A dynamical invariant for Sierpiński cardioid Julia sets

Paul Blanchard ¹   ; Daniel Cuzzocreo ¹   ; Robert L. Devaney ¹   ; Elizabeth Fitzgibbon ¹   ; Stefano Silvestri ¹

¹ Department of Mathematics Boston University 111 Cummington Mall Boston, MA 02215, U.S.A.

Fundamenta Mathematicae, Tome 226 (2014) no. 3, pp. 253-277

Résumé

For the family of rational maps $z^n + \lambda /z^n$ where $n \geq 3$, 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.

Détail

DOI : 10.4064/fm226-3-5

Keywords: family rational maps lambda where geq known there infinitely many small copies mandelbrot set buried parameter plane extend outer boundary set parameters lying main cardioids these mandelbrot sets corresponding julia sets always sierpi ski curves homeomorphic another however known only those cardioids symmetrically located parameter plane have conjugate dynamics produce dynamical invariant explains why these maps have different dynamics