An explosion point for the set of endpoints of the Julia set of λ exp (z)
An explosion point for the set of endpoints of the Julia set of λ exp (z)
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λ exp (z) Julia 集的端点集的爆炸点
DOI：
10.1017/s0143385700005460
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发表时间：
1990
影响因子：
0.9
通讯作者：
J. Mayer
中科院分区：
文献类型：
--
作者：
J. Mayer
摘要对于真实的参数λ（0 < λ < 1/e），复指数函数Eλ：z → λez的Julia集Jλ已知是从Jλ端点集Aλ延伸到∞的康托射线束。由于Aλ包含Eλ的所有排斥周期点，因此Jλ = Cl（Aλ）。我们证明了Aλ是复平面λ的一个完全不连通子空间，但如果在∞处加上一点，则A λ是黎曼球面的一个连通子空间。作为推论，Aλ的拓扑维数为1。因此，∞是λ在拓扑意义上的爆炸点。值得注意的是，具有爆炸点的空间以这种方式“自然地”发生。
Abstract The Julia set Jλ of the complex exponential function Eλ: z → λez for a real parameter λ(0 < λ < 1/e) is known to be a Cantor bouquet of rays extending from the set Aλ of endpoints of Jλ to ∞. Since Aλ contains all the repelling periodic points of Eλ, it follows that Jλ = Cl (Aλ). We show that Aλ is a totally disconnected subspace of the complex plane ℂ, but if the point at ∞ is added, then is a connected subspace of the Riemann sphere . As a corollary, Aλ has topological dimension 1. Thus, ∞ is an explosion point in the topological sense for Âλ. It is remarkable that a space with an explosion point occurs ‘naturally’ in this way.