Faculty of Science, University of Hradec Králové, Rokitanského 62, 500 03 Hradec Králové, Czech Republic.
Laboratory of Dynamical Systems and Applications, National Research University Higher School of Economics, ul. Bolshaya Pecherskaya, 25/12, Nizhny Novgorod 603155, Russia.
Chaos. 2020 Oct;30(10):103116. doi: 10.1063/5.0021596.
Chaotic foliations generalize Devaney's concept of chaos for dynamical systems. The property of a foliation to be chaotic is transversal, i.e, depends on the structure of the leaf space of the foliation. The transversal structure of a Cartan foliation is modeled on a Cartan manifold. The problem of investigating chaotic Cartan foliations is reduced to the corresponding problem for their holonomy pseudogroups of local automorphisms of transversal Cartan manifolds. For a Cartan foliation of a wide class, this problem is reduced to the corresponding problem for its global holonomy group, which is a countable discrete subgroup of the Lie automorphism group of an associated simply connected Cartan manifold. Several types of Cartan foliations that cannot be chaotic are indicated. Examples of chaotic Cartan foliations are constructed.
混沌叶状结构将 Devaney 关于动力系统的混沌概念推广。叶状结构的混沌性质是横向的，即取决于叶状结构的叶空间的结构。Cartan 叶状结构的横向结构模拟 Cartan 流形。研究混沌 Cartan 叶状结构的问题可归结为其横向 Cartan 流形局部自同构的同伦伪群的相应问题。对于广泛的一类 Cartan 叶状结构，这个问题可归结为其整体同伦群的相应问题，后者是相关单连通 Cartan 流形的 Lie 自同构群的可数离散子群。指出了几种不可能是混沌的 Cartan 叶状结构。构造了混沌 Cartan 叶状结构的例子。
