Existence of Attractors of Foliations, Pseudogroups and Groups of Transformations

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.