On a class of topological conjugacy with a homothety

We consider a class $H(\mathbb{R}^n)$ of orientation-preserving homeomorphisms of Euclidean space $\mathbb{R}^n$ such that for any homeomorphism $h\in H(\mathbb{R}^n)$ and for any point $x\in \mathbb{R}^n$ a condition $\lim \limits_{n\to +\infty}h^n(x)\to O$ holds, were $O$ is the origin. It is proved that for any $n\geq 1$ an arbitrary homeomorphism $h\in H(\mathbb{R}^n)$ is topologically conjugated with the homothety $a_n: \mathbb{R}^n\to \mathbb{R}^n$, given by $a_n(x_1,\dots,a_n)=(\dfrac12 x_1,\dots,\dfrac12 x_n)$. 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 $n\notin \{4,5\}$ this fact is proven in several works of 20th century, but authors do not know any papers where it would be proven for $n\in \{4,5\}$. This paper fills this gap.