Bifurcations of Periodic Points of Holomorphic Maps From C2 into C2
Bifurcations of Periodic Points of Holomorphic Maps From C2 into C2
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DOI：
10.1112/s0024611599011910
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发表时间：
1999-09
影响因子：
1.8
通讯作者：
G. Zhang
中科院分区：
文献类型：
--
作者：
G. Zhang
让F: Cn→Cn全纯映射,颗F的k迭代,和p∈Cn是一个周期的F (k。也就是说,颗(p) = p,但对于任何正整数j与< k, Fj (p)≠p。如果p是双曲线,即如果国际(p)没有模量的特征值1,然后众所周知,F的动力学行为是稳定的周期轨道附近Γ= {p F (p)…,颗−1 (p)}。但如果Γ不是双曲的，则F在Γ附近的动力学行为可能是非常复杂和不稳定的。在这种情况下,一个非常有趣的分歧现象可能发生即使Γ可能是唯一周期轨道在某些地区Γ:对于给定M∈N \{1},可能存在一个Cr弧{英国《金融时报》:t∈[0,1]}(r∈N或r =∞)在空间H (Cn)的全纯映射从Cn Cn,这样F0 = F, t∈(0,1),英国《金融时报》有一个可周期性轨道Γt和d(Γt,Γ)=增刊∈Γtinfq∈Γ∥p−∥→0 t→0时。因此，在Cr‐小的扰动下，周期增加了M倍！如果这样的一个Ft确实存在，那么Γ，以及p，被称为M -元分岔。这个定义与r无关。
Let F:Cn → Cn be a holomorphic map, Fk be the kth iterate of F, and p ∈ Cn be a periodic point of F of period k. That is, Fk(p) = p, but for any positive integer j with j < k, Fj(p) ≠ p. If p is hyperbolic, namely if DFk(p) has no eigenvalue of modulus 1, then it is well known that the dynamical behaviour of F is stable near the periodic orbit Γ = {p, F(p),…, Fk−1(p)}. But if Γ is not hyperbolic, the dynamical behaviour of F near Γ may be very complicated and unstable. In this case, a very interesting bifurcational phenomenon may occur even though Γ may be the only periodic orbit in some neighbourhood of Γ: for given M ∈ N\{1}, there may exist a Cr‐arc {Ft: t ∈ [0,1]} (where r ∈ N or r = ∞) in the space H(Cn) of holomorphic maps from Cn into Cn, such that F0 = F and, for t ∈ (0,1], Ft has an Mk‐periodic orbit Γt with d(Γt,Γ)=supp∈Γtinfq∈Γ∥p−q∥→0 as t → 0. The period thus increases by a factor M under a Cr‐small perturbation! If such an Ft does exist, then Γ, as well as p, is said to be M‐tupling bifurcational. This definition is independent of r.