Fast growth of the number of periodic points arising from heterodimensional connections
Fast growth of the number of periodic points arising from heterodimensional connections
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DOI：
10.1112/s0010437x21007405
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发表时间：
2018-08
影响因子：
1.8
通讯作者：
Masayuki Asaoka;Katsutoshi Shinohara;D. Turaev
中科院分区：
文献类型：
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作者：
Masayuki Asaoka;Katsutoshi Shinohara;D. Turaev
我们考虑紧凑光滑流形的 $C^{r}$-微分同胚 ($1 \leq r \leq +\infty$)，该流形具有两对不同索引的双曲周期点，这些点允许横向异宿点并通过混合器连接。我们证明，通过在周期点附近给出任意$C^{r}$小扰动，我们可以产生一个周期点，其中中心方向上的第一个返回图与高达$r$阶的恒等图重合，前提是横向异宿点满足涉及其中心方向上的过渡图的高阶导数的某些自然条件。因此，我们证明了所考虑的微分同胚的一个小邻域中的 $C^{r}$-通用微分同胚表现出周期点数量的超指数增长。我们还给出了一些例子来表明我们假设的条件的必要性。
We consider $C^{r}$-diffeomorphisms ($1 \leq r \leq +\infty$) of a compact smooth manifold having two pairs of hyperbolic periodic points of different indices which admit transverse heteroclinic points and are connected through a blender. We prove that, by giving an arbitrarily $C^{r}$-small perturbation near the periodic points, we can produce a periodic point for which the first return map in the center direction coincides with the identity map up to order $r$, provided the transverse heteroclinic points satisfy certain natural conditions involving higher derivatives of their transition maps in the center direction. As a consequence, we prove that $C^{r}$-generic diffeomorphisms in a small neighborhood of the diffeomorphism under consideration exhibit super-exponential growth of number of periodic points. We also give examples which show the necessity of the conditions we assume.