论文标题
$ \ mathbb {p}^2 $的朱莉娅结构
Structure of Julia sets for post-critically finite endomorphisms on $\mathbb{p}^2$
论文作者
论文摘要
令$ f $为$ \ mathbb {p}^2 $上的严格有限的内态（简称PCF地图），令$ j_1 $表示朱莉娅套装，让$ j_2 $表示对最大熵的措施的支持。在本文中，我们表明：1。$ J_1 \ setMinus j_2 $包含在关键组分周期的（有限多个）盆地的联合中，以及零星超级垫圈循环的稳定歧管。 2。对于J_2 $中的每一个$ x \，它不包含在零星超级循环的稳定歧管中，没有包含$ x $的FATOU磁盘。这里零星意味着超级循环不包含在关键的组件周期中。 在额外的假设下，$ PC（F）$的所有分支都是平滑的，并且在横向上相交，我们表明没有零星的超级助剂周期。因此，在这种情况下，$ j_1 \ setMinus j_2 $包含在关键组件循环的盆地联合中，并且在J_2 $中的每一个$ x \中，都没有包含$ x $的FATOU磁盘。结果是我们的结果：1。我们回答一些关于$ \ Mathbb {p}^2 $在没有散发性超级助攻周期的pcf映射的非随机设置的问题。 2。我们在$ \ mathbb {p}^2 $上为$ j_1 \ setminus j_2 $ in $ j_1 \ setMinus j_2 $ he $ j_1 \ setminus j_2 $提供了新的证明。 3。我们表明，对于$ \ mathbb {p}^2 $，在且仅当它不包含关键点的情况下，不变的紧凑型集进行扩展，并且我们在$ \ mathbb {p}^2 $上获得了PCF映射的特征，而在$ J_2 $或满足Axiom A. A. A. A. A. A. A. A. A. A. A.
Let $f$ be a post-critically finite endomorphism (PCF map for short) on $\mathbb{P}^2$, let $J_1$ denote the Julia set and let $J_2$ denote the support of the measure of maximal entropy. In this paper we show that: 1. $J_1\setminus J_2$ is contained in the union of the (finitely many) basins of critical component cycles and stable manifolds of sporadic super-saddle cycles. 2. For every $x\in J_2$ which is not contained in the stable manifold of a sporadic super-saddle cycle, there is no Fatou disk containing $x$. Here sporadic means that the super-saddle cycle is not contained in a critical component cycle. Under the additional assumption that all branches of $PC(f)$ are smooth and intersect transversally, we show that there is no sporadic super-saddle cycle. Thus in this case $J_1\setminus J_2$ is contained in the union of the basins of critical component cycles, and for every $x\in J_2$ there is no Fatou disk containing $x$. As consequences of our result: 1.We answer some questions of Fornaess-Sibony about the non-wandering set for PCF maps on $\mathbb{P}^2$ with no sporadic super-saddle cycles. 2. We give a new proof of de Thélin's laminarity of the Green current in $J_1\setminus J_2$ for PCF maps on $\mathbb{P}^2$. 3. We show that for PCF maps on $\mathbb{P}^2$ an invariant compact set is expanding if and only if it does not contain critical points, and we obtain characterizations of PCF maps on $\mathbb{P}^2$ which are expanding on $J_2$ or satisfy Axiom A.
