Dynamics Seminar

Event Information Story of Julia sets of positive area, through the renormalization lens
14:10 on Monday March 04, 2024
15:00 on Monday March 04, 2024
BA6183, Bahen Center, 40 St. George St.
Mikhail Lyubich
Stony Brook University

It was a classical problem going back to Fatou of whether a Julia set of a polynomial may have positive area. It was realized in the 1980s that it is intimately related to the problem of ``wild" attractors. Examples of entire functions of this kind were constructed in the 1980s, but it took another 20 years to produce polynomial ones. First examples, of Cremer polynomials, were constructed by Buff and Cheritat. More abunndant and robust class of examples, of Feigenbaum polynomials, was later produced by Avila and the author. Moreover, these maps admit a perturbation to polynomial automorphisms of C^2. The whole story is based upon several renormalization theories that allow one to control precisely small scale geometry of various dynamical systems and bifurcation loci.