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A dynamical invariant for Sierpiński cardioid Julia sets

Paul Blanchard, Daniel Cuzzocreo, Robert L. Devaney, Elizabeth Fitzgibbon, Stefano Silvestri (2014)

Fundamenta Mathematicae

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...

A family of critically finite maps with symmetry.

Scott Crass (2005)

Publicacions Matemàtiques

The symmetric group Sn acts as a reflection group on CPn-2 (for n>=3).Associated with each of the (n2) transpositions in Sn is an involution on CPn-2 that pointwise fixes a hyperplane -the mirrors of the action. For each such action, there is a unique Sn-symmetric holomorphic map of degree n+1 whose critical set is precisely the collection of hyperplanes. Since the map preserves each reflecting hyperplane, the members of this family are critically-finite in a very strong sense. Considerations...

A parabolic Pommerenke-Levin-Yoccoz inequality

Xavier Buff, Adam L. Epstein (2002)

Fundamenta Mathematicae

In a recent preprint [B], Bergweiler relates the number of critical points contained in the immediate basin of a multiple fixed point β of a rational map f: ℙ¹ → ℙ¹, the number N of attracting petals and the residue ι(f,β) of the 1-form dz/(z-f(z)) at β. In this article, we present a different approach to the same problem, which we were developing independently at the same time. We apply our method to answer a question raised by Bergweiler. In particular, we prove that when there are only...

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