Robert P. Munafo, 2008 Feb 20.

In 1918, Fatou and Julia, while working towards a prize to be awarded for "study of iteration from a global point of view", studied the iteration of f(c) = c² + K for complex c with K constant. They each independently discovered many of the properties of Julia sets and Fatou dusts.

Julia did not produce any images of the Julia sets himself; the first such image appeared in a later paper by Cremer.

Mandelbrot began using the term fractal, a contraction of fractional in reference to "fractional dimension", in 1975.

In 1978, Brooks and Matelski, published the first image of the Mandelbrot set's interior. They did not investigate it to anywhere near the extent that Mandelbrot would a few years later, and did not know much about its properties. In particular, they omit the filaments and (probably) the islands.

A year or two later, mathematician Benoit Mandelbrot discovered the Mandelbrot set during his research of the Julia and Fatou sets. His discovery was independent of the work by Brooks and Matelski. Mandelbrot describes the Julia sets and the Mandelbrot Set (which he calls the "mu map") in his book The Fractal Geometry of Nature. He also gives a brief description of its original discovery.

The discovery made its way throughout the mathematics community and exploration of the Mandelbrot Set was eventually taken on by Peitgen and Richter, ARTMATRIX, and others. The images of Peitgen and Richter became known to A. K. Dewdney who then used them as the subject for his column in Scientific American, which appeared in the August 1985 issue, featuring one of the images on the cover.

The article was seen by many people and immediately created a widespread interest in Mandelbrot Set computer programs.

J. Hubbard and A. Douady proved that the Mandelbrot set is connected. In 1991, M. Shishikura proved that the boundary of the Mandelbrot set is a fractal with a Hausdorff dimension of 2.0 .

Various errorts to determine the area of the Mandelbrot Set occurred during the 1990's.

Sources:

Lecture notes from Harvard mathematics course 118r "Dynamical Systems" (Oliver Knill), Spring 2005.

MacTutor History of Mathematics, entries for Fatou and Julia.

WikiVisual, Mandelbrot set.