AUM_2012_66_2_a4/

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[2] Bshouty, D., Hengartner, W., Lyzzaik, A. and Weitsman, A., Valency of harmonic mappings onto bounded convex domains, Comput. Methods Funct. Theory 1 (2001), 479-499.

[3] Duren, P., Harmonic Mappings in the Plane, Cambridge University Press, Cambridge, 2004.

[4] Markushevich, A. I., Theory of functions of a complex variable. vol. III, English edition translated and edited by Richard A. Silverman, Prentice-Hall Inc., N. J., 1967.