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On the classification of critically fixed rational maps. (English) Zbl 1384.37054

The authors consider {critically fixed rational maps}, that is rational maps \(f:\widehat{\mathbb{C}}\to \widehat{\mathbb{C}}\) where each critical point \(c\) of \(f\) is also a fixed point.
The authors first provide a nice survey on {Thurston maps}, that are orientation-preserving branched coverings \(f: S^2\to S^2\) of the \(2\)-sphere \(S^2\) for which the postcritical set finite, i.e., each critical point has a finite orbit. Thurston maps may be thought of as topological analogues of rational maps, and it is possible to recover several data from a Thurston map \(f\). The authors consider six different categories associated to Thurston maps and among them is the {critical orbit portrait}, which captures the dynamics of the critical points. Formally the critical orbit portrait is the restriction of \(f\) to the union of the orbits of all critical points, together with the information of the local degree of each critical point. The authors discuss relations between the six categories, and consider also Hurwitz factorizations, the iterated monodromy group, and Tischler graphs.
The authors then focus on critical orbit portraits, where each critical point is a fixed point. Let \(m_1,\dots, m_n\) be the multiplicity of the critical points. If \(d\) is the degree of the corresponding map, then it follows that \(m_j\leq d-1\). The Riemann-Hurwitz formula shows that \(\sum_{j=1}^n m_j=2d-2\). Under these assumptions the authors prove the existence of a rational function with this critical orbit portrait if and only if \(n\leq d\). The key idea in the proof is to start with a suitable planar graph, where each vertex represents a critical point, and the degree at each vertex equals \(m_j\) and then blow up the arcs in this graph as introduced by the fourth author and T. Lei [Ergodic Theory Dyn. Syst. 18, No. 1, 221–245 (1998; Zbl 0915.58043)].
The authors also discuss a large class of examples and give a description of the iterated monodromy group.

MSC:

37F20 Combinatorics and topology in relation with holomorphic dynamical systems
05C10 Planar graphs; geometric and topological aspects of graph theory
57M12 Low-dimensional topology of special (e.g., branched) coverings
57M15 Relations of low-dimensional topology with graph theory
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable

Citations:

Zbl 0915.58043

Software:

GitHub; IMG; MathOverflow; FR
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Full Text: DOI arXiv

References:

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