## Abstract

We consider the problem of classifying the dynamics of complex polynomials f: W C → C restricted to the basins of inFInity X(f). We synthesize existing combinatorial tools-tableaux, trees, and laminations-into a new invariant of basin dynamics we call the pictograph. For polynomials with all critical points escaping to inFInity, we obtain a complete description of the set of topological conjugacy classes with given pictograph. For arbitrary polynomials, we compute the total number of topological conjugacy classes of basins (f;X(f)) with a given pictograph. We also deFIne abstract pictographs and prove that every abstract pictograph is realized by a polynomial. Extra details are given in degree 3, and we provide examples that show the pictograph is a FIner invariant than both the tableau of [5] and the tree of [10].

Original language | English (US) |
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Pages (from-to) | 799-877 |

Number of pages | 79 |

Journal | Annales Scientifiques de l'Ecole Normale Superieure |

Volume | 50 |

Issue number | 4 |

DOIs | |

State | Published - Jul 1 2017 |

## ASJC Scopus subject areas

- Mathematics(all)