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**Thurston equivalence of topological polynomials.**
*(English)*
Zbl 1176.37020

Consider a complex polynomial \(f(z)=z^2+a\) such that its critical point \(z=0\) has an orbit of finite length. Let \(T\) be the Dehn twist of \(\mathbb{C}\) around two points of the orbit. The main problem is to define to which polynomial or rational map is equivalent the composition of \(f\) with the \(m\)-th power \(T^m\), \(m\in {\mathbb Z}\). In the paper the following two particular cases are considered.

1) The Hubbard problem, see [K. M. Pilgrim, Proc. Am. Math. Soc. 131, No. 11, 3527–3534 (2003; Zbl 1113.37029)]. The following polynomials have periodic orbits of length \(3\): the rabbit polynomial \(f_R(z)\approx z^2+(-0.1226+0.7449i)\), the corabbit polynomial \(f_C(z)\approx z^2+(-0.1226-0.7449i)\), and the airplane polynomial \(f_A(z)\approx z^2+-1.7549\). Let \(T\) be the Dehn twist \(T\) of C around the two non-critical points of the orbit. The composition \(T^m f_R\) is equivalent to one of the polynomials \(f_R\), \(f_C\) or \(f_A\). The authors give the following answer to the problem. Let \(m=\sum_{k=1}^\infty m_k4^k\), \(m_k\in \{0,1,2,3\}\) and almost all \(m_k=0\) if \(m\) is positive, and almost all \(m_k=-3\) if \(m\) is negative. If at least one of \(m_k\)’s equals \(1\) or \(2\), then \(T^m f_R\) is equivalent to \(f_A\). Otherwise, \(T^mf_R\sim f_R\) for non-negative \(m\), and \(T^mf_R\sim f_C\) for negative \(m\).

2) The problem by A. Douady and J. H. Hubbard [Acta Math. 171, No. 2, 263–297 (1993; Zbl 0806.30027)]. Let \(f_i=z^2+i\), \(f_{-i}=z^2-i\). The orbit of the critical point \(z=0\) is \(0\mapsto i\mapsto i-1\mapsto-i\mapsto i-1\). It is known that any branched covering with that ramification graph is equivalent to \(f_i\), \(f_{-i}\) or is not equivalent to a rational map. In the last case we say that it is obstructed. For a Dehn twist \(D\) it is determined when \(f_i\cdot D\) is equivalent to \(f_i\), \(f_{-i}\) or it is obstructed (Theorem 6.1). It depends on the image of the Dehn twist \(D\) in a finite group of order \(100\). In Corollary 6.11 it is proved that there are infinitely many inequivalent obstructed maps with the same ramification graph as \(f_i\). The authors give an algorithm to determine when two obstructed maps are equivalent.

1) The Hubbard problem, see [K. M. Pilgrim, Proc. Am. Math. Soc. 131, No. 11, 3527–3534 (2003; Zbl 1113.37029)]. The following polynomials have periodic orbits of length \(3\): the rabbit polynomial \(f_R(z)\approx z^2+(-0.1226+0.7449i)\), the corabbit polynomial \(f_C(z)\approx z^2+(-0.1226-0.7449i)\), and the airplane polynomial \(f_A(z)\approx z^2+-1.7549\). Let \(T\) be the Dehn twist \(T\) of C around the two non-critical points of the orbit. The composition \(T^m f_R\) is equivalent to one of the polynomials \(f_R\), \(f_C\) or \(f_A\). The authors give the following answer to the problem. Let \(m=\sum_{k=1}^\infty m_k4^k\), \(m_k\in \{0,1,2,3\}\) and almost all \(m_k=0\) if \(m\) is positive, and almost all \(m_k=-3\) if \(m\) is negative. If at least one of \(m_k\)’s equals \(1\) or \(2\), then \(T^m f_R\) is equivalent to \(f_A\). Otherwise, \(T^mf_R\sim f_R\) for non-negative \(m\), and \(T^mf_R\sim f_C\) for negative \(m\).

2) The problem by A. Douady and J. H. Hubbard [Acta Math. 171, No. 2, 263–297 (1993; Zbl 0806.30027)]. Let \(f_i=z^2+i\), \(f_{-i}=z^2-i\). The orbit of the critical point \(z=0\) is \(0\mapsto i\mapsto i-1\mapsto-i\mapsto i-1\). It is known that any branched covering with that ramification graph is equivalent to \(f_i\), \(f_{-i}\) or is not equivalent to a rational map. In the last case we say that it is obstructed. For a Dehn twist \(D\) it is determined when \(f_i\cdot D\) is equivalent to \(f_i\), \(f_{-i}\) or it is obstructed (Theorem 6.1). It depends on the image of the Dehn twist \(D\) in a finite group of order \(100\). In Corollary 6.11 it is proved that there are infinitely many inequivalent obstructed maps with the same ramification graph as \(f_i\). The authors give an algorithm to determine when two obstructed maps are equivalent.

Reviewer: Samyon R. Nasyrov (Kazan’)

### MSC:

37F20 | Combinatorics and topology in relation with holomorphic dynamical systems |

20F65 | Geometric group theory |

05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |

30F60 | Teichmüller theory for Riemann surfaces |

57M12 | Low-dimensional topology of special (e.g., branched) coverings |

### Keywords:

Thurston’s criterion; rabbit polynomial; Dehn twist; branched covering; rational maps; ramification graph
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\textit{L. Bartholdi} and \textit{V. Nekrashevych}, Acta Math. 197, No. 1, 1--51 (2006; Zbl 1176.37020)

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