A coincidence theorem for orthogonal maps

Abstract

A Borsuk-Ulam type theorem for orthogonal maps acting in finite-dimensional Euclidean spaces is obtained. This result is equivalent to the fact that Z is BOrsuk-Ulam group with respect to orthogonal representations. As a corollary, the nonexistence of a semiconjugacy between somestandart linear dynamical systems on spheres is proved. Finally, it is shown that every group of the form $G=A \oplus Z^m \oplus R^n \oplus T^k$, where $A$ in a finite Abelian group, is a Borsuk-Ulam group with respect to orthogonal representations.