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MSC2010 37С15

On a class of topological conjugacy with a homothety

Е. Gurevich1, A. Makarov2

AnnotationWe 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.
Keywordstopological classification of homeomorphisms, topological conjugacy with dilatation, factor-space, homothety

1Elena Ya. Gurevich, Associate Professor, National Research University Higher School of Economics (25/12 Bolshaya Pecherskaya St., Nizhnii Novgorod 603155, Russia), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0003-1815-3120, egurevich@hse.ru

2Aleksey A. Makarov, Undergraduate, National Research University Higher School of Economics (25/12 Bolshaya Pecherskaya St., Nizhnii Novgorod 603155, Russia), ORCID: http://orcid.org/0000-0003-2447-4836, leomakar@mail.ru

Citation: Е. Gurevich, A. Makarov, "[On a class of topological conjugacy with a homothety]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,22:3 (2020) 261–267 (In Russian)

DOI 10.15507/2079-6900.22.202003.261-267