Affiliation:
1. HSE University, Bolshaya Pecherskaya 25/12, Nizhny Novgorod, Russia, 603155
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>
Publisher
American Institute of Mathematical Sciences (AIMS)
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,Analysis
Reference21 articles.
1. S. Kh. Aranson, V. Z. Grines.The topological classification of cascades on closed two-dimensional manifolds, Russian Math. Surveys, 45 (1990), 1-35.
2. S. Kh. Aranson and V. Z. Grines, Cascades on surfaces, in Dynamical Systems IX (eds. D. V. Anosov), Springer, (1995), 141–175.
3. S. Kh. Aranson, R. V. Plykin, A. Yu. Zhirov, E. V. Zhuzhoma.Exact upper bounds for the number of one-dimensional basic sets of surface A-diffeomorphisms, Journal of Dynamical and Control Systems, 3 (1997), 1-18.
4. R. Bowen.Periodic points and measures for Axiom a diffeomorphisms, Transactions of the American Mathematical Society, 154 (1971), 377-397.
5. V. Z. Grines.The topological conjugacy of diffeomorphisms of a two-dimensional manifold on one-dimensional orientable basic sets. I(in Russian), Tr. Mosk. Mat. Obs., 32 (1975), 35-60.