论文标题
代数标准，不变s子变量和算术动力学的超越问题
Algebraicity criteria, invariant subvarities and transcendence problems from arithmetic dynamics
论文作者
论文摘要
我们引入了代数标准。它具有以下形式：在某些条件下，如果包含许多$ k $ points，则在全球字段$ k $上的某些代数品种的分析子变量，那么它是$ k的代数。当我们拥有动态系统时，这种情况通常会出现，因为我们通常可以通过迭代从一个点产生许多点。 梳理这一标准和对不变亚变化的研究，我们对算术动力学的超越性得到了一些结果。我们获得了Böttcher坐标的产物或多项式动力学配对的乘积高度的产物或代数为代数的。为此，我们研究了内态性产物的不变亚不同。特别是，我们部分生成了梅德韦杰夫 - 扫描官的分类，将分裂多项式图的不变亚变量分类为$（\ mathbb {p}^1）^n $在任何特征中。我们还通过引入独立概念来获得一些高度的部分概括。然后，我们在$ \ mathbb {a}^n $上研究代数$ d \ geq 2 $的主字段。我们表明，在大多数情况下（例如，当这种内态扩展到$ \ Mathbb {p}^n $上的内态性时，有许多以无限为中心的分析曲线是定期的。我们表明，对于大多数人来说，仅当它包含一个代数点时，它是代数。我们还研究了周期性曲线。我们表明，对于大多数$ f $，所有周期性曲线最多都有$ 2 $。当$ n = 2 $时，我们将获得更精确的分类结果。我们表明，如果$ f $有无限的周期性曲线，那么在一般$ f $的条件下，$ f $是同质的。
We introduce an algebraicity criteria. It has the following form: under certain conditions, an analytic subvariety of some algebriac variety over a global field $K$, if it contains many $K$-points, then it is algebraic over $K.$ This gives a way to show the transcendence of points via the transcendence of analytic subvarieties. Such a situation often appears when we have a dynamical system, because we can often produce infinitely many points from one point via iterates. Combing this criteria and the study of invariant subvarieties, we get some results on the transcendence in arithmetic dynamics. We get a characterization of products of Böttcher coordinates or products of multiplicative canonical heights for polynomial dynamical pairs to be algebraic. For this, we studied the invariant subvarieties for product of endomorphisms. In particular, we partially generate Medvedev-Scanlon's classification of invariant subvarieties of split polynomial maps to separable endomorphisms on $(\mathbb{P}^1)^N$ in any characteristic. We also get some hight dimensional partial generalization via introducing of a notion of independence. We then study dominant endomorphisms $f$ on $\mathbb{A}^N$ over a number field of algebraic degree $d\geq 2$. We show that in most of the cases (e.g. when such an endomorphism extends to an endomorphism on $\mathbb{P}^N$), there are many analytic curves centered at infinity which are periodic. We show that for most of them, it is algebraic if and only if it contains one algebraic point. We also study the periodic curves. We show that for most of $f$, all periodic curves has degree at most $2$. When $N=2$, we get a more precise classification result. We show that under a condition which is satisfied for a general $f$, if $f$ has infinitely many periodic curves, then $f$ is homogenous up to a changing of origin.
