Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds
Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds
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DOI：
10.1090/s0002-9947-2012-05541-3
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发表时间：
2009-07
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通讯作者：
Helena Mihaljević-Brandt
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作者：
Helena Mihaljević-Brandt
令f为次双曲的先验全映射，即Fatou集F(f)与后奇异集P(f)的交集是紧的，Julia集J(f)与P(f)的交集是有限的。假设 f 的渐近值不属于 J(f)，并且 J(f) 中所有点处的 f 的局部次数受某个有限常数的限制。我们证明存在一个具有连通 Fatou 集的双曲映射 g（对于某些复数 b，其形式为 g(z)=f(bz)），使得 f 和 g 在其 Julia 集上是半共轭的。此外，我们证明当限制于 g 的转义集 I(g) 时，该半共轭是共轭。在 f 可以写成有限阶映射的有限组合的情况下，我们的定理以及双曲映射 Julia 集的最新结果意味着 J(f) 是收缩的康托花束，由动态射线及其端点组成。我们的结果似乎也首次完整地描述了整个超越映射的拓扑动力学，其朱莉娅集是整个复平面。
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.