## Institute for Mathematical Sciences

## Preprint ims05-03

** Jeremy Kahn, Mikhail Lyubich**
* Local connectivity of Julia sets for unicritical polynomials*

Abstract: We prove that the Julia set $J(f)$ of at most finitely renormalizable unicritical polynomial $f:z\mapsto z^d+c$ 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 Principle Nest of puzzle pieces.
The proof of a priori bounds makes use of new analytic tools developed in IMS Preprint #2005/02
that give control of moduli of annuli under maps of high degree.

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