The goal of the PhD program is to carefully test the hypotheses of the uniqueness theorems for the final states of black holes. This will be done within the framework of mathematical general relativity. This means first of all a careful examination of the assumptions of the results on the Kerr spacetime. According to well-known arguments, the solution of all stationary black holes of the electro-vacuum Einstein equations can be characterized by the values of three observable classical (asymptotic) parameters: mass, electric charge, and angular momentum. Previous studies show that, for example, in the case of Kerr black holes, the correctness of several of the conditions of the theorems is questionable, so that the proofs of the black hole uniqueness theorems for general configurations are not fully satisfactory. The most important rotating black hole solution of Einstein's theory is the Kerr spacetime, so the first line of the proposed research is directed towards understanding the fundamental properties of this solution. The details of the arguments for the uniqueness of the Kerr solution as well as the results on the stability of Kerr black holes will be reviewed. Special emphasis will be given to the study of how perturbations affect the final state properties of the rotating black hole spacetime. How they can lead from one Kerr state to another state with different asymptotic parameters. It is expected that these investigations will lead to the formulation of a dynamical version of the black hole ambiguity theorem. The results of the planned studies will be crucial in deciding whether the Kerr family of black holes indeed satisfies the no-hair theorem, and if so, in what sense.