Systems of Markov functions generated by graphs and the asymptotics of their Hermite-Pade approximants

The paper considers Hermite-Pade approximants to systems of Markov functions defined by means of directed graphs. The minimization problem for the energy functional is investigated for a vector measure whose components are related by a given interaction matrix and supported in some fixed system of intervals. The weak asymptotics of the approximants are obtained in terms of the solution of this problem. The defining graph is allowed to contain undirected cycles, so the minimization problem in question is considered within the class of measures whose masses are not fixed, but allowed to 'flow' between intervals. Strong asymptotic formulae are also obtained. The basic tool that is used is an algebraic Riemann surface defined by means of the supports of the components of the extremal measure. The strong asymptotic formulae involve standard functions on this Riemann surface and solutions of some boundary value problems on it. The proof depends upon an asymptotic solution of the corresponding matrix Riemann-Hilbert problem.