For the family of rational maps zⁿ + λ/zⁿ where n ≥ 3, it is known that there are infinitely many small copies of the Mandelbrot set that are buried in the parameter plane, i.e., they do not extend to the outer boundary of this set. For parameters lying in the main cardioids of these Mandelbrot sets, the corresponding Julia sets are always Sierpiński curves, and so they are all homeomorphic to one another. However, it is known that only those cardioids that are symmetrically located in the parameter plane have conjugate dynamics. We produce a dynamical invariant that explains why these maps have different dynamics.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm226-3-5,
author = {Paul Blanchard and Daniel Cuzzocreo and Robert L. Devaney and Elizabeth Fitzgibbon and Stefano Silvestri},
title = {A dynamical invariant for Sierpi\'nski cardioid Julia sets},
journal = {Fundamenta Mathematicae},
volume = {227},
year = {2014},
pages = {253-277},
zbl = {06314012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm226-3-5}
}
Paul Blanchard; Daniel Cuzzocreo; Robert L. Devaney; Elizabeth Fitzgibbon; Stefano Silvestri. A dynamical invariant for Sierpiński cardioid Julia sets. Fundamenta Mathematicae, Tome 227 (2014) pp. 253-277. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm226-3-5/