2020/06/18 by Geyer, Lukas, Hlushchanka, Mikhail · 1 citation
#37F10 (Primary) 30D05 (Secondary) #37F20 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2006.10788
We provide a complete combinatorial classification of critically fixed anti-Thurston maps, i.e., orientation-reversing branched covers of the 2-sphere that fix every critical point. The first step in the proof, and an interesting result in its own right, is a combinatorial classification of critically fixed anti-rational maps as "Schottky maps" associated to certain plane graphs. Both of these classification results heavily rely on an orientation-reversing version of Thurstons's theory, including the canonical decomposition of anti-Thurston maps, which we develop in this paper. Lastly, we give some applications to the global curve attractor and twisting problems, as well as to anti-rational maps with symmetries and to critically fixed anti-polynomials.