Gyorgy Ligeti

An early description of the influence of fractal geometry on Ligeti's work was given in John Rockwell's November 11, 1986, New York Times article about Ligeti's receiving the Grawemeyer Award for Music Composition.
"I do not use direct mathematical translation into my music, like Xenaki. The influence is poetic: fractals are the most complex ornaments ever, in all the arts, like the Book of Kells or the Alhambra. They provide exactly what I want to discover in my own music, a kind of organic development."
The citation for his 2001 Kyoto Award emphasized, among other things, the influence of fractals on Ligeti's work.
See Josh Rosen's site for details about Ligeti.