Rational Functions with a Unique Critical Point

Details Rational Functions with a Unique Critical Point Rational Functions with a Unique Critical Point

2011-02-07

arxiv.org/abs/1102.1433v2

Mathematics

Number Theory

13 pages; extended to include a discussion of the geometry of the space of unicritical rational functions and some of its quot

Scientific paper

Over an algebraically closed field of positive characteristic, there exist rational functions with only one critical point. We give an elementary characterization of these functions in terms of their continued fraction expansions. Then we use this tool to discern some of the basic geometry of the space of unicritical rational functions, as well as its quotients by the SL(2)-actions of conjugation and postcomposition. We also give an application to dynamical systems with restricted ramification defined over non-Archimedean fields of positive residue characteristic.

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