On the Finiteness of Attractors for One-Dimensional Maps with   Discontinuities
  Since the proof, at the end of the 80's, of the finiteness of the number of attractors for $C^3$ maps of the interval having negative Schwarzian derivative, it has been generally considered that the same result could be true for maps with discontinuities. In the present paper we show that this is indeed the case. 
Paulo Brand\~ao, Jacob Palis and Vilton Pinheiro
math.DS