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.