Following Losik's approach to Gelfand formal geometry, certain

characteristic classes for codimension one foliations coming from

Gelfand-Fuchs cohomology are considered. Sufficient conditions for

non-triviality in terms of the dynamical properties of generators

of the holonomy groups are found. The non-triviality for the Reeb

foliation is shown; this is in contrast with some classical theorems

on the Godbillon-Vey class, e.g, the Mizutani-Morita-Tsuboi Theorem

about triviality of the Godbillon-Vey class of foliations almost

without holonomy is not true for the classes under consideration. It

is shown that the considered classes are trivial for a large class

of foliations without holonomy. The question of triviality is

related to ergodic theory of dynamical systems on the circle and to

the problem of smooth conjugacy of local diffeomorphisms. Certain

classes are obstructions for the existence of... more