On Dynamical Cancellation

Abstract

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 2nd author, Meng, Shibata, and Zhang, who asks whether the tower of $K$-points $Y(K)\subseteq (f^{-1}(Y))(K)\subseteq (f^{-2}(Y))(K)\subseteq \cdots $ eventually stabilizes, where $Y\subset X$ 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 $s_0$, depending only on $X$ and $K$, such that whenever $x, y \in X(K)$ have the property that $f^{s}(x) = f^{s}(y)$ for some $s \geqslant 0$, we necessarily have $f^{s_{0}}(x) = f^{s_{0}}(y)$. 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 ${\mathbb {P}}^1$ where we allow for composition by multiple different maps $f_1,\dots ,f_r$.

Dynamical cancellation of polynomials

“…Let $$\Delta _K \subseteq (\mathbb {P}^1_K\times \mathbb {P}^1_K)$$ be the diagonal subvariety. By [1, Lemma 5.7], all the possible irreducible curves over $$\mathbb {C}$$ different from $$\Delta$$, having a Zariski dense set of $$K$$‐points and living inside the preimages of $$\Delta$$ under elements of the form $$(f,f)$$ with $$f \in S$$, are in a finite set $$\Sigma$$. Also, by the proof of [1, Theorem 1.6], there is a finite set $$Z \subset \mathbb {P}^1_K \times {\mathbb {P}}^1_K$$ with the property that if $$C \in \Sigma$$ and $$C^{\prime }$$ is an irreducible component of $$(f,f)^{-1}(C)$$ not in $$\Sigma$$, then $$C^{\prime }(K) \subseteq Z$$.…”

“…Then, there exists a sequence of curves $$\lbrace C_0, \dots , C_{k_1 -1}\rbrace \subseteq \Sigma ^{\prime }$$ such that $$C_{i-1} \subseteq (\phi _i, \phi _i)^{-1}(C_i)$$, $$C_0 = C$$ and $$C_{k_1} = \Delta$$, $$0 \leqslant i \leqslant k_1$$. The fact that all of these curves have to be genus 0 follows from the genus estimation lemma [1, Proposition 4.1 ] saying that for any irreducible curve $$C^{\prime }$$ over $$\mathbb {C}$$ and irreducible curve $$C^{\prime \prime } \subseteq (f, f)^{-1}(C^{\prime })$$, $$f \in S$$, genus of $$C^{\prime \prime }$$ is not less than genus of $$C^{\prime }$$. So, if there is a curve $$C_j$$, …”

“…A series of observations on the increasing geometric complexity of the iterated preimages and how, in principal, the geometric structure should exert significant influence on the underlying arithmetic structure gives rise to the expectation that the tower of $$K$$‐points $$$$\begin{equation*} Y(K) \subseteq (f^{-1}(Y))(K) \subseteq (f^{-2}(Y))(K) \subseteq \cdots \end{equation*}$$$$should eventually stabilize. This expectation has been made precise as the Preimages Question in [5, Question 8.4(1)] and solved with an affirmative answer when $$X$$ is a smooth variety with non‐negative Kodaira dimension and $$f$$ is étale [1, Theorem 1.2].…”

“…In this case, the Preimages Question is translated to a cancellation problem; that is, it says that there should exist a natural number $$s$$ such that for all $$x,y \in X(K)$$, if $$g^n(x) = g^n(y)$$ for some $$n > s$$, then we have $$g^s(x) = g^s(y)$$. The precise statement is given by [1, Conjecture 1.5] and in the same paper [1, Theorem 1.3], the case when $$X$$ is a curve is proved by applying $$p$$‐adic uniformization techniques of Rivera–Letelier [9]. Since significantly more is known about polynomial self‐maps on $$\mathbb {P}^1$$ than surjective self‐morphisms of projective varieties in general, the authors also prove a more general cancellation result allowing multiple polynomial self‐maps on $$\mathbb {P}^1$$ as follows.…”

“…Theorem 1.4 [1,Theorem 1.7]. Let 𝐾 be a number field and let 𝜙 1 , … , 𝜙 𝑟 be polynomial maps on ℙ 1 𝐾 of degree at least two.…”