Approximation properties of the poles of diagonal Padé approximants for certain generalizations of Markov functions

A non-linear system of differential equations (‘`generalized Dubrovin system") is obtained to describe the behaviour of the zeros of polynomials orthogonal on several intervals that lie in lacunae between the intervals. The same system is shown to describe the dynamical behaviour of zeros of this kind for more general orthogonal polynomials: the denominators of the diagonal Padé approximants of meromorphic functions on a real hyperelliptic Riemann surface.
On the basis of this approach several refinements of Rakhmanov’s results on the convergence of diagonal Padé approximants for rational perturbations of Markov functions are obtained.