We show how the well-known concept of external rays in polynomial dynamics may be extended throughout the Julia set of certain rational maps. These new types of rays, which we call internal rays, meet the Julia set in a Cantor set of points, and each of these rays crosses infinitely many other internal rays at many points. We then use this construction to show that there are infinitely many disjoint copies of the Mandelbrot set in the parameter planes for these maps.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm206-0-9,
author = {Robert L. Devaney},
title = {Intertwined internal rays in Julia sets of rational maps},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {139-159},
zbl = {1187.37067},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm206-0-9}
}
Robert L. Devaney. Intertwined internal rays in Julia sets of rational maps. Fundamenta Mathematicae, Tome 205 (2009) pp. 139-159. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm206-0-9/