Sierpiński and non‐Sierpiński curve Julia sets in families of rational maps
Sierpiński and non‐Sierpiński curve Julia sets in families of rational maps
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DOI：
10.1112/jlms/jdn030
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发表时间：
2008-10
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影响因子：
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通讯作者：
N. Steinmetz
中科院分区：
文献类型：
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作者：
N. Steinmetz
本文讨论了具有少数临界轨道的某些有理映射族的动力学和参数平面的结构。我们的范式是族Rt（z）=<$（1 +（4/27）z3/（1 − z）），动力学由后临界轨道（R<$ n（<$））n∈ <$的行为控制。特别地，证明了如果R逃逸（即R n（R n）趋于无穷大），则R的Julia集是Cantor集，或Sierpienski曲线，或具有一个或无穷多个割点的曲线;这些情况实际上都发生了。𝔱
We discuss the dynamics as well as the structure of the parameter plane of certain families of rational maps with few critical orbits. Our paradigm is the family Rt(z) = 𝔱 (1 + (4/27)z3/(1 − z)), with dynamics governed by the behaviour of the postcritical orbit (R𝔱n(𝔱))n∈ℕ. In particular, it is shown that if 𝔱 escapes (that is, R𝔱n(𝔱) tends to infinity), then the Julia set of R𝔱 is a Cantor set, or a Sierpiński curve, or a curve with one or else infinitely many cut‐points; each of these cases actually occurs.