In Europe, with high energy prices and cities that have been rebuilt multiple times, the rich people live in the city center and poor people live in some of the suburbs. For example, see Paris.
Energy was cheap in the US and this enabled the development of the car commuting culture as people built homes farther and farther away from factories and offices in the city center. Later, the factories and offices tended to also move to "edge cities".
This evolution has only occurred once for most US cities. For example, just after WW II, Minneapolis only extended to about its current southern boundary. Subsequently, Richfield, Bloomington, and Apple Valley have been built as Cedar Avenue and I-35 provided commuting corridors to the south. Housing nearest the downtown aged and wealthier citizens moved out as areas became more run down.
Gentrification is the beginning of a second wave of rebuilding and development, as expensive housing is built or rehabbed in the city center. Poor inhabitants will be forced to migrate outwards into the next ring of run down housing, and the cycle of rebuilding will continue.
Meanwhile, energy prices are going to rise, so this cycle is likely to be permanent, with US cities evolving towards the European settlement pattern, with the upper class downtown, and the middle class and poor in separate sectors around the periphery.