Abstract: The universality in one-dimensional dynamics was discovered numerically by Feigenbaum and Coullet-Tresser in the late 1970s. (Very soon, similar observations were made for some important high-dimensional non-linear dynamical systems such as Lorenz system of differential equations.) It is described by fixed points of renormalization operators R of the form
RH=a\circ H^p \circ a^{-1},
where a is a re-scaling. For every real number c>1, the universality is observed in a space of maps H with a single critical point of order c. In a recent joint work with Greg Swiatek, we prove that the family H_c of corresponding fixed-point maps of R for different criticalities c converges as c goes to infinity.

In the talk, which is going to be elementary, we describe the history of the area including a rigorous computer-assisted approach to the above problem by Eckmann and Wittwer, 1985.