The authors deal with the classical question: whether one can approximate each polynomial by a hyperbolic polynomial of the same degree. The authors give an answer to this question under the additional assumption that the original polynomial \(f\) is non-renormalizable and has only hyperbolic periodic points. They derive this result by proving that topologically conjugate non-renormalizable polynomials are quasi-conformally conjugate, while dealing with the additional complication that there may be several critical points.
In addition the authors prove that under the same non-renormalizable assumption (a) each point in the Julia set is contained in arbitrarily small puzzle pieces; (b) if the Julia set is connected, then it is locally connected and (c) \(f\) has no invariant linefields on its Julia set. This result implies the conjecture of B. Branner and J. H. Hubbard, stated in the cubic case [Acta Math. 169, No. 3–4, 229–325 (1992; Zbl 0812.30008)].