On the Supports of Vector Equilibrium Measures in the Angelesco Problem with Nested Intervals

Abstract

A vector logarithmic-potential equilibrium problem with the Angelesco interaction matrix is considered for two nested intervals with a common endpoint. The ratio of the lengths of the intervals is a parameter of the problem, and another parameter is the ratio of the masses of the components of the vector equilibrium measure. Two cases are distinguished, depending on the relations between the parameters. In the first case, the equilibrium measure is described by a meromorphic function on a three-sheeted Riemann surface of genus zero, and the supports of the components do not overlap and are connected. In the second case, a solution to the equilibrium problem is found in terms of a meromorphic function on a six-sheeted surface of genus one, and the supports overlap and are not connected.