Superstructures over Cartesian products of orientation-preserving rough circle transformations

Abstract

One of the constructions for obtaining flows on a manifold is building a superstructure over a cascade. In this case, the flow is non-singular, that is, it has no fixed points. C. Smale showed that superstructures over conjugate diffeomorphisms are topologically equivalent. The converse statement is not generally true, but under certain assumptions the conjugacy of diffeomorphisms is tantamount to equivalence of superstructures. Thus, J. Ikegami showed that the criterion works in the case when a diffeomorphism is given on a manifold whose fundamental group does not admit an epimorphism into the group Z . He also constructed examples of non-conjugate diffeomorphisms of a circle whose superstructures are equivalent. In the work of I. V. Golikova and O. V. Pochinka superstructures over diffeomorphisms of circles are examined. It is also proven in this paper that the complete invariant of the equivalence of superstructures over orientation-preserving diffeomorphisms is the equality of periods for periodic points generating their diffeomorphisms. For the other side, it is known from the result of A.G. Mayer that the coincidence of rotation numbers is also necessary for conjugacy of orientation-preserving diffeomorphisms. At the same time, superstructures over orientation-changing diffeomorphisms of circles are equivalent if and only if the corresponding diffeomorphisms of circles are topologically conjugate. Work of S. Kh. Zinina and P. I. Pochinka proved that superstructures over orientation-changing Cartesian products of diffeomorphisms of circles are equivalent if and only if the corresponding diffeomorphisms of tori are topologically conjugate. In this paper a classification result is obtained for superstructures over Cartesian products of orientation-preserving diffeomorphisms of circles.