2021/06/22 by Bell, Jason P., Matsuzawa, Yohsuke, Satriano, Matthew · 1 citation
#14G05 #37P55 #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2106.11544
Let X be a projective variety and let f be a dominant endomorphism of X, both of which are defined over a number field K. We consider a question of the second author, Meng, Shibata, and Zhang, which asks whether the tower of K-points Y(K)⊆ (f-1(Y))(K)⊆ (f-2(Y))(K)⊆ ⋯ eventually stabilizes, where Y⊂ X is a subvariety invariant under f. We show this question has an affirmative answer when the map f is étale. We also look at a related problem of showing that there is some integer s0, depending only on X and K, such that whenever x, y ∈ X(K) have the property that fs(x) = fs(y) for some s ≥ 0, we necessarily have f^s0(x) = f^s0(y). We prove this holds for étale morphisms of projective varieties, as well as self-morphisms of smooth projective curves. We also prove a more general cancellation theorem for polynomial maps on ℙ1 where we allow for composition by multiple different maps f1,…,fr.