Biaccessibility in quadratic Julia sets
Biaccessibility in quadratic Julia sets
复制标题
DOI：
10.1017/s0143385700001024
复制
发表时间：
2000-12
影响因子：
0.9
通讯作者：
S. Zakeri
中科院分区：
文献类型：
--
作者：
S. Zakeri
本文由两个几乎独立的部分组成，这两个部分讨论了二次多项式f：z\mapsto z^2+c$的Julia集J$上的双可达点的共同问题。在第一部分中，我们假设$J$是局部连通的。证明了J$中双可达点集（通过无穷吸引域）的Brolin测度为零，除非f（z）=z^2-2$是切比雪夫映射，其相应测度为1.作为推论，我们证明了一个局部连通的二次Julia集不是一个可数的嵌入弧的并集，除非它是一条直线或一条Jordan曲线。在第二部分中，我们假设f有一个无理中立不动点\alpha。如果$z$是$J$中的一个双可达点，我们证明了$z$的轨道在Siegel情形下最终到达$f$的临界点，在Cremer情形下最终到达不动点$\alpha$。作为一个推论，可以得出J$中的双可达点集的Brolin测度为零。
This paper consists of two nearly independent parts, both of which discuss the common theme of biaccessible points in the Julia set $J$ of a quadratic polynomial $f:z\mapsto z^2+c$. In Part I, we assume that $J$ is locally-connected. We prove that the Brolin measure of the set of biaccessible points (through the basin of attraction of infinity) in $J$ is zero except when $f(z)=z^2-2$ is the Chebyshev map for which the corresponding measure is one. As a corollary, we show that a locally-connected quadratic Julia set is not a countable union of embedded arcs unless it is a straight line or a Jordan curve. In Part II, we assume that $f$ has an irrationally indifferent fixed point $\alpha$. If $z$ is a biaccessible point in $J$, we prove that the orbit of $z$ eventually hits the critical point of $f$ in the Siegel case, and the fixed point $\alpha$ in the Cremer case. As a corollary, it follows that the set of biaccessible points in $J$ has Brolin measure zero.