论文标题
无限轨道关系族的无穷大的方程式的理性图
Equations at infinity for critical-orbit-relation families of rational maps
论文作者
论文摘要
我们开发了使用Hurwitz空间的压缩的技术来研究理性地图$ \ Mathbb {p}^1 \ to \ Mathbb {p}^1 $由关键轨道关系定义的。我们在两个设置中应用这些技术：我们表明参数空间$ \ mathrm {per} _ {d，n} $ of Leg $ - $ d $ - $ d $两智能映射，带有标记的4个周期关键点是$ d^2 $ d^$ punctionus $ \ frac $ \ frac {（d-d-1）（D-1）（D-1）（D-1）（D-1）（D-2）} $ {2）我们还表明，参数空间$ \ mathrm {per} _ {2,5} $ 2级有理于5个周期的临界点是一个10个函数的椭圆曲线，我们确定其在$ \ mathbb {q} $上的$ \ mathbb {q} $。我们对$ \ mathrm {per} _ {2,5} $的动态定义点（例如PCF点或穿刺）与基础椭圆曲线的组结构之间的相互作用进行了实验研究。
We develop techniques for using compactifications of Hurwitz spaces to study families of rational maps $\mathbb{P}^1\to\mathbb{P}^1$ defined by critical orbit relations. We apply these techniques in two settings: We show that the parameter space $\mathrm{Per}_{d,n}$ of degree-$d$ bicritical maps with a marked 4-periodic critical point is a $d^2$-punctured Riemann surface of genus $\frac{(d-1)(d-2)}{2}$. We also show that the parameter space $\mathrm{Per}_{2,5}$ of degree-2 rational maps with a marked 5-periodic critical point is a 10-punctured elliptic curve, and we identify its isomorphism class over $\mathbb{Q}$. We carry out an experimental study of the interaction between dynamically defined points of $\mathrm{Per}_{2,5}$ (such as PCF points or punctures) and the group structure of the underlying elliptic curve.
