$\boldsymbol{\text{ℂ}^2}$的Axiom A多项式斜积的可计算性
Computability for Axiom A Polynomial Skew Products of $\mathbb{C}^2$
AI总结：
本文首次研究多复变映射可计算性，针对$\text{ℂ}^2$的Axiom A多项式斜积证明其链回归集可计算，算法能识别不同类型双曲集，还得出Axiom A半可判定性及双曲性轨迹下半可计算性的结论。
摘要：
近年来，黎曼球面上有理映射Julia集的可计算性得到了深入研究（例如，参见https://doi.org/10.17323/1609-4514-2008-8-2-185-231、https://doi.org/10.1090/conm/797/15936以获取综述）。例如，根据Braverman的结果（https://doi.org/10.1016/j.entcs.2004.06.031、https://doi.org/10.1088/0951-7715/19/6/009），双曲和抛物Julia集在多项式时间内是可计算的。本文首次开展了与多复变映射相关的可计算性研究。我们考察了$\text{ℂ}^2$的一类多项式自同态，即多项式斜积；也就是形如$f(z,w)=(p(z),q(z,w))$的映射，其中$p$和$q$是次数$d\neq2$的复多项式。我们证明：若多项式斜积是Axiom A的，则其链回归集（等于非游荡集，也等于周期轨道的闭包）是可计算的。我们的算法还能识别不同类型的双曲集，即扩张型、吸引型和鞍型双曲集。我们结果的一个推论是：Axiom A在Axiom A多项式斜积轨迹的闭包上是半可判定的。最后，我们提出了一个算法，用于建立固定次数多项式斜积的双曲性轨迹的下半可计算性。
英文摘要：
The computability of Julia sets of rational maps on the Riemann sphere has been intensively studied in recent years (see, e.g. https://doi.org/10.17323/1609-4514-2008-8-2-185-231, https://doi.org/10.1090/conm/797/15936) for an overview. For example, by Braverman's results (https://doi.org/10.1016/j.entcs.2004.06.031, https://doi.org/10.1088/0951-7715/19/6/009), hyperbolic and parabolic Julia sets are computable in polynomial time. In this paper, we present the first work on computability related to maps of more than one complex dimension. We examine a family of polynomial endomorphisms of $\mathbb{C}^2$, the polynomial skew products; i.e., maps of the form $f(z,w) = (p(z), q(z,w)),$ where $p$ and $q$ are complex polynomials of the same degree $d\geq 2$. We show that if a polynomial skew product is Axiom A, then its chain recurrent set, which is equal to its non-wandering set and also equal to the closure of the periodic orbits, is computable. Our algorithm also identifies the various hyperbolic sets of different types, i.e., expanding, attracting, and hyperbolic sets of saddle-type. One consequence of our results is that Axiom A is a semi-decidable property on the closure of the Axiom A polynomial skew product locus. Finally, we introduce an algorithm that establishes the lower semi-computability of the hyperbolicity locus of polynomial skew products of a fixed degree.
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