Local connectivity of Julia sets for unicritical polynomials

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...

Local connectivity of Julia sets: expository lectures

Rigidity and non local connectivity of Julia sets of some quadratic polynomials

For an infinitely renormalizable quadratic map fc : z 7→ z 2+c with the sequence of renormalization periods {nm} and the rotation numbers {tm = pm/qm}, we prove that if lim supn−1 m log |pm| > 0, then the Mandelbrot set is locally connected at c. We prove also that if lim sup |tm+1| 1/qm < 1 and qm → ∞, then the Julia set of fc is not locally connected provided c is the limit of corresponding c...

Fekete Polynomials and Shapes of Julia Sets

We prove that a nonempty, proper subset S of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if S is bounded and Ĉ \ int(S) is connected. The proof that such a set is approximable by filled Julia sets is constructive and relies on Fekete polynomials. Illustrative examples are presented. We also prove an estimate for the rate of approximation i...

Combinatorial Rigidity for Unicritical Polynomials

We prove that any unicritical polynomial fc : z 7→ z+c which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. It implies that the connectedness locus (the “Multibrot set”) is locally connected at the corresponding parameter values. It generalizes Yoccoz’s Theorem for quadratics to the higher degree case. Stony Brook IMS Preprint #2005/05 July 2005