Spaces of Rational Maps: Dynamics, combinatorcs and topology

• Applicant: Professor Dr. Laurent Bartholdi
• Subject Area: Mathematics
• Term: from 2013 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 240186133

• Final Report Year: 2018

Final Report Abstract

The dynamics of complex rational maps is both very rich, and accessible to study. For this reason, they have been thoroughly investigated since the last 100 years, and particularly so recently, in part thanks to their computational nature. In this proposal, we develop an algebraic language, that of “bisets” (sets endowed with two commutating actions of a group), to encode and manipulate rational maps as dynamical systems. In fact, this language naturally captures much more: all topological branched coverings of the Riemann sphere may be encoded by bisets. When the maps involved are “post-critically finite”, these bisets are computable, and futhermore equivalence of bisets is algorithmically decidable. Furthermore, these bisets are computable both in theory and in pratice; implementations yield valuable data on “slices” of the space of rational maps, as well as hints of its global structure. This covers in particular the matings, and the famous slice “V3” (degree-2 rational maps with a cycle 0 ⇒ ∞ → 1 → 0). Thanks to these new results, a wealth of new research projects has emerged: in particular, an extension of all these results to “relatively hyperbolic” maps and bisets; decomposition of post-critically finite maps according to cut points in their Julia set; links between the encoding, by bisets, of Julia sets of maps with monodromy actions in the shift locus in parameter space; decomposition of the parameter space of Hénon maps using automata. These will all use the present work as a building block.

Publications

• Algorithmic construction of Hurwitz maps, Exp. Math. 24 (2015), no. 1, 76–92
  Laurent Bartholdi, Xavier Buff, Hans-Christian Graf von Bothmer, and Jakob Kröker
  
• Algorithmic aspects of branched coverings II/V. Sphere bisets and their decompositions (2016)
  Laurent Bartholdi and Dzmitry Dudko
  
• Algorithmic aspects of branched coverings, Ann. Fac. Sci. Toulouse 5 (2017), 1219–1296
  Laurent Bartholdi and Dzmitry Dudko
  
• Algorithmic aspects of branched coverings I/V. Van Kampen’s theorem for bisets, Groups Geom. Dyn. 12 (2018), no. 1, 121–172
  Laurent Bartholdi and Dzmitry Dudko
  
• Algorithmic aspects of branched coverings III/V. Erasing maps, orbispaces, and the Birman exact sequence (2018)
  Laurent Bartholdi and Dzmitry Dudko
  
• Algorithmic aspects of branched coverings IV/V. Expanding maps, Trans. Amer. Math. Soc. 370 (2018), 7679-7714
  Laurent Bartholdi and Dzmitry Dudko