Michael Barnsley is the principal researcher (or entrepreneur) who has worked on fractal compression, and he holds several patents on the technology. The most widely known practical fractal compression algorithm was invented by Arnaud Jacquin in 1988, although Barnsley and Alan Sloan took out the patent (US. 5,065,447) on this method also. See also fractal transform.
As of 2001, fractal compression seems to be one of those technologies with "a great future behind it". It promised much in the late 1980s, when in some circumstances it appeared to compress much better than JPEG, its main competitor in those days.
However, fractal compression never did see much use. The patent issue may have been a problem (JPEG can be used without any patent royalties), and fractal compression is much slower to compress than JPEG is. (It decompresses at a similar rate or even a bit faster.) Also, the improved compression ratio may have been an illusion. Fractal compression only has a large advantage over JPEG at low image quality levels, which is not what most people want. The claim that fractal compressed images, when enlarged beyond their original size, looked better than similarly enlarged JPEG images seems also to have been an irrelevant distinction.
Today fractal compression seems to be even less relevant, with Wavelet compression outperforming it in most applications for those willing to brave the patent problems, and JPEG still out there and working well enough for many people.