On structure of one dimensional basic sets of endomorphisms of surfaces

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.