  The paper is focused on the existence problem of attractors for foliations.
Since the existence of an attractor is a transversal property of the foliation,
it is natural to consider foliations admitting transversal geometric
structures. As transversal structures are chosen Cartan geometries due to their
universality. The existence problem of an attractor on a complete Cartan
foliation is reduced to a similar problem for the action of its structure Lie
group on a certain smooth manifold. In the case of a complete Cartan foliation
with a structure subordinated to a transformation group, the problem is reduced
to the level of the global holonomy group of this foliation. Each countable
automorphism group preserving a Cartan geometry on a manifold and admitting an
attractor is realized as the global holonomy group of some Cartan foliation
with an attractor. Conditions on the linear holonomy group of a leaf of a
reductive Cartan foliation sufficient for the existence of an attractor (and a
global attractor) which is a minimal set are found. Various examples are
considered.
