On structure of one dimensional basic sets of endomorphisms of surfaces
V. Z. Grines1, E. D. Kurenkov.2
| Annotation | 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. |
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| Keywords | axiom $A$, endomorphism, basic set |
1Professor of Department of fundamental mathematics, Higher School of Economics, Nizhny Novgorod; vgrines@hse.ru
2Laboratory TAPRADESS, National Research University Higher School of Economics; ekurenkov@hse.ru
Citation: V. Z. Grines, E. D. Kurenkov., "[On structure of one dimensional basic sets of endomorphisms of surfaces]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,18:2 (2016) 16–24 (In Russian)
