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 Zhurnal SVMO, 2016, Volume 18, Number 2, Pages 16–24 (Mi svmo589)

Mathematics

On structure of one dimensional basic sets of endomorphisms of surfaces

V. Z. Grines, E. D. Kurenkov

State University – Higher School of Economics in Nizhnii Novgorod

Abstract: This paper deals with the study of the dynamics in the neighborhood of one-dimensional basic sets of $C^k$, $k \geq 1$, endomorphism satisfying axiom of $A$ and given on surfaces. It is established that if one-dimensional basic set of endomorphism $f$ has the type $(1, 1)$ and is a one-dimensional submanifold without boundary, then it is an attractor smoothly embedded in ambient surface. Moreover, there is a $k \geq 1$ such that the restriction of the endomorphism $f^k$ to any connected component of the attractor is expanding endomorphism. It is also established that if the basic set of endomorphism $f$ has the type $(2, 0)$ and is a one-dimensional submanifold without boundary then it is a repeller and there is a $k \geq 1$ such that the restriction of the endomorphism $f^k$ to any connected component of the basic set is expanding endomorphism.

Keywords: axiom $A$, endomorphism, basic set

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Citation: V. Z. Grines, E. D. Kurenkov, “On structure of one dimensional basic sets of endomorphisms of surfaces”, Zhurnal SVMO, 18:2 (2016), 16–24

Citation in format AMSBIB
\Bibitem{GriKur16} \by V.~Z.~Grines, E.~D.~Kurenkov \paper On structure of one dimensional basic sets of endomorphisms of surfaces \jour Zhurnal SVMO \yr 2016 \vol 18 \issue 2 \pages 16--24 \mathnet{http://mi.mathnet.ru/svmo589} \elib{https://elibrary.ru/item.asp?id=26322687} 

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