Abstract: The MLC Conjecture was put forward by Douady and Hubbard in the mid-1980s and, since then, has remained a central question in Complex Dynamics. The universality of period-doubling cascade (renormalization theory) was discovered in the mid-70s by Feigenbaum (the parameter part) and independently by Coullet and Tresser (the dynamical part). In the early 1990s, the MLC conjecture was closely linked to the renormalization theory in the work of Yoccoz by reducing the conjecture to the infinitely renormalizable case. We will first discuss various motivations and developments around the MLC conjecture, and then we outline some ideas of the proof that the MLC holds at the classical period-doubling Feigenbaum parameter.
Joint work with Misha Lyubich.