Local Connectivity of the Julia Set of Real Polynomials

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 ...

Local Connectivity of Julia Sets

We prove that the Julia set J(f) of at most finitely renormalizable unicritical polynomial f : z 7→ z + 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 [KL] that give control of moduli...

The Polynomials Associated With A Julia Set

We prove that, with two exceptions, the set of polynomials with Julia set J has the form {σ pn : n ∈ N , σ ∈ Σ} , where p is one of these polynomials and Σ is the symmetry group of J . The exceptions occur when J is a circle or a straight line segment. Several papers [1, 2, 3, 5] have appeared dealing with the relation between polynomials having the same Julia set J (for notation the reader is ...

Local connectivity of Julia sets: expository lectures

Commuting Polynomials and Polynomials with Same Julia Set

It has been known since Julia that polynomials commuting under composition have the same Julia set. More recently in the works of Baker and Eremenko, Fernández, and Beardon, results were given on the converse question: When do two polynomials have the same Julia set? We give a complete answer to this question and show the exact relation between the two problems of polynomials with the same Juli...