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Seeing the monodromy group of a Blaschke product. (English) Zbl 1467.30035

This paper is a piece of art, both mathematically and aesthetically. It is a universal wisdom that the best way to get a first feeling or even sufficient understanding of a function is to look at its graph. However, this is famously not trivial for functions mapping \(\mathbb{C}\) into \(\mathbb{C}\). The author is an expert in overcoming this difficulty and uses his know-how and imaginative faculty, known at least from his paper with G. Semmler [Notices Am. Math. Soc. 58, No. 6, 768–780 (2011; Zbl 1227.00032)] and his book [Visual complex functions. An introduction with phase portraits. Basel: Birkhäuser (2012; Zbl 1264.30001)], to visualize some interesting phenomena connected with finite Blaschke products. The functions themselves are visualized by phase plots and variations thereof, e.g., tiled phase plots. It is again and again a pleasure to see the phase plot of a Blaschke product of degree \(50\) in the complex plane or on the Riemann sphere (Figures 5 and 6 of the paper).
A Blaschke product is not injective, so it has a Riemann surface, and this surface in turn defines a monodromy group. The paper is about the question whether one can also see this monodromy group in phase plots. Although there are several obstacles that prevent us from seeing the monodromy group in one form or another completely, the author shows that one can see important pieces of information, such as the generators of the group, by inspection of the basins of attraction of the zeros, which are bordered by the stable manifolds of the critical (saddle) points with respect to the phase flow. All this can be seen in tiled phase plots. The final section of the paper is about thinking of tiled phase plots as certain discrete structures whose appropriately defined monodromy groups can be determined by counting exercises.
Another question considered is how to read off from phase plots whether a given Blaschke product \(f\) of degree \(mn\) is the composition \(f=g \circ h\) of two Blaschke products \(g\) and \(h\) of degrees \(m\) and \(n\). A side result of the corresponding journey through many eye-catching pictures is that the monodromy group of a composition of finite Blaschke products is (isomorphic to) the direct product of the monodromy groups of its factors. As cyclic groups are monodromy groups of Blaschke products with their zeros in the vertices of a regular polygon, it follows that every abelian group is the monodromy group of some Blaschke product. On the other hand, as the author told the reviewer, if one zero is at the origin and \(n-1\) zeros are on a circle around the origin, then the monodromy group is the full symmetric group \(\mathbb{S}_n\). The paper closes with raising the intriguing open question of identifying the subgroups of \(\mathbb{S}_n\) which are monodromy groups of Blaschke products and of how to construct the associated Blaschke products.
The paper is very well written. The reader is gently introduced into phase plots, Blaschke products, phase flows, the Riemann surface of a Blaschke product, and its monodromy group. The paper is therefore accessible to everyone with basic knowledge in complex function theory.

MSC:

30J10 Blaschke products
20F05 Generators, relations, and presentations of groups
30F20 Classification theory of Riemann surfaces
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