Post-Critically Finite Maps on ℙⁿ for n≥2 are Sparse

2019/10/24 by Ingram, Patrick, Ramadas, Rohini, Silverman, Joseph H.
#37F45 #37P45 #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Primary: 37P05 #Secondary: 37F10

paper · doi:10.48550/arxiv.1910.11290

Abstract

Let f:\mathbb Pⁿ→\mathbb Pⁿ be a morphism of degree d≥2. The map f is said to be post-critically finite (PCF) if there exist integers k≥1 and ℓ≥0 such that the critical locus Crit_f satisfies f^k+ℓ(Crit_f)⊆f^ℓ(Crit_f). The smallest such ℓ is called the tail-length. We prove that for d≥3 and n≥2, the set of PCF maps f with tail-length at most 2 is not Zariski dense in the the parameter space of all such maps. In particular, maps with periodic critical loci, i.e., with ℓ=0, are not Zariski dense.

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