ze^(z+λ) 的动力系统

The Dynamical System of ze z+λ

目的研究整函数族ｆλ（ｚ）＝ｚｅｚ＋λ（λ∈Ｃ｜）的动力系统．方法用Ｂａｋｅｒ研究ｅλｚ所采用的方法．结果与结论设Ｄ＾ｐ＝｛λ∈Ｃ｜｜ｆλ（ｚ）有一个ｐ阶的吸性周期点｝，Ｄｐ表示Ｄ＾ｐ的一个连通分支．说明了Ｄ＾１由无穷多个分支Ｄ１，Ｄ１ｋ，ｋ＝０，±１，±２，…组成．对任意的ｐ≥２和任意的Ｄ阶单位根η，根存在Ｄｐ切Ｄ１ｋ于１＋ｅ２ηπｉ＋２ｋπｉ．文中还证明了｛ｆｎｐλ（－ｅλ－１），ｎ∈ＮＩ｝在Ｄｐ收敛于ｐ阶吸性周期点ｚ（λ），并且ｚ（λ）关于λ解析，并证明了Ｄｐ是单连通的．

Aim To discuss the dynamics of the family f λ(z)=ze z+λ ,where λ∈C Methods To use the method of applied by Baker to his study of the family e λz Results and Conclusion Let D ^ p={λ∈C|f λ(z) )have an attractive periodic point of order p } and D p denote the connected component of D ^ p It is found that D ^ 1 consists of infinite components D 1,D 1 k,k=0,±1, ±2, ,and for any p≥2 and for each p th root η of unity there is a domain D p which is tangent to D 1 k at 1+e 2ηπi +2kπi Then it is proved that {f np λ(-e λ-1 ),n∈N} converges to one of the attractive periodic points z(λ) of order p,which is analytic in λ,It follows that D p is simply connected