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.