Hermite-Padé approximants for systems of Markov functions generated by cyclic

The Cauchy transform of a positive measure with the support on some interval of the real axis is called a Markov-type function. Systems of Markov functions are the basic models for understanding the analytic properties and the asymptotic behavior of the Hermite-Padé approximants. In 1980 E. M. Nikishin has put forward a special system of Markov functions with supports on the same interval which now is called the Nikishin system. It appears that this model system nicely reflects the general features of analytic functions (from the point of view of the Hermite-Padé approximants). Convergence of Hermite-Padé approximants for Nikishin system formed by two functions has been proven by Nikishin himself. The convergence result for Nikishin system formed by arbitrary number of functions has been proven by G. López Lagomasino and J. Bustamante in 1992. In 1997 A. Gonchar, E. Rakhmanov and V. Sorokin have introduced a notion of a generalized Nikishin system — a system of Markov functions generated by a graph-tree; they investigated also convergence and asymptotic properties of Hermite-Padé approximants for this system. In our lecture we discuss about further developments in this topic.