标题： Axiom A多项式斜积的可计算性$\mathbb{C}^2$ 显示英文标题

标题： Computability for Axiom A Polynomial Skew Products of $\mathbb{C}^2$

摘要： 有理映射在黎曼球面上的Julia集的可计算性近年来受到了深入研究（参见，例如https://doi.org/10.17323/1609-4514-2008-8-2-185-231, https://doi.org/10.1090/conm/797/15936）以供概述。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），双曲和抛物型Julia集可以在多项式时间内计算。在本文中，我们提出了关于多复维映射可计算性的第一项工作。我们研究了一类多项式自映射的$\mathbb{C}^2$，即多项式斜积；即形式为$f(z,w) = (p(z), q(z,w)),$的映射，其中$p$和$q$是相同次数$d\geq 2$的复多项式。我们证明，如果一个多项式斜积满足Axiom A，则其链回归集（等于其非游荡集，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、吸引和鞍型双曲集。我们结果的一个后果是，Axiom A在Axiom A多项式斜积区域的闭包上是一个半可判定性质。最后，我们介绍了一个算法，用于确立固定次数多项式斜积的双曲性区域的下半可计算性。 显示英文摘要

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.