### ?

## Топологическая сопряженность n-кратных декартовых произведений грубых преобразований окружности

It is known from the 1939 work of A. G. Mayer that rough transformations of the circle are limited to the

diffeomorphisms of Morse – Smale. A topological conjugacy class of orientation-preserving diffeomorphism is entirely determined by its rotation number and the number of its periodic orbits, while for orientation-changing diffeomorphism the topological invariant will be only the number of periodic orbits. Thus, the purpose of this study is to find topological invariants of 𝑛-fold Cartesian products of diffeomorphisms of a circle. Methods. This paper explores the rough Morse – Smale diffeomorphisms on the 𝑛-torus surface. To prove the main result, additional constructions and formation of subsets of considered sets were used. Results. In this paper, a numerical topological invariant is introduced for 𝑛-fold Cartesian products of rough circle transformations. Conclusion.The criterion of topological conjugacy of 𝑛-fold Cartesian products of rough transformations of a circle is formulated.