Kharchenko N. Spectral and topological properties of structurally simmilar measures

In the dissertation the spectral and topological properties of structurally simmilar measures are investigated. In the space of structurally simmilar measures are defined dynamical conflict systems. It is proved that all trajectories of dynamical systems in measures space converge to fixed invariant points with one spectral component in the Lebesgue decomposition: purely point, purely absolutely continuous or purely singular continuous. In terms of the original measure found sufficient conditions for the limiting measures to belong to a certain spectral type. Also in the dissertation the topological properties of the carriers limiting measures are discribed and criteries of belonging limiting measures to C-, S- and P-type are established. The inverse problem of a dynamical conflict system is solved. In particular, in terms of pairs of structurally similar measures all measures, that are limiting for trajectories of dynamical systems of conflict, are described and all pairs of structurally similar measures, which are basic for the trajectories converge to the forward given limiting pair of measures, are parameterized. The structure and properties of the spectrum of limiting purely point measures are studied. A method of constructing purely point measure of forward given spectral distribution is proposed.