Local Connectivity of Julia Sets

On Fibers and Local Connectivity of Compact Sets in C

A frequent problem in holomorphic dynamics is to prove local connectivity of Julia sets and of many points of the Mandelbrot set local connectivity has many interesting implications The intention of this paper is to present a new point of view for this problem we introduce bers of these sets De nition and the goal becomes to show that bers are trivial i e they consist of single points The idea ...

Local connectivity of Julia sets: expository lectures

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. ...

Topological, Geometric and Complex Analytic Properties of Julia Sets

In this paper, we discuss several aspects of Julia sets, as well as those of the Mandelbrot set. We are interested in topological properties such as connectivity and local connectivity, geometric properties such as Hausdorff dimension and Lebesgue measure, and complex analytic properties such as holomorphic removability. As one can easily see from the pictures of numerical experiments, there is...

A note on the structure of quadratic Julia sets

In a series of papers, Bandt and the author have given a symbolic and topological description of locally connected quadratic Julia sets by use of special closed equivalence relations on the circle called Julia equivalences. These equivalence relations reflect the landing behaviour of external rays in the case of local connectivity, and do not apply completely if a Julia set is connected but fai...

Graphical exploration of the connectivity sets of alternated Julia sets M , the set of disconnected alternated Julia sets

Using computer graphics and visualization algorithms, we extend in this work the results obtained analytically in [1], on the connectivity domains of alternated Julia sets, defined by switching the dynamics of two quadratic Julia sets. As proved in [1], the alternated Julia sets exhibit, as for polynomials of degree greater than two, the disconnectivity property in addition to the known dichoto...