Topological Equivalence of Suspensions over Cartesian Product of Rough Transformations of the Circle

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 clacc 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 rough circle’s transformations n-ary product. Additional constructions and formation of subsets of the considered sets were used to prove the main result. The present paper introduced a numerical topological invariant for the n-ary Cartesian products of rough circles transformations. The criterion of topological conjugacy of rough circle’s transformations n-ary product is formulated.