Pólya–Schur 理论中的一个反问题。II. 精确可解算子与复动力学
An inverse problem in Pólya--Schur theory. II. Exactly solvable operators and complex dynamics
AI总结：
本文研究精确可解线性常微分算子在固定次数多项式上的复平面不变子集，揭示其与经典及多值复动力学的联系，并恢复有理函数的 Julia 集。
摘要：
本文是《Pólya–Schur 理论中的一个反问题。I. 非退化与退化算子》的续篇，研究一类具有多项式系数的线性常微分算子，称为“精确可解”算子；这类算子把每个次数充分高的多项式映射为同次数的多项式。我们关注此类算子在其作用被限制于固定次数多项式时复平面上的不变子集，并发现这一主题与经典复动力学及其多值对应理论之间存在联系。作为不变集合的一个非常特殊的情形，我们恢复了有理函数的 Julia 集。
英文摘要：
This paper, being the sequel of [An inverse problem in Polya-Schur theory. I. Non-genegerate and degenerate operators], studies a class of linear ordinary differential operators with polynomial coefficients called \emph{exactly solvable}; such an operator sends every polynomial of sufficiently large degree to a polynomial of the same degree. We focus on invariant subsets of the complex plane for such operators when their action is restricted to polynomials of a fixed degree and discover a connection between this topic and classical complex dynamics and its multi-valued counterpart. As a very special case of invariant sets we recover the Julia sets of rational functions.
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