Kaleidoscopic decomposition, classification of finite branched coverings of the sphere and path-lifting algorithms
The aim of this work is two-fold: On the one hand, to study the finite branched coverings of the Riemann sphere, providing original contributions with regard to their structure and properties, with a particular focus on rational maps and polynomials. On the other hand, to employ the introduced theory and original results of this memoir to design and implement novel numerical path-lifting procedures. In order to meet these purposes, we study finite branched coverings between compact Riemann surfaces, introducing an original canonical decomposition, which we call kaleidoscopic decomposition, that will be employed throughout the memoir to present a novel analysis of different topological aspects as the monodromy action or the regularity. Moreover, we study rational maps and polynomials as branched self-coverings of the Riemann sphere by considering an original approach based on finite combinatorial groupoids, which allows us to prove novel results addressing some of their topological properties (like the monodromy or the group of automorphisms), to completely characterize their regularity in simple algebraic terms, and to study and relate different notions of classification for these families of maps. As a consequence of the aforementioned study, we design novel algorithms supported by the path-lifting properties that rational and polynomial maps exhibit, which are devoted to different purposes, like the approximation of roots, the computation of the monodromy action, or the description of the group of automorphisms. The implementation of such algorithms is tested through two different cases of study, yielding an exploration of some numerical aspects of the considered path-lifting methods.
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