
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 onedimensional basic sets of $C^k$, $k \geq 1$, endomorphism satisfying axiom of $A$ and given on surfaces. It is established that if onedimensional basic set of endomorphism $f$ has the type $ (1, 1)$ and is a onedimensional 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 onedimensional 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 1624
\mathnet{http://mi.mathnet.ru/svmo589}
\elib{https://elibrary.ru/item.asp?id=26322687}
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