Preimages Question for Surjective Endomorphisms on (ℙ¹)ⁿ

2023/11/07 by Zhong, Xiao
#14G05 #37P55 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2311.04349

Abstract

Let K be a number field and let f : (ℙ¹)ⁿ → (ℙ¹)ⁿ be a dominant endomorphism defined over K. We show that if V is an f-invariant subvariety (that is, f(V)=V) then there is a positive integer s₀ such that (f^-s-1(V)∖ f^-s(V))(K) = ∅ for every integer s ≥ s₀, answering the Preimages Question of Matsuzawa, Meng, Shibata, and Zhang in the case of (ℙ¹)ⁿ.

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