We show that there exist polynomial endomorphisms of C 2 , possessing a wandering Fatou component. These mappings are polynomial skewproducts, and can be chosen to extend holomorphically of P 2 (C). We also find real examples with wandering domains in R 2 . The proof relies on parabolic implosion techniques, and is based on an original idea of M. Lyubich.

We prove the existence of quadratic polynomials having a Julia set with positive Lebesgue measure. We find such examples with a Cremer fixed point, with a Siegel disk, or with infinitely many satellite renormalizations.

Let f : P 1 → P 1 be a rational map with finite postcritical set P f . Thurston showed that f induces a holomorphic map σ f : Teich(P 1 , P f ) → Teich(P 1 , P f ) of the Teichmüller space to itself.