• DOI:10.5802/AFST.1357
• Corpus ID: 14386972

A CLASSIFICATION OF BICRITICAL RATIONAL MAPS WITH A PAIR OF PERIOD TWO SUPERATTRACTING CYCLES.

@article{Epstein2012ACO,
  title={A CLASSIFICATION OF BICRITICAL RATIONAL MAPS WITH A PAIR OF PERIOD TWO SUPERATTRACTING CYCLES.},
  author={Adam L. Epstein and Thomas Sharland},
  journal={Annales de la Facult{\'e} des Sciences de Toulouse},
  year={2012},
  volume={21},
  pages={907-934}
}

• A. Epstein, Thomas Sharland
• Published 2012
• Mathematics
• Annales de la Faculté des Sciences de Toulouse

We give a Thurston classification of those bicritical rational maps which have two period two superattracting cycles. We also show that all such maps are constructed by the mating of two unicritical degree d polynomials. In this note we study bicritical rational maps whose critical points lie in distinct period two cycles. These maps are completely classified by a natural combinatorial invariant. A common strategy used in the classification of rational maps is to invoke Thurston's Theorem (DH93). This will not be necessary in this note, due to the availability of a purely algebraic classification of the maps in question. However, we will still tackle the question of Thurston equivalence at the end of the paper, since it motivates a discussion of the combinatorial description of these maps, which is a characteristic that will be much more useful than the algebraic classification in general. We refer the reader to (Mil06) for background material on the dynamics of rational maps. 1.1. Thurston's Criterion. Let F : � → � be an orientation-preserving branched self-covering of a topological 2-sphere. We denote by F the critical set of F and define PF = ( n>0 F ◦n ( F)