Initial data regions leading to collisions of asteroid “Apophis” with the Earth in XXI century are investigated. Their detection and their properties research are necessary to be the most important task in order to predict dangerous collisions and to ward off them. Asteroid “Apophis” orbit was precised accurately in 2013 after its observations. Nevertheless many (more then 100) possible collisions exist in future, the asteroid can be regarded as a dangerous object as before. Locations and sizes of gaps leading to collisions are calculated and presented for different time instants. For example on Web site [9] characteristics of 9 possible impact collisions are presented in time scale of XXI century: they seem to be the more dangerous from the total collisions number. They were compared with our result, the comparison shows the satisfactionily to our predictions. The method of collisions trajectories discovery is discussed. The characteristics stability is estimated at small variation of motion theory, such as alternate models of planetary motions, numerical integration methods and so on. Similar structure of fractal kind can be predicted for initial data leading to collisions with other dangerous asteroids as the result of resonances orbital returns of their trajectories.