You see this effect anywhere that supply of rental housing is a problem. If you implement price controls, people with high incomes will seek out lower rents. They'll still often end up driving out the lower-income folks, because if you're a landlord with two applications for a vacant apartment, you're going to favor the one with better credit. Rent controls also deter landlords from investing, both in existing properties and in new ones.
IMO the right answer (and this is not a popular sentiment) is to let rents float and have certain desirable areas of certain cities end up effectively "off limits" to lower income renters. Is it fair? Not particularly, but the side effects of this situation are that the people who can afford to shoulder the costs associated with their way of life are the ones actually doing so. They are going to crowd low-income renters out of the neighborhood eventually anyway, and hopefully a housing / worker shortage would prompt the city to actually develop a low income housing solution rather than forcing landlords to just deal with rents too low for even basic maintenance.