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.


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
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