Local Connectivity of Limit Sets

On the local connectivity of limit sets of Kleinian groups

AMS no. 30F40 The main result here is the fact that the limit set of a geometrically finite Kleinian group with connected limit set is locally connected. For an analytically finite Kleinian group with connected limit set, we give necessary and sufficient conditions for the limit set to be locally connected.

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

On the Local Connectivity Number of Stationary Random Closed Sets

Random closed sets (RACS) in the d–dimensional Euclidean space are considered, whose realizations belong to the extended convex ring. A family of nonparametric estimators is investigated for the simultaneous estimation of the vector of all specific Minkowski functionals (or, equivalently, the specific intrinsic volumes) of stationary RACS. The construction of these estimators is based on a repr...

On Fibers and Local Connectivity of Mandelbrot and Multibrot Sets

We give new proofs that the Mandelbrot set is locally connected at every Misiurewicz point and at every point on the boundary of a hyperbolic component. The idea is to show “shrinking of puzzle pieces” without using specific puzzles. Instead, we introduce fibers of the Mandelbrot set (see Definition 3.2) and show that fibers of certain points are “trivial”, i.e., they consist of single points. ...