Surgery for partially hyperbolic dynamical systems I. Blow-ups of invariant submanifolds
Surgery for partially hyperbolic dynamical systems I. Blow-ups of invariant submanifolds
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部分双曲动力系统的手术 I. 不变子流形的放大
DOI：
10.2140/gt.2018.22.2219
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
A. Gogolev
中科院分区：
文献类型：
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作者：
A. Gogolev
我们提出了一种构造部分双曲微分同胚的新例子的方法。我们从部分双曲微分同胚$f\冒号M\到M$开始，它留下一个不变的子流形$N\子集M$。我们假设$N$是$f$的Anosov子流形，即限制$f|_N$是Anosov微分同胚且中心分布横切于$TN\子集TM$。通过用垂直于$N$的直线的射影空间(实数或复数)代替$N$中的每个点，我们得到了爆破$\HAT M$。用$\HAT M$替换$M$相当于对$N$的邻域进行手术，从而改变了流形的拓扑结构。微分同胚$f$诱导了一个典型的微分同胚$\HAT f\冒号\H M\到\HAT M$。我们证明了在一定的假设下，f$在$N$的局部动力下的微分同胚也是部分双曲的。我们还给出了一些修改，如连通和结构，它允许将两个部分双曲微分同胚“粘贴”在一起以获得新的一个。最后，我们给出了几个应用我们的结果的例子。
We suggest a method to construct new examples of partially hyperbolic diffeomorphisms. We begin with a partially hyperbolic diffeomorphism $f\colon M\to M$ which leaves invariant a submanifold $N\subset M$. We assume that $N$ is an Anosov submanifold for $f$, that is, the restriction $f|_N$ is an Anosov diffeomorphism and the center distribution is transverse to $TN\subset TM$. By replacing each point in $N$ with the projective space (real or complex) of lines normal to $N$ we obtain the blow-up $\hat M$. Replacing $M$ with $\hat M$ amounts to a surgery on the neighborhood of $N$ which alters the topology of the manifold. The diffeomorphism $f$ induces a canonical diffeomorphism $\hat f\colon \hat M\to \hat M$. We prove that under certain assumptions on the local dynamics of $f$ at $N$ the diffeomorphism $\hat f$ is also partially hyperbolic. We also present some modifications such as the connected sum construction which allows to "paste together" two partially hyperbolic diffeomorphisms to obtain a new one. Finally, we present several examples to which our results apply.