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A dynamical invariant for Sierpiński cardioid Julia sets

Volume 226 / 2014

Paul Blanchard, Daniel Cuzzocreo, Robert L. Devaney, Elizabeth Fitzgibbon, Stefano Silvestri Fundamenta Mathematicae 226 (2014), 253-277 MSC: Primary 37F10; Secondary 37F45. DOI: 10.4064/fm226-3-5

Abstract

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.

Authors

  • Paul BlanchardDepartment of Mathematics
    Boston University
    111 Cummington Mall
    Boston, MA 02215, U.S.A.
  • Daniel CuzzocreoDepartment of Mathematics
    Boston University
    111 Cummington Mall
    Boston, MA 02215, U.S.A.
  • Robert L. DevaneyDepartment of Mathematics
    Boston University
    111 Cummington Mall
    Boston, MA 02215, U.S.A.
    e-mail
  • Elizabeth FitzgibbonDepartment of Mathematics
    Boston University
    111 Cummington Mall
    Boston, MA 02215, U.S.A.
  • Stefano SilvestriDepartment of Mathematics
    Boston University
    111 Cummington Mall
    Boston, MA 02215, U.S.A.

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