It stands to reason, then, that if the children of top earners are likely to stay in that income bracket, that people in the top income bracket are also likely to stay there.
Notice that the top 4% mentioned in this article fit quite comfortably in the top 20% of income earners. And through extrapolation, I imagine that the movement among the top 4% is even more severely diminished. It would be truly exceptional for even the child of someone in the top 4% to find themselves in poverty or for that matter the middle class.
[1] http://www.economicmobility.org/assets/pdfs/EMP_Across_Gener... (PDF! page 5)