On decomposition of ambient surfaces admitting $ A $-diffeomorphisms with non-trivial attractors and repellers

It is well-known that there is a close relationship between the dynamics of diffeomorphisms satisfying the axiom \begin{document}$ A $\end{document} and the topology of the ambient manifold. In the given article, this statement is considered for the class \begin{document}$ \mathbb G(M^2) $\end{document} of \begin{document}$ A $\end{document} -diffeomorphisms of closed orientable connected surfaces, the non-wandering set of each of which consists of \begin{document}$ k_f\geq 2 $\end{document} connected components of one-dimensional basic sets (attractors and repellers). We prove that the ambient surface of every diffeomorphism \begin{document}$ f\in \mathbb G(M^2) $\end{document} is homeomorphic to the connected sum of \begin{document}$ k_f $\end{document} closed orientable connected surfaces and \begin{document}$ l_f $\end{document} two-dimensional tori such that the genus of each surface is determined by the dynamical properties of appropriating connected component of a basic set and \begin{document}$ l_f $\end{document} is determined by the number and position of bunches, belonging to all connected components of basic sets. We also prove that every diffeomorphism from the class \begin{document}$ \mathbb G(M^2) $\end{document} is \begin{document}$ \Omega $\end{document} -stable but is not structurally stable.