In 1990--1998, my main focus was the dynamics in the real quadratic family, which I approached by complex methods.
The main results are:
  • No wild attractors [2] (solution of Milnor's Problem in the quadratic case);
  • Density of hyperbolic maps [3];
  • Proof of the Feigenbaum-Coullet-Tresser Renormalization Conjecture [5];
  • Typical non-regular and non-renormalizable maps are stochastic (satisfying the Martens-Nowicki criterion) [6];
  • Construction of the Full Renormalization Horseshoe [7].

    The last two papers imply the Regular or Stochastic Theorem : Almost any real quadratic map is either regular or stochastic.

    Note [8] is a survey on the subject based on my plenary talk at the AMS meeting in Washington, DC (January 2000).