Siegel捕获多项式在参数空间中的位置
Location of Siegel capture polynomials in parameter spaces
摘要：
一个三次多项式$f$与一个周期性的西格尔圆盘包含一个临界点的最终图像被称为\emph{Siegel capture polynomial}。如果西格尔圆盘是不变的，我们称$f$为\emph{IS-capture polynomial}（或只是IS捕获; IS代表不变西格尔）。我们研究了IS-捕获多项式在所有三次多项式的参数空间中的位置，并证明了任何IS-捕获都是在由映射的有理层所确定的唯一双曲分量的边界上。我们还将IS-捕获与三次主双曲域及其闭包联系起来（根据定义，\emph{cubic Principal Hyperbolic Domain}由三次双曲多项式与Jordan曲线Julia集组成），并证明，在三次多项式的切片中，由一个固定的乘数在一个不动点处给出，三次主双曲域的闭包可能只有有界补域$U$，使得（1）$f\in U$的临界点是不同的，属于$J(f)$;（2）$J(f)$具有正的勒贝格测度，并带有不变的线场。
英文摘要：
A cubic polynomial $f$ with a periodic Siegel disk containing an eventual image of a critical point is said to be a \emph{Siegel capture polynomial}. If the Siegel disk is invariant, we call $f$ a \emph{IS-capture polynomial} (or just an IS-capture; IS stands for Invariant Siegel). We study the location of IS-capture polynomials in the parameter space of all cubic polynomials and show that any IS-capture is on the boundary of a unique hyperbolic component determined by the rational lamination of the map. We also relate IS-captures to the cubic Principal Hyperbolic Domain and its closure (by definition, the \emph{cubic Principal Hyperbolic Domain} consists of cubic hyperbolic polynomials with Jordan curve Julia sets) and prove that, in the slice of cubic polynomials given by a fixed multiplier at one of the fixed points, the closure of the cubic principal hyperbolic domain might possibly only have bounded complementary domains $U$ such that (1) critical points of $f\in U$ are distinct and belong to $J(f)$, and (2) $J(f)$ has positive Lebesgue measure and carries an invariant line field.
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