Надстройки над декартовыми произведениями сохраняющих ориентацию грубых преобразований окружности

Журнал Средневолжского математического общества. 2023. Т. 25. № 4. С. 273–283.

Зинина С. Х., Nozdrinov A., Shmukler V.

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 orientationpreserving
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.

Keywords: число вращения, manifold, многообразие, superstructure over a diffeomorphism, orientation-preserving diffeomorphism of a circle, number of rotations, Cartesian product of diffeomorphisms, надстройка над диффеоморфизмом, сохраняющий ориентацию диффеоморфизм окружности, декартово произведение диффеоморфизмов