Extreme growth rate of periodic orbits for equivalent differentiable flows
Extreme growth rate of periodic orbits for equivalent differentiable flows
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等效可微流的周期轨道的极端增长率
DOI：
10.1016/j.jmaa.2019.01.002
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发表时间：
2019
影响因子：
1.3
通讯作者：
Sun Wenxiang
中科院分区：
文献类型：
--
作者：
Liao Gang;Liu Sixu;Sun Wenxiang
我们证明了当所有不动点都是双曲时，可微流的周期轨道的极值增长率由Lipschitz等价保持，从而它在C^r$（1\leq r\leq \infty）$流的开稠密子集$\mathcal{X}^r_h$中是不变量.相比之下，对于任意的$1\leq r\leq \infty$，存在两个与$\mathcal{X}^r_h$不相交的等价$C^r$流的无穷维连通子集，使得周期轨道的增长率分别达到无穷大和零.
We prove that the extreme growth rate of periodic orbits of differentiable flows is preserved by Lipschitz equivalence when all the fixed points are {\bf hyperbolic} and thus it is an invariant in an open dense subset $\mathcal{X}^r_h$ of the set of $C^r$ $(1\leq r\leq \infty)$ flows. By contrast, for any $1\leq r\leq \infty$, there exist two infinite dimensional connected subsets of equivalent $C^r$ flows disjoint from $\mathcal{X}^r_h$ such that the growth rate of periodic orbits attains infinity and zero respectively.