Local connectivity of Julia sets for unicritical polynomials

We prove that the Julia set J (f) of at most finitely renormalizable unicritical polynomial f : z {upwards arrow from bar} z^d + c with all periodic points repelling is locally connected. (For d = 2 it was proved by Yoccoz around 1990.) It follows from a priori bounds in a modified Principal Nest of puzzle pieces. The proof of a priori bounds makes use of new analytic tools developed in [KL09] that give control of moduli of annuli under maps of high degree.