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О классе топологической сопряженности с гомотетией
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 provided 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 $a_n(x_1,\dots,a_n)=(\frac12 x_1,\dots,\frac12 x_n)$. For a smooth case under the condition that all eighenvalues of the differetial of the map $h$ 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 centure, but authors do not know any papers where it would be prooven for $n\in \{4,5\}$. This paper fills this gap.