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

  • * Corresponding author: Dmitrii Mints

    * Corresponding author: Dmitrii Mints

This work was financially supported by the Russian Science Foundation (project 21-11-00010), except for the proofs of Lemma 3.2 and Theorem 2. The proof of Lemma 3.2 was obtained with the financial support from the Academic Fund Program at the HSE University in 2021-2022 (grant 21-04-004). The proof of Theorem 2 was obtained with the financial support from the Laboratory of Dynamical Systems and Applications NRU HSE, of the Ministry of science and higher education of the RF grant (ag. 075-15-2019-1931)

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  • It is well-known that there is a close relationship between the dynamics of diffeomorphisms satisfying the axiom $ A $ and the topology of the ambient manifold. In the given article, this statement is considered for the class $ \mathbb G(M^2) $ of $ A $-diffeomorphisms of closed orientable connected surfaces, the non-wandering set of each of which consists of $ k_f\geq 2 $ connected components of one-dimensional basic sets (attractors and repellers). We prove that the ambient surface of every diffeomorphism $ f\in \mathbb G(M^2) $ is homeomorphic to the connected sum of $ k_f $ closed orientable connected surfaces and $ l_f $ 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 $ l_f $ 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 $ \mathbb G(M^2) $ is $ \Omega $-stable but is not structurally stable.

    Mathematics Subject Classification: 37D20.


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  • Figure 1.  Phase portrait of the diffeomorphism a) $ f_1 $; b) $ f_2 $

    Figure 2.  Construction of the characteristic curve for the bunch of degree 4

    Figure 3.  Construction of the surface $ M_{\Lambda} $

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