Persistence of heterodimensional cycles
Persistence of heterodimensional cycles
复制标题
异维循环的持续存在
作者：
Dongchen Li;D. Turaev
异维循环是动力系统的不变集，由两个具有不同维度的不稳定流形的双曲周期轨道和一对连接它们的轨道组成。对于至少 $C^{2}$ 的系统
C
2
，我们证明了通用 2 参数族内的 coindex-1 异维循环的分叉创建了鲁棒的异维动力学，即一对具有不同数量的正 Lyapunov 指数的非平凡双曲基本集，使得每个集合的不稳定流形与第二组的稳定流形相交，并且这些交集对于参数值的开放集合持续存在。我们还给出了任何正则类中 coindex-1 异维环的所谓局部稳定问题的解决方案 $r=2,ldots ,infty ,omega $
r
=
2
,
……
,
无穷大
,
ω
。结果基于以下观察：具有异维循环的系统的拓扑共轭模的算术性质决定了 Bonatti-Díaz 混合器的出现。
A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least $C^{2}$
C
2
, we show that bifurcations of a coindex-1 heterodimensional cycle within a generic 2-parameter family create robust heterodimensional dynamics, i.e., a pair of non-trivial hyperbolic basic sets with different numbers of positive Lyapunov exponents, such that the unstable manifold of each of the sets intersects the stable manifold of the second set and these intersections persist for an open set of parameter values. We also give a solution to the so-called local stabilization problem of coindex-1 heterodimensional cycles in any regularity class $r=2,ldots ,infty ,omega $
r
=
2
,
…
,
∞
,
ω
. The results are based on the observation that arithmetic properties of moduli of topological conjugacy of systems with heterodimensional cycles determine the emergence of Bonatti-Díaz blenders.
影响因子：
1.7
作者：
Pierre Berger
Sylvain Crovisier
Enrique Pujals
通讯作者：
Pierre Berger
Sylvain Crovisier
Enrique Pujals