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 huge variety of "shapes" of Julia sets even for polynomials of a simple form z + c. And as the parameter c varies, the Julia set can drastically change its shape. In the Mandelbrot set, the blow-ups of different places in the Mandelbrot set can look totally different. Some parts look like the entire Mandelbrot set, and other parts sometimes look like the shape of the corresponding Julia sets. These sets often look very complicated but one can see rich structures inside them. They provide typical examples of "fractals''. Since Douady and Hubbard [DH] started their work on quadratic polynomials, there have been many developments with many new techniques in this field. In this paper, we try to summarize some results from the point of view of the above mentioned properties.