Abstract

Local connectivity is an important concept in holomorphic dynamics. The local

connectivity of a Julia set is a sign that, despite being fractal, the Julia set is understandable.

If the boundary of a simply connected Fatou component is locally connected, it even means that

this boundary can be parametrized by the unit circle. One of the flag conjectures in holomorphic dynamics

is in fact the local connectivity of the Mandelbrot set.

In this talk we explain what is known about local connectivity of Julia sets in different contexts and

we show in detail the proof of local connectivity of the simplest case. We also emphasize the difficulties

in other cases  and state some new results for transcendental functions.