Julia sets of complex Hénon maps
Julia sets of complex Hénon maps
复制标题
复杂 Hénon 地图的 Julia 集
DOI：
10.1142/s0129167x18500477
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发表时间：
2017
影响因子：
0.6
通讯作者：
Han Peters
中科院分区：
文献类型：
--
作者：
Lorenzo Guerini;Han Peters
对于复杂的Hénon映射，Julia集有两个自然的定义：[公式：见正文]和[公式：见正文]。这两个集合是否总是相等的是该领域的主要开放问题之一。我们证明平等时，地图双曲行为的先验较小的集[公式：见正文]，在额外的假设下，大量dissipativity。这一结果在[J. E. Fornæss，The julia set of hénon maps，Math.Ann.334（2）（2006）457-464]，但证明是不完整的。我们的证明密切遵循[J. E. Fornæss，The julia set of hénon maps，Math.Ann.334（2）（2006）457-464]，在两点处偏离，其中使用实质耗散性。我们表明，[公式：见正文]也成立时，双曲性是由两个较弱的条件之一。第一个是拟双曲性，在[E.贝德福德和J. Smillie，多项式的同构[公式：见正文]。八.准膨胀124（2）（2002）221-271]，半双曲性的一维概念的自然推广。第二个是[公式：见正文]上的支配分裂的存在。实质上耗散，Hénon地图承认一个主导分裂的可能更大的集[公式：见文字]最近研究[M。Lyubich和H. Peters，部分双曲hénon映射的结构，ArXiv e-prints（2017）。
There are two natural definitions of the Julia set for complex Hénon maps: the sets [Formula: see text] and [Formula: see text]. Whether these two sets are always equal is one of the main open questions in the field. We prove equality when the map acts hyperbolically on the a priori smaller set [Formula: see text], under the additional hypothesis of substantial dissipativity. This result was claimed, without using the additional assumption, in [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457–464], but the proof is incomplete. Our proof closely follows ideas from [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457–464], deviating at two points, where substantial dissipativity is used. We show that [Formula: see text] also holds when hyperbolicity is replaced by one of the two weaker conditions. The first is quasi-hyperbolicity, introduced in [E. Bedford and J. Smillie, Polynomial diffeomorphisms of [Formula: see text]. VIII. Quasi-expansion. Amer. J. Math. 124(2) (2002) 221–271], a natural generalization of the one-dimensional notion of semi-hyperbolicity. The second is the existence of a dominated splitting on [Formula: see text]. Substantially dissipative, Hénon maps admitting a dominated splitting on the possibly larger set [Formula: see text] were recently studied in [M. Lyubich and H. Peters, Structure of partially hyperbolic hénon maps, ArXiv e-prints (2017)].