We are interested in the dynamics of cubic complex polynomials. Branner an Hubbard reduced the study of dynamics of cubic polynomials to the ones with a periodic critical point of a given period. In the corresponding moduli space such polynomials form degree one complex algebraic set.

We will explain how Jan Kiwi and I recently used Mary Rees' work on dynamics on Teichmüller space (inspired by Thurston's work on the characterization of analytic topological branched covers of the sphere ) to prove that our algebraic set is irreducible as conjectured by Milnor 25 years ago.