Chaos in Cartan foliations
Chaotic foliations generalize Devaney's concept of chaos for
dynamical systems. The property of a foliation to
be chaotic is transversal. The existence problem of chaos for a Cartan foliation
is reduced to the corresponding problem for its holonomy pseudogroup of
local automorphisms of a transversal manifold. Chaotic foliations with transversal Cartan
structures are investigated.
A Cartan $(\Phi,X)$-foliation $(M, F)$ that admits an Ehresmann connection is
covered by a locally trivial bundle, and the global holonomy group of $(M, F)$
is defined. In this case, the problem is reduced to the level of
the global holonomy group of the foliation, which is a countable discrete subgroup of the
Lie group of automorphisms of some simply connected Cartan $(\Phi,X)$-manifold.
Several classes of Cartan foliations that cannot be chaotic, are indicated.
Examples of chaotic Cartan foliations are constructed.