论文标题
$ \ Mathbb {p}^1 $上的动力学
Dynamics on $\mathbb{P}^1$: preperiodic points and pairwise stability
论文作者
论文摘要
在[dky]中，有人推测有一个统一的$ b $，仅取决于$ d $，以便任何一对holomorphic maps $ f，g：\ mathbb {p}^1 \ to \ mathbb {p}^1 $带有学位$ d $的$ d $，或者分享所有$ d $的共同点或大多数$ b $。在这里，我们表明，此统一界限为Zariski打开，并在所有对的空间中设置为$ \ mathrm {rat} _d \ times \ times \ mathrm {rat} _d $，每个度量$ d \ geq 2 $。该证明涉及算术交集理论和复杂动力学结果的结合，尤其是Gauthier-Vigny，Yuan-Zhang和Mavraki-Schmidt最近开发的。此外，我们还提供了Demarco-Krieger-ye和Poineau的最新结果的其他证据。实际上，我们证明了在动态系统和椭圆曲线的混合环境中，Bogomolov-Fu-tschinkel的猜想的概括。
In [DKY], it was conjectured that there is a uniform bound $B$, depending only on the degree $d$, so that any pair of holomorphic maps $f, g :\mathbb{P}^1\to\mathbb{P}^1$ with degree $d$ will either share all of their preperiodic points or have at most $B$ in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, $\mathrm{Rat}_d \times \mathrm{Rat}_d$, for each degree $d\geq 2$. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier-Vigny, Yuan-Zhang, and Mavraki-Schmidt. In addition, we present alternate proofs of recent results of DeMarco-Krieger-Ye and of Poineau. In fact we prove a generalization of a conjecture of Bogomolov-Fu-Tschinkel in a mixed setting of dynamical systems and elliptic curves.
