Geodesic
On structure of one dimensional basic sets of endomorphisms of surfaces
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 18 (2016) no. 2, pp. 16-24

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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.
Mots-clés : axiom $A$, endomorphism
Keywords: basic set. 
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     author = {V. Z. Grines and E. D. Kurenkov},
     title = {On structure of one dimensional basic sets of endomorphisms of surfaces},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {16--24},
     publisher = {mathdoc},
     volume = {18},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2016_18_2_a2/}
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V. Z. Grines; E. D. Kurenkov. On structure of one dimensional basic sets of endomorphisms of surfaces. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 18 (2016) no. 2, pp. 16-24. http://geodesic.mathdoc.fr/item/SVMO_2016_18_2_a2/