On maximal transitive sets of generic diffeomorphisms
On maximal transitive sets of generic diffeomorphisms
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DOI：
10.1007/s10240-003-0008-0
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发表时间：
2003-05
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通讯作者：
C. Bonatti;L. Díaz
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文献类型：
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作者：
C. Bonatti;L. Díaz
本文构造了具有不含周期点的极大可迁Cantor集的3-流形的局部一般C1-代数同态。构造的局部类属同胚也表现出强烈的病态特征，推广了纽豪斯现象（无限多个汇或源的共存）。其中两个功能是：无穷多个非平凡（双曲和非双曲）吸引子和排斥子共存，无穷多个非平凡（非双曲）同宿类共存。我们证明这些现象与同宿类H（P，f）的存在有关，该同宿类H（P，f）具有两个特定性质：-以C1-鲁棒的方式，同宿类H（P，f）不允许任何支配分裂，-存在周期点P同宿相关P，使得P和P的雅可比量分别大于和小于1。
We construct locally generic C1-diffeomorphisms of 3-manifolds with maximal transitive Cantor sets without periodic points. The locally generic diffeomorphisms constructed also exhibit strongly pathological features generalizing the Newhouse phenomenon (coexistence of infinitely many sinks or sources). Two of these features are: coexistence of infinitely many nontrivial (hyperbolic and nonhyperbolic) attractors and repellors, and coexistence of infinitely many nontrivial (nonhyperbolic) homoclinic classes. We prove that these phenomena are associated to the existence of a homoclinic class H (P, f) with two specific properties:–in a C1-robust way, the homoclinic class H (P, f) does not admit any dominated splitting,–there is a periodic point P homoclinically related to P such that the Jacobians of P and P are greater than and less than one, respectively.