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 is to show shrinking of puzzle pieces without using speci c puzzles This implies local connectivity at these points but triviality of bers is a somewhat stronger property than local connectivity Local connectivity proofs in holomorphic dynamics often actually yield that bers are trivial and this extra knowledge is sometimes useful Since we believe that bers may be useful in further situations we discuss their properties for arbitrary compact connected and full sets in the complex plane This allows to use them for connected lled in Julia sets of polynomials and we deduce for example that in nitely renormalizable polynomials of the form z d c have the property that the impression of any dynamic ray at a rational angle is a single point An appendix reviews known topological properties of compact connected and full sets in the plane The de nition of bers grew out of a new brief proof that the Mandelbrot set is locally connected at every Misiurewicz point and at every point on the boundary of a hyperbolic component This proof works also for Multibrot sets which are the higher degree cousins of the Mandelbrot set These sets are discussed in a self contained sequel S Finally we relate triviality of bers to tuning and renormalization in S