C^2 的 Axiom A 多项式斜积及其后临界集
Axiom A polynomial skew products of C^2 and their postcritical sets
AI总结：
本文研究 C^2 上多项式斜积的 Axiom A 性质，证明其临界轨道要么逃逸到无穷远、要么累积于吸引集，并构造新例子展示多样的后临界行为。
摘要：
C^2 的多项式斜积是指形如 f(z,w) = (p(z), q(z,w)) 的映射，其中 p 和 q 为多项式，且 f 是次数 d >= 2 的正则映射。对于 C 上的多项式映射，双曲性等价于后临界集的闭包与 Julia 集不相交；进一步地，临界点要么迭代到吸引循环，要么迭代到无穷远。对于多项式斜积，Jonsson（Math. Ann., 1999）建立了如下结论：f 是 Axiom A 当且仅当后临界集的闭包与 Julia 集的右类比集不相交。本文给出类似的结论：临界轨道要么逃逸到无穷远，要么累积于一个吸引集。此外，我们构造了新的 Axiom A 映射例子，展示了各种不同的后临界行为。
英文摘要：
A polynomial skew product of C^2 is a map of the form f(z,w) = (p(z), q(z,w)), where p and q are polynomials, such that f is regular of degree d >= 2. For polynomial maps of C, hyperbolicity is equivalent to the condition that the closure of the postcritical set is disjoint from the Julia set; further, critical points either iterate to an attracting cycle or infinity. For polynomial skew products, Jonsson (Math. Ann., 1999) established that f is Axiom A if and only if the closure of the postcritical set is disjoint from the right analog of the Julia set. Here we present the analogous conclusion: critical orbits either escape to infinity or accumulate on an attracting set. In addition, we construct new examples of Axiom A maps demonstrating various postcritical behaviors.
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