复Hénon映射的Julia集的可计算性：吸引与中性周期轨道的作用

中文摘要 AI 辅助

我们研究了复平面$\boldsymbol{\text{C}}^2$上动力度$d>1$的多项式微分同胚的Julia集的可计算性，其典型例子是复Hénon映射。在之前的工作中，我们在双曲性（公理A）的假设下建立了可计算性。在此，我们将该结果扩展到其Fatou分支为吸引盆地的映射，允许不存在吸引盆地或存在无穷多个盆地的情况。这使得几类非双曲映射的Julia集具有可计算性，包括Lyubich-Peters类中某些高度耗散的映射，以及某些拟双曲映射。我们的证明基于一种将动力学划分为逃逸、吸引和鞍点区域的算法。一个关键要素是利用鞍点周期轨道的稳定流形，通过反向迭代来近似前向Julia集。我们首先针对广义Hénon映射证明该结果，然后通过将任意多项式微分同胚表示为广义Hénon映射的有限次复合，将其扩展到任意多项式微分同胚。最后，我们给出了存在中性动力学时不可计算的例子，包括具有可计算系数且表现出半Siegel行为的Hénon映射。这些例子表明，一维复动力学中与中性动力学相关的可计算性/不可计算性二分法，在高维中部分存在。

英文摘要

We study the computability of Julia sets for polynomial diffeomorphisms of $\mathbb{C}^2$ with dynamical degree $d>1$, whose prototypical examples are complex Hénon maps. In previous work, we established computability under the assumption of hyperbolicity (Axiom A). Here, we extend this result to maps whose Fatou components are attracting basins, allowing for the possibility of no attracting basins or infinitely many basins. This yields computability of the Julia set for several classes of non-hyperbolic maps, including certain substantially dissipative maps in the Lyubich-Peters class, and certain quasi-hyperbolic maps. Our proof is based on an algorithm that separates the dynamics into escaping, attracting, and saddle regimes. A key ingredient is the use of stable manifolds of saddle periodic points to approximate the forward Julia set via backward iteration. We first prove the result for generalized Hénon mappings and then extend it to arbitrary polynomial diffeomorphisms by expressing them as finite compositions of generalized Hénon maps. Finally, we present examples of non-computability in the presence of neutral dynamics, including Hénon maps with computable coefficients exhibiting semi-Siegel behavior. These examples show that the computability/non-computability dichotomy associated with neutral dynamics in one-dimensional complex dynamics in part persists in higher dimensions.

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18 pages