Absorbing Cantor sets in dynamical systems: Fibonacci maps
  In this paper we shall show that there exists a polynomial unimodal map f: [0,1] -> [0,1] which is   1) non-renormalizable(therefore for each x from a residual set, $\omega(x)$ is equal to an interval),   2) for which $\omega(c)$ is a Cantor set, and   3) for which $\omega(x)=\omega(c)$ for Lebesgue almost all x.   So the topological and the metric attractor of such a map do not coincide. This gives the answer to a question posed by Milnor. 
Henk Bruin, Gerhard Keller, Tomasz Nowicki, Sebastian van Strien