论文标题
经典和动态的mordell-lang猜想的融合变体是积极特征的
A fusion variant of the classical and dynamical Mordell-Lang conjectures in positive characteristic
论文作者
论文摘要
我们研究了一个阳性特征的经典和动态莫德尔 - 朗格之间的相互作用。让$ k $是一个积极特征的代数封闭的字段，让$ g $是$ k $的乘法组的有限生成的子组，让$ x $是定义在$ k $上定义的（不可约）的quasiprojective品种。我们考虑表格$ a_n：= f（φ^n（x_0））$的$ k $值序列，其中$φ\ colon x \ colon x \ rightarrow x $和$ f \ colon x \ colon x \ rightarrow \ rightarrow \ mathbb {p}^1 $是在$ k $和$ k $ y y y in x $ in x $ in的coptie of the x $ in x $ in x $ in o in x $ in x $ in x a点上$φ$和$ f $。我们表明，G $中的$ a_n \ in g $的$ n $是算术进步的有限结合，以及一组上层Banach密度零。此外，我们表明，如果每$ n $ $ a_n \ in g $ in g $ in $ n $，$ x $的$φ$轨道为zariski weens in $ x $，则{有}有多种圆环$ \ mathbb {g} _mm^d $ and maps $ and maps $ and maps $ c $ and maps $ψ $ g：\ mathbb {g} _m^d \ to \ mathbb {g} _m $，以至于$ a_n = g \ circ Cint ocircinbb {y y \ in \ mathbb {g} _m^d $。然后，我们描述结果的各种应用。
We study an open question at the interplay between the classical and the dynamical Mordell-Lang conjectures in positive characteristic. Let $K$ be an algebraically closed field of positive characteristic, let $G$ be a finitely generated subgroup of the multiplicative group of $K$, and let $X$ be a (irreducible) quasiprojective variety defined over $K$. We consider $K$-valued sequences of the form $a_n:=f(φ^n(x_0))$, where $φ\colon X\rightarrow X$ and $f\colon X\rightarrow\mathbb{P}^1$ are rational maps defined over $K$ and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $φ$ and $f$. We show that the set of $n$ for which $a_n\in G$ is a finite union of arithmetic progressions along with a set of upper Banach density zero. In addition, we show that if $a_n\in G$ for every $n$ and the $φ$ orbit of $x$ is Zariski dense in $X$ then {there is} a multiplicative torus $\mathbb{G}_m^d$ and maps $Ψ:\mathbb{G}_m^d \to \mathbb{G}_m^d$ and $g:\mathbb{G}_m^d \to \mathbb{G}_m$ such that $a_n = g\circ Ψ^n(y)$ for some $y\in \mathbb{G}_m^d$. We then describe various applications of our results.
