Non-wandering sets with non-empty interiors

被引:21
作者
Abdenur, F
Bonatti, C
Díaz, LJ
机构
[1] Inst Nacl Matemat Pura & Aplicada, IMPA Estrada Da, BR-22460320 Rio De Janeiro, RJ, Brazil
[2] Univ Bourgogne, Lab Topolog, UMR 5584, CNRS, F-21078 Dijon, France
[3] Dep Matemat PUC Rio, BR-22453900 Rio De Janeiro, RJ, Brazil
关键词
D O I
10.1088/0951-7715/17/1/011
中图分类号
O29 [应用数学];
学科分类号
070104 [应用数学];
摘要
We study diffeomorphisms of a closed connected manifold whose non-wandering set has a non-empty interior and conjecture that C-1-generic diffeomorphisms whose non-wandering set has a non-empty interior are transitive. We prove this conjecture in three cases: hyperbolic diffeomorphisms, partially hyperbolic diffeomorphisms with two hyperbolic bundles, and tame diffeomorphisms (in the first case, the conjecture is folklore; in the second one, it follows by adapting the proof in Brin (1975 Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature Funct. Anal. Appl. 9 9-19)). We study this conjecture without global assumptions and prove that, generically, a homoclinic class with non-empty interior is either the whole manifold or else accumulated by infinitely many different homoclinic classes. Finally, we prove that, generically, homoclinic classes and non-wandering sets with non-empty interiors are weakly hyperbolic (the existence of a dominated or a volume hyperbolic splitting).
引用
收藏
页码:175 / 191
页数:17
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