A classification of degree 2 semi-stable rational maps ℙ²→ℙ² with large finite dynamical automorphism group

2016/07/19 by Manes, Michelle, Silverman, Joseph H.
#37P05 (Secondary) #37P45 (Primary) #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1607.05772

Abstract

Let K be an algebraically closed field of characteristic 0. In this paper we classify the PGL₃(K)-conjugacy classes of semi-stable dominant degree 2 rational maps f:\mathbb P²_K\dashrightarrow\mathbb P²_K whose automorphism group Aut(f):=\ϕ\inPGL₃(K): ϕ^-1∘ f∘ϕ=f\ is finite and of order at least 3. In particular, we prove that #Aut(f)≤24 in general, that #Aut(f)≤21 for morphisms, and that #Aut(f)≤6 for all but finitely many conjugacy classes of f.

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