论文标题
具有极渐近值的异晶函数
Meromorphic Functions with a Polar Asymptotic Value
论文作者
论文摘要
本文是复杂动力学中的一般程序的一部分，以了解具有许多单数值有限的先验图的参数空间。 此类功能的最简单家族具有两个渐近值，没有临界值。这些家庭，直到仿射共轭，都取决于两个复杂的参数。了解它们的参数空间是了解具有更多渐近价值的家族的关键，就像了解二次多项式有关的是理性地图。 研究的第一个家庭是指数家庭的一维切片，$ \ exp（z） + a $和切线家庭$λ\ tan z $。指数案例表现出对有理图的现象：动态和参数空间中的cantor花束，没有有界的双曲分量。带有两个有限渐近值$ \ pmλi$的切线案例更接近于理性案例，这是后者的一种无限度版本。 在本文中，我们考虑了一个通用家庭，该家庭在$ \ exp（z） + a $和$λ\ tan z $之间插值。我们的新家族有两个渐近值和一个一维切片，其中一个渐近值被约束为极点，标题的“极性渐近值”。我们展示了这种切片的动态和参数平面表现出的行为，这是$ \ exp（z） + a $ a $ and $λ\ tan z $ family之间令人惊讶的精致相互作用。
This paper is part of a general program in complex dynamics to understand parameter spaces of transcendental maps with finitely many singular values. The simplest families of such functions have two asymptotic values and no critical values. These families, up to affine conjugation, depend on two complex parameters. Understanding their parameter spaces is key to understanding families with more asymptotic values, just as understanding quadratic polynomials was for rational maps more generally. The first such families studied were the one-dimensional slices of the exponential family, $\exp(z) + a$, and the tangent family $λ\tan z$. The exponential case exhibited phenomena not seen for rational maps: Cantor bouquets in both the dynamic and parameter spaces, and no bounded hyperbolic components. The tangent case, with its two finite asymptotic values $\pm λi$, is closer to the rational case, a kind of infinite degree version of the latter. In this paper, we consider a general family that interpolates between $\exp(z) + a$ and $λ\tan z$. Our new family has two asymptotic values and a one-dimensional slice for which one of the asymptotic values is constrained to be pole, the "polar asymptotic value" of the title. We show how the dynamic and parameter planes for this slice exhibit behavior that is a surprisingly delicate interplay between that of the $\exp(z) + a$ and $λ\tan z$ families.
