Existence of Attractors of Foliations, Pseudogroups and Groups of Transformations

Russian Journal of Nonlinear Dynamics. 2025. Vol. 21. No. 1. P. 85–102.

Dedaev R., Zhukova N.

In this work, by a dynamical system we mean a pair (S, X), where S is either a pseudogroup
of local diffeomorphisms, or a transformation group, or a smooth foliation of the manifold X.
The groups of transformations can be both discrete and nondiscrete. We define the concepts of
attractor and global attractor of the dynamical system (S, X) and investigate the properties of
attractors and the problem of the existence of attractors of dynamical systems (S, X). Compactness
of attractors and ambient manifolds is not assumed. A property of the dynamical system is
called transverse if it can be expressed in terms of the orbit space or the leaf space (in the case of
foliations). It is shown that the existence of an attractor of a dynamical system is a transverse
property. This property is applied by us in proving two subsequent criteria for the existence of an
attractor (and global attractor): for foliations of codimension q on an n-dimensional manifold,
0 < q < n, and for foliations covered by fibrations. A criterion for the existence of an attractor
that is a minimal set for an arbitrary dynamical system is also proven. Various examples of both
regular attractors and attractors of transformation groups that are fractals are constructed.

Keywords: foliation, слоение, attractor, аттрактор, global attractor, глобальный аттрактор, global holonomy group, глобальная группа голономии, pseudogroup, псевдогруппа