Dynamical diophantine approximation exponents in characteristic p $p$

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-10-17 DOI:10.1112/blms.13168

Wade Hindes

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Abstract

Let $ϕ (z)$ be a non-isotrivial rational function in one-variable with coefficients in ${\bar{F}}_{p} (t)$ and assume that $γ \in P^{1} ({\bar{F}}_{p} (t))$ is not a post-critical point for $ϕ$ . Then we prove that the diophantine approximation exponent of elements of $ϕ^{- m} (γ)$ are eventually bounded above by $⌈ deg {(ϕ)}^{m} / 2 ⌉ + 1$ . To do this, we mix diophantine techniques in characteristic $p$ with the adelic equidistribution of small points in Berkovich space. As an application, we deduce a form of Silverman's celebrated limit theorem in this setting. Namely, if we take any wandering point $a \in P^{1} ({\bar{F}}_{p} (t))$ and write $ϕ^{n} (a) = a_{n} / b_{n}$ for some coprime polynomials $a_{n}, b_{n} \in {\bar{F}}_{p} [t]$ , then we prove that

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特征p$ p$的动态丢番图近似指数

设φ (z) $\phi (z)$为系数为F¯p的单变量非等平凡有理函数(t) $\overline{\mathbb {F}}_p(t)$，假设γ∈p1 (F¯p (t)) $\gamma \in \mathbb {P}^1(\overline{\mathbb {F}}_p(t))$不是φ $\phi$的后临界点。然后我们证明了φ - m (γ) $\phi ^{-m}(\gamma)$的元素的丢芬图近似指数最终在上有界≤φ≤m / 2≤+ 1 $\lceil \deg (\phi)^m/2\rceil +1$。为此，我们将特征p $p$中的丢芬图技术与Berkovich空间中小点的adelic等分布相结合。作为应用，我们在这种情况下推导出了著名的西尔弗曼极限定理的一种形式。即，如果取任意游走点a∈p1 (F¯P （）T)) $a\in \mathbb {P}^1(\overline{\mathbb {F}}_p(t))$，写φ n (a) =A n / b n $\phi ^n(a)=a_n/b_n$对于一些素数多项式A n，b n∈F¯p [t] $a_n,b_n\in \overline{\mathbb {F}}_p[t]$，则证明

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