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

Abstract

<p style='text-indent:20px;'>It is well-known that there is a close relationship between the dynamics of diffeomorphisms satisfying the axiom <inline-formula><tex-math id="M2">\begin{document}$ A $\end{document}</tex-math></inline-formula> and the topology of the ambient manifold. In the given article, this statement is considered for the class <inline-formula><tex-math id="M3">\begin{document}$ \mathbb G(M^2) $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M4">\begin{document}$ A $\end{document}</tex-math></inline-formula>-diffeomorphisms of closed orientable connected surfaces, the non-wandering set of each of which consists of <inline-formula><tex-math id="M5">\begin{document}$ k_f\geq 2 $\end{document}</tex-math></inline-formula> connected components of one-dimensional basic sets (attractors and repellers). We prove that the ambient surface of every diffeomorphism <inline-formula><tex-math id="M6">\begin{document}$ f\in \mathbb G(M^2) $\end{document}</tex-math></inline-formula> is homeomorphic to the connected sum of <inline-formula><tex-math id="M7">\begin{document}$ k_f $\end{document}</tex-math></inline-formula> closed orientable connected surfaces and <inline-formula><tex-math id="M8">\begin{document}$ l_f $\end{document}</tex-math></inline-formula> two-dimensional tori such that the genus of each surface is determined by the dynamical properties of appropriating connected component of a basic set and <inline-formula><tex-math id="M9">\begin{document}$ l_f $\end{document}</tex-math></inline-formula> is determined by the number and position of bunches, belonging to all connected components of basic sets. We also prove that every diffeomorphism from the class <inline-formula><tex-math id="M10">\begin{document}$ \mathbb G(M^2) $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M11">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>-stable but is not structurally stable.</p>