Dynamic classification of escape time Sierpiński curve Julia sets

Robert L. Devaney ¹   ; Kevin M. Pilgrim ²

¹ Department of Mathematics Boston University Boston, MA 02215, U.S.A.
² Department of Mathematics Indiana University Bloomington, IN 47405, U.S.A.

Fundamenta Mathematicae, Tome 202 (2009) no. 2, pp. 181-198

Résumé

For $n \geq 2$, the family of rational maps $F_\lambda(z) = z^n + \lambda/z^n$ contains a countably infinite set of parameter values for which all critical orbits eventually land after some number $\kappa$ of iterations on the point at infinity. The Julia sets of such maps are Sierpiński curves if $\kappa \geq 3$. We show that two such maps are topologically conjugate on their Julia sets if and only if they are Möbius or anti-Möbius conjugate, and we give a precise count of the number of topological conjugacy classes as a function of $n$ and $\kappa$.

Détail

DOI : 10.4064/fm202-2-5

Keywords: geq family rational maps lambda lambda contains countably infinite set parameter values which critical orbits eventually land after number kappa iterations point infinity julia sets maps sierpi ski curves kappa geq maps topologically conjugate their julia sets only bius anti m bius conjugate precise count number topological conjugacy classes function kappa