On a class of topological conjugacy with a homothety

We consider a class H(Rn) of orientation-preserving homeomorphisms of Euclidean space Rn such that for any homeomorphism h∈H(Rn) and for any point x∈Rn a condition limn→+∞hn(x)→O holds, were O is the origin. It is proved that for any n≥1 an arbitrary homeomorphism h∈H(Rn) is topologically conjugated with the homothety an:Rn→Rn, given by an(x1,…,an)=(12x1,…,12xn). For a smooth case under the condition that all eigenvalues of the differential of the mapping h have absolute values smaller than one, this fact follows from the classical theory of dynamical systems. In the topological case for n∉{4,5} this fact is proven in several works of 20th century, but authors do not know any papers where it would be proven for n∈{4,5}. This paper fills this gap.