关于动态抵消
On Dynamical Cancellation
摘要：
设$X$是一个射影簇，$f$是$X$的一个支配自同态，它们都定义在数域$K$上.我们考虑一个问题的第二作者，孟，柴田，张，这问塔的$K$ -点$Y(K)\subseteq (f^{-1}(Y))(K)\subseteq (f^{-2}(Y))(K)\subseteq \cdots$最终稳定，其中$Y\subset X$是一个不变的$f$下的子品种。当$f$为étale时，我们证明了这个问题有肯定的答案。我们还研究了一个相关的问题，即存在某个整数$s_0$，它只依赖于$X$和$K$，因此只要$x, y \in X(K)$具有$f^{s}(x) = f^{s}(y)$对于某个$s \geq 0$的性质，我们就必然有$f^{s_{0}}(x) = f^{s_{0}}(y)$。我们证明了这适用于射影簇的etale态射，以及光滑射影曲线的自态射。我们还证明了一个更一般的取消定理多项式映射$\mathbb{P}^1$，我们允许由多个不同的地图$f_1,\dots,f_r$组成。
英文摘要：
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 $Y(K)\subseteq (f^{-1}(Y))(K)\subseteq (f^{-2}(Y))(K)\subseteq \cdots$ 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 $s \geq 0$, 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 $f_1,\dots,f_r$.
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