##
**A family of cubic rational maps and matings of cubic polynomials.**
*(English)*
Zbl 0969.37020

Let \(g\) and \(f\) be two monic polynomials of same degree \(d\), and extend each of them to \({\mathbb C}\cup\{\infty\cdot e^{2\pi is}\}\) by mapping \(\infty\cdot e^{2\pi is}\mapsto\infty\cdot e^{2\pi ids}\). Glueing these two circles at infinity in opposite directions gives rise to a topological sphere. The mating of \(g\) and \(f\) is the branched covering of this sphere which is given by \(g\) on one hemisphere and \(f\) on the other. If this map is equivalent to a rational map up to isotopy relative to the post-critical set and topological conjugacy, then \(g\) and \(f\) are said to be matable. In this paper, the authors take \(g\) and \(f\) cubic polynomials with respectively one double critical point and one period-three orbit containing two simple critical points, a situation which can be reduced to \(g(z)=g_a(z):=z^3+a\) and \(f\) one of eight specific polynomials \(P_i,\widetilde{P_i} (i=1,\dots,4)\), and give criteria for the matability of \(g_a\) and \(P_i\) in the case where \(g_a\) is critically finite. In particular, they exhibit new phenomena which don’t occur for quadratic maps, for instance matings which have Thurston obstructions but no Levy cycles. This very nicely written article contains a good survey of Thurston and Levy’s theory of branched coverings of the sphere, as well as the general theory of matings. It concludes with an appendix containing experimental observations which give rise to various conjectures.

Reviewer: Line Baribeau (Quebec)

### MSC:

37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |

PDF
BibTeX
XML
Cite

\textit{M. Shishikura} and \textit{T. Lei}, Exp. Math. 9, No. 1, 29--53 (2000; Zbl 0969.37020)

### References:

[1] | Bielefeld B., J. Amer. Math. Soc. 5 (4) pp 721– (1992) |

[2] | Carleson L., Complex dynamics (1993) · Zbl 0782.30022 |

[3] | Douady A., Real and complex dynamical systems (Hillerød, 1993) pp 65– (1995) |

[4] | Douady R., Algébre et théories galoisiennes 2 (1979) |

[5] | Douady A., Étude dynamique des polynômes complexes, I (1984) · Zbl 0552.30018 |

[6] | Douady A., Étude dynamique des polynômes complexes, II (1985) · Zbl 0552.30018 |

[7] | Douady A., Acta Math. 171 (2) pp 263– (1993) · Zbl 0806.30027 |

[8] | Gantmacher F. R., The theory of matrices (1959) · Zbl 0085.01001 |

[9] | Levy S., Ph.D. thesis, in: Critically finite rational maps (1985) |

[10] | Moore R. L., Trans. Amer. Math. Soc. 27 pp 416– (1925) |

[11] | Pilgrim K., Ph.D. thesis, in: Cylinders for iterated rational maps (1994) |

[12] | Pilgrim K. M., Ergodic Theory Dynam. Systems 18 (1) pp 221– (1998) · Zbl 0915.58043 |

[13] | Poirier A., ”On post critically finite polynomials” (1993) |

[14] | Rees M., ”Realization of matings of polynomials as rational maps of degree two” (1986) |

[15] | Rees M., Acta Math. 168 (1) pp 11– (1992) · Zbl 0774.58035 |

[16] | Shishikura M., The Mandelbrot set: theme and variations pp 289– (2000) |

[17] | Ergodic Theory Dynamical Systems 12 (3) pp 589– (1992) |

[18] | Fund. Math. 154 (3) pp 207– (1997) |

[19] | Thurston W., ”The combinatorics of iterated rational maps” (1983) |

[20] | Wittner B., Ph.D. thesis, in: On the bifurcation loci of rational maps of degree two (1986) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.