半共轭、压缩Cantor丛和双曲轨道折叠
Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds
摘要：
设f是一个超越整映射，它是次双曲的，即，法图集F（f）与后奇异集P（f）的交是紧的，朱利亚集J（f）与P（f）的交是有限的.设f的任何渐近值都不属于J（f），且f在J（f）中所有点的局部度都有某个有限常数的界.本文证明了存在一个具有连通法图集的双曲映射g（对于某个复数b，其形式为g（z）=f（bz）），使得f和g在它们的朱利亚集上共轭.此外，我们证明了当限制在g的逃逸集I（g）上时，这个双共轭性是一个共轭性。在f可表示为有穷级映射的有限合成的情形下，我们的定理与最近关于双曲映射的朱利亚集的结果一起，意味着J（f）是一个由动态射线及其端点组成的pinched Cantor丛.我们的结果似乎也给出了朱利亚集是整个复平面的整个超越映射的拓扑动力学的第一个完整描述。
英文摘要：
Let f be a transcendental entire map that is subhyperbolic, i.e., the intersection of the Fatou set F(f) and the postsingular set P(f) is compact and the intersection of the Julia set J(f) and P(f) is finite. Assume that no asymptotic value of f belongs to J(f) and that the local degree of f at all points in J(f) is bounded by some finite constant. We prove that there is a hyperbolic map g (of the form g(z)=f(bz) for some complex number b) with connected Fatou set such that f and g are semiconjugate on their Julia sets. Furthermore, we show that this semiconjugacy is a conjugacy when restricted to the escaping set I(g) of g. In the case where f can be written as a finite composition of maps of finite order, our theorem, together with recent results on Julia sets of hyperbolic maps, implies that J(f) is a pinched Cantor bouquet, consisting of dynamic rays and their endpoints. Our result also seems to give the first complete description of topological dynamics of an entire transcendental map whose Julia set is the whole complex plane.
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