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Article

О классе топологической сопряженности с гомотетией

Гуревич Е. Я., Макаров А. А.

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.