ROBUST HETERODIMENSIONAL CYCLES AND $C^1$-GENERIC DYNAMICS
ROBUST HETERODIMENSIONAL CYCLES AND $C^1$-GENERIC DYNAMICS
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DOI：
10.1017/s1474748008000030
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发表时间：
2007-12
影响因子：
0.9
通讯作者：
C. Bonatti;L. Díaz
中科院分区：
文献类型：
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作者：
C. Bonatti;L. Díaz
如果存在（可传递的）双曲集$\varLambda$和$\varSigma$具有不同的指标（不稳定束的维数），使得$\varLambda$的不稳定流形满足$\varSigma$的稳定流形，反之亦然，则微分同构$f$具有一个异维循环。如果$\mathop{\mathrm{index}}(\varLambda)=\mathop{\mathrm{index}}(\varSigma)\pm1$。如果$ $g$接近$ $f$, $ $\varLambda$和$ $\varSigma$对于$g$有一个异维循环，那么这个循环是鲁棒的。证明了与双曲鞍对相关的任何协指数$1$异维环产生$C^1$-鲁棒异维环。因此，在三维空间中，每个异维循环都会产生鲁棒循环。我们还从这个结果中得出了$C^1$-泛型动力学（在任何维度）的一些结果。其中两个后果如下。对于驯服的微分同态（具有有限多个链式递推类的一般微分同态），存在以下二分法：系统要么是双曲的，要么是鲁棒异维循环的。此外，任何包含不同指标鞍的链递归类都具有鲁棒循环。
A diffeomorphism $f$ has a heterodimensional cycle if there are (transitive) hyperbolic sets $\varLambda$ and $\varSigma$ having different indices (dimension of the unstable bundle) such that the unstable manifold of $\varLambda$ meets the stable one of $\varSigma$ and vice versa. This cycle has co-index $1$ if $\mathop{\mathrm{index}}(\varLambda)=\mathop{\mathrm{index}}(\varSigma)\pm1$. This cycle is robust if, for every $g$ close to $f$, the continuations of $\varLambda$ and $\varSigma$ for $g$ have a heterodimensional cycle. We prove that any co-index $1$ heterodimensional cycle associated with a pair of hyperbolic saddles generates $C^1$-robust heterodimensioal cycles. Therefore, in dimension three, every heterodimensional cycle generates robust cycles. We also derive some consequences from this result for $C^1$-generic dynamics (in any dimension). Two of such consequences are the following. For tame diffeomorphisms (generic diffeomorphisms with finitely many chain recurrence classes) there is the following dichotomy: either the system is hyperbolic or it has a robust heterodimensional cycle. Moreover, any chain recurrence class containing saddles having different indices has a robust cycle.