Journal of Approximation Theory vol:191 pages:1-37
Curves in the complex plane that satisfy the S-property were first introduced by Stahl and they were further studied by Gonchar and Rakhmanov in the 1980s.
Rakhmanov recently showed the existence of curves with the S-property in a harmonic external field by means of a max-min variational problem in logarithmic potential theory.
This is done in a fairly general setting, which however does not include the important special case of an external field Re V where V is a polynomial of degree at least 2.
In this paper we give a detailed proof of the existence of a curve with the S-property in the external field Re V within the collection of all curves that connect two
or more pre-assigned directions at infinity in which Re V tends to infinity.
Our method of proof is very much based on the works of Rakhmanov on the max-min variational problem and of Martinez-Finkelshtein and Rakhmanov on critical measures.