A dynamical invariant for Sierpiński cardioid Julia sets by

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

Research output: Contribution to journalArticle

Abstract

For the family of rational maps zn+ λ=znwhere 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.

Original languageEnglish (US)
Pages (from-to)253-277
Number of pages25
JournalFundamenta Mathematicae
Volume226
Issue number3
DOIs
StatePublished - Jan 1 2014

Keywords

  • Cantor necklace
  • Julia set
  • Mandelbrot set
  • McMullen domain
  • Sierpiński curve

ASJC Scopus subject areas

  • Algebra and Number Theory

Fingerprint Dive into the research topics of 'A dynamical invariant for Sierpiński cardioid Julia sets by'. Together they form a unique fingerprint.

  • Cite this

    Blanchard, P., Cuzzocreo, D., Devaney, R. L., Fitzgibbon, E., & Silvestri, S. (2014). A dynamical invariant for Sierpiński cardioid Julia sets by. Fundamenta Mathematicae, 226(3), 253-277. https://doi.org/10.4064/fm226-3-5