论文标题
关于分裂理性图的家族的动态Bogomolov猜想
On the dynamical Bogomolov conjecture for families of split rational maps
论文作者
论文摘要
我们证明了张的动态bogomolov猜想沿着$ 1 $ - 参数统一的理性拆分图和曲线统一。这提供了Dimitrov-Gao-Habegger和Kühne最近结果的动态类似物。实际上，我们证明，在相对的Bogomolov猜想的精神上，对Bogomolov型的更强的型型结果有效。因此，我们提供了Baker和Demarco对更高维度进行猜想的概括的第一个实例。我们的证明既包含算术和分析成分。我们建立了曲线的表征，这些曲线是在非偏见的理性内态$（f，g）$（f，g）$（\ mathbb {p}^1 _ {\ mathbb {c}}）^2 $的$ f $ f $ f $ g $ g $ g $ g $ g $ ge的量度方面的$（f，g）$（f，g）的表征。我们进一步建立了分裂图和品种家族的高度不等式，比较了纤维呼叫式式式式规范高度的值，而基数的高度为高度，并且在非per虫品种的大多数点上有效。这通过Habegger对结果进行了动态概括，并将呼叫 - 丝尔曼和贝克的结果推广到更高的维度。特别是，我们建立了一个几何bogomolov定理，用于拆分有理图和任意维度的品种。
We prove that Zhang's dynamical Bogomolov conjecture holds uniformly along $1$-parameter families of rational split maps and curves. This provides dynamical analogues of recent results of Dimitrov-Gao-Habegger and Kühne. In fact, we prove a stronger Bogomolov-type result valid for families of split maps in the spirit of the relative Bogomolov conjecture. We thus provide first instances of a generalization of a conjecture by Baker and DeMarco to higher dimensions. Our proof contains both arithmetic and analytic ingredients. We establish a characterization of curves that are preperiodic under the action of a non-exceptional split rational endomorphism $(f,g)$ of $(\mathbb{P}^1_{\mathbb{C}})^2$ with respect to the measures of maximal entropy of $f$ and $g$, extending a previous result of Levin-Przytycki. We further establish a height inequality for families of split maps and varieties comparing the values of a fiber-wise Call-Silverman canonical height with a height on the base and valid for most points of a non-preperiodic variety. This provides a dynamical generalization of a result by Habegger and generalizes results of Call-Silverman and Baker to higher dimensions. In particular, we establish a geometric Bogomolov theorem for split rational maps and varieties of arbitrary dimension.
