A pr 1 99 5 Local connectivity of the Julia set of real polynomials

In particular, the Julia set of z + c1 is locally connected if c1 ∈ [−2, 1/4] and totally disconnected if c1 ∈ R \ [−2, 1/4] (note that [−2, 1/4] is equal to the set of parameters c1 ∈ R for which the critical point c = 0 does not escape to infinity). This answers a question posed by Milnor, see [Mil1]. We should emphasize that if the ω-limit set ω(c) of the critical point c = 0 is not minimal then it very easy to see that the Julia set is locally connected, see for example Section 10. Yoccoz [Y] already had shown that each quadratic polynomial which is only finitely often renormalizable (with non-escaping critical point and no neutral periodic point) has a locally connected Julia set. Moreover, Douady and Hubbard [DH1] already had shown before that each polynomial of the form z 7→ z + c1 with an attracting or neutral parabolic cycle has a locally connected Julia set. As will become clear, the difficult case is the infinitely renormalizable case. In fact, using the reduction method developed in Section 3 of this paper, it turns out that in the non-renormalizable case the Main Theorem follows from some results in [Ly3] and [Ly5], see the final section of this paper.